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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf,len=8476
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='00305v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='CT]  31 Dec 2022  UNIVERSITY OF CALGARY  The functorial semantics of Lie theory  by  Benjamin MacAdam  A THESIS  SUBMITTED TO THE FACULTY OF GRADUATE STUDIES  IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE  DEGREE OF DOCTOR OF PHILOSOPHY  GRADUATE PROGRAM IN COMPUTER SCIENCE  CALGARY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ALBERTA  JUNE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 2022  © Benjamin MacAdam 2022  Abstract  Ehresmann’s introduction of differentiable groupoids in the 1950s may be seen as a starting point for  two diverging lines of research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' many-object Lie theory (the study of Lie algebroids and Lie groupoids)  and sketch theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This thesis uses tangent categories to build a bridge between these two lines of  research, providing a structural account of Lie algebroids and the Lie functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To accomplish this, we develop the theory of involution algebroids, which are a tangent-categorical  sketch of Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We show that the category of Lie algebroids is precisely the category of invo-  lution algebroids in smooth manifolds, and that the category of Weil algebras is precisely the classi-  fying category of an involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This exhibits the category of Lie algebroids as a tangent-  categorical functor category, and the Lie functor via precomposition with a functor  ∂ : Weil1 → �Gpd,  bringing Lie algebroids and the Lie functor into the realm of functorial semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ii  Preface  This thesis is the original work of the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This thesis project arose from attempts to broadly understand differential geometry, and in partic-  ular the differential geometry of mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First steps were taken with Jonathan Gallagher in under-  standing how the enriched perspective on category theory introduced in Garner (2018) might relate to  differential geometric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The basic structures in this thesis, however, came out of discussions  and collaboration with Matthew Burke in the Lie theory context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Chapter 1 is an introduction to the theory of tangent categories, and contains no new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Chap-  ter 2 began as a joint project with Matthew, who ultimately had to leave the project due to time con-  straints, although by that time we had already found the basic structure of the proof that differential  bundles are vector bundles in the category of smooth manifolds (proved in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The cur-  rent structure of the chapter, particularly with the emphasis on associative coalgebras of the weak tan-  gent comonad (T,ℓ) and the tight connection to Grabowski’s previous work on the Euler Vector Field  construction (Grabowski and Rotkiewicz (2009)), is original work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The basic results in that chapter ap-  pear in MacAdam (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' however, the results have been streamlined (there was originally a notion of  a strong differential bundle, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 it is observed that all differential bundles are strong), and  there are some new observations about linear connections (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Chapter3 isa rewrite ofa preprintwritten with Matthew on involution algebroids(Burke and MacAdam  (2019)), while Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 is due to conversations with Richard Garner (we expect to release a new pa-  per on involution algebroids based on these results as a joint work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The original idea of augmenting  an anchored bundle with an involution map is entirely due to Matthew, and the isomorphism on ob-  jects between involution and Lie algebroids follows calculations shared by Richard Garner (Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' My own contribution in this section comes from connecting this to the work of Martínez (2001),  as well as giving the bijection on morphisms for involution and Lie algebroids (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first three chapters provide the background required for the deeper results in Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In  the work of Weinstein (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Martínez (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' de León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2005) on Lie algebroids, it was observed  that the “prolongation” of a Lie algebroid acted like a tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 makes this  intuition precise by showing the prolongation is a second tangent structure on the category of Lie al-  gebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The main theorem in this chapter (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8) shows that involution algebroids in a  tangent category � are equivalent to tangent functors (A,α) from Weil1 to � so that the functor A pre-  serves transverse limits and the natural transformation α is T -cartesian (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These are  entirely new observations about Lie algebroids and are original work of the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Finally, Chapter 5 puts the first four chapters into the language of enriched category theory, us-  ing Garner’s enriched perspective on tangent categories (Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first original results in  this chapter demonstrate that differential bundles and anchored bundles are models of � -sketches  (Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10), where � is the site of enrichment for tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Next, the  enriched theories framework of Bourke and Garner (2019) is used to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14, that invo-  lution algebroids are models of a nervous theory, which is the enriched version of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  thesis concludes with the Lie Realization, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13, which is a new characterization of Lie differ-  iii  entiation and introduces an entirely new way to construct adjunctions between categories of “smooth  groupoids” and categories of “Lie algebroids” using purely enriched-categorical methods (this is in  contrast to the geometric approach used in Crainic and Fernandes (2003) and the homotopy theoretic  approach in Sullivan (1977)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This thesis touches on relatively advanced topics in two areas of math: differential geometry (Lie  algebroids) and enriched category theory (enriched nerve constructions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' I have striven to keep it as  self-contained as possible, introducing the category of smooth manifolds and tangent categories and  including an appendix with the basics of enriched category theory and locally presentable category the-  ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Material on foundational category theory (which is to say, anything that can be found in MacLane  (1988)) and basic differential calculus (see any calculus textbook) is used without citation or introduc-  tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this includes limit, adjunction, monads and monadicity theorems, and the calculus of Kan exten-  sions and coends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Some facts about horizontal composition spans are used in Chapter 4, but nothing  that goes beyond the basic definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  iv  Acknowledgments  I would first like to thank my supervisor, Robin Cockett, for guiding this project and for his investment  of time in reading various preprints and walking through proofs with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' I also owe a special debt of  thankstoMatthew Burke, whowasa close collaboratoron thisprojectand spearheaded the application  of tangent categories to Lie theory, and to Jonathan Gallagher for extensive discussions about tangent  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' I am also grateful to Rory Lucyshyn-Wright for spending a summer teaching me enriched  categorytheory, and toKristine Bauerforhelpful conversationsoverthe years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' I would alsolike tothank  all of the members of the Peripatetic Seminar and others over the years for friendly discussion and  feedback: Chad Nester, Prashant Kumar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lemay, Priyaa Srinivasan, Cole Comfort, Daniel Satanove,  Rachel Hardeman, and Geoff Vooys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Finally, I am grateful to my parents and family, and to my wife Niloofar for her support as I wrote  this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  v  For my wife, Niloofar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  vi  Notation  We start with a table of symbols:  Notation  �,�,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A (usually tangent) category, treated as a general context for mathematics, de-  noted using mathbb  � ,� ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A small category treated as a mathematical object, denoted by mathcal  A,B,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Objects in a category and also functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' written using capital letters  f ,φ  Morphisms in a category and natural transformations: lower-case Roman and  Greek letters  f ◦ g  The composition of two maps g : A → B, f : B → C (applicative notation)  πi  The projection from the i t h component of an n-fold pullback A0q0×q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='qn−1×qnAn  or a product  �n Ai  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G  Composition of two functors, F : � → �,G : � → � (applicative notation)  φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G  Whiskering of a natural transformation  ⊗  Tensor product in a monoidal category  ⊠  A restricted notion of span composition, introduced in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  T,p,0,+,ℓ,c  The data for an arbitrary tangent category, introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  D,⊙,0,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',δ  The data for an infinitesimal object, introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7  T n  Iterated application of an endofunctor T  Tn  The n-fold pullback power of p : T → id for a tangent category, T p ×p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ×p T  vii  We also provide a table of categories:  Notation  SMan  The category of smooth manifolds  �ℓ  The category of lifts in a tangent category �, introduced in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  NonSing(�)  The category of non-singular lifts in a tangent category �, introduced in Defini-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  PDiff(�)  The category of pre-differential bundles in a tangent category �, introduced in  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1(i)  Diff(�)  The category of differential bundles in a tangent category �, introduced in Defi-  nition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='1(ii)  LieAlgd  The category of Lie algebroids, introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='1  Anc(A)  The category of anchored bundles in a tangent category �, introduced in Defini-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='1  Anc� (A)  The category of involution algebroids with chosen prolongations in a tangent cat-  egory �  Weil1  The category of Weil algebras, introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  W  Notation for the Weil algebra �[x]/x 2  �  The category of transverse-limit-preserving functors Weil1 → Set, introduced in  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Weiln  1  The category of Weil algebras with width n, used in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='8  Weil∗  1  The full subcategory of Weil1 spanned by {�,W }  �  The classifying � -category of an anchored bundles and all of its prolongations,  introduced in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='12  viii  Table of Contents  Abstract .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' 140  x  Introduction  The study of Lie groupoids and Lie algebroids goes back to Charles Ehresmann and his student Jean  Pradines in the late 1950s, building upon Sophus Lie’s original research into the application of groups of  smooth symmetries to solving ordinary differential equations (Lie (1893)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Motivated by partial differ-  ential equations, Ehresmann (1959) introduced the notion of a differentiable groupoid, which models  the internal symmetries of a smooth manifold, in contrast to the external symmetries given by a Lie  group (a group object in the category of smooth manifolds) or Lie group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Pradines (1967) ex-  tended the Lie functor (which sends Lie groups and Lie group actions to Lie algebras and Lie algebra  actions) from external to internal symmetries and introduced the notion of a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In doing  so, he identified some shortcomings in Ehresmann’s original definition of differentiable groupoids, in-  troducing the modern notion of a Lie groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Ehresmann’s investigations into differentiable groupoids initiated one of the major differential ge-  ometry research programmes of the second half of the twentieth century, the study of Lie groupoids  and Lie algebroids (which one may refer to as many-object Lie theory to distinguish it from the “single-  object” Lie groups and Lie algebras that had classically been studied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Research into many-object Lie  theory in the 80s and 90s focused on extending Lie’s second theorem and the Cartan–Lie theorem,  which in modern terms state that Lie algebras form a coreflective subcategory of Lie groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, the  Lie functorhasa fullyfaithful leftadjoint(Mackenzie and Xu (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Moerdijk and Mrˇcun (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nistor  (2000)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The leftadjointisoften called Lie integration, and in theirfamouspaperCrainic and Fernandes  (2003) found the exactconditionsgoverningwhethera Lie algebroid integratestoa Lie groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Weinstein  (1996) initiated a line of research into classical mechanics on Lie algebroids and groupoids, extending  Poincarè’s development of mechanics on a space with a Lie group action and the Euler–Poincarè equa-  tions (Poincaré (1901);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' see Marle (2013) for a modern treatment);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this has been further developed by  Eduardo Martinez and his collaborators (de León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Martínez (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Martínez (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Fusca  (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  However, Ehresmann’s work in differentiable groupoids signalled a change in focus for his own re-  search, as he increasingly focused within the then-new area of category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Over the course of  the 60s and 70s Ehresmann published a string of influential papers in the nascent area of functorial  semantics, developing the formalism of sketch theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Sketch theory has proven to be highly influen-  tial in mathematical logic, and has been active for some 40 years, being extended to syntax/semantics  adjunctions Gabriel and Ulmer(2006), enriched category theory Kelly (1982) and generalized limit doc-  trines Adámek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The influence of sketch theory can still be seen on Lie theory in the work  of Kirill Mackenzie and his collaborators to develop the theory of double Lie algebroids, Lie-algebroid  groupoids, double-vector bundles, and other tensor-product theories (Mackenzie (1992, 2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This thesis aims to provide a structural account of the Lie functor from Lie groupoids to Lie al-  gebroids, using tangent categories Cockett and Cruttwell (2014) to unify Ehresmann’s many-object Lie  theory and sketch theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tangent categories provide a syntactic description of tangent structure based  on Kock and Lawvere’s synthetic differential geometry (Kock (2006), Lawvere (1979)) and the Weil func-  tor formalism (a comprehensive account may be found in Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993), while the explicit link to  1  abstract tangent structure is found in Leung (2017), another line of research in differential geometry  that has run parallel to modern Lie theory (albeit with some exchange, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kolár (2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recent work  has recast tangent categories as a class of enriched categories (Garner (2018)), making it possible for  modern techniques from sketch theory and functorial semantics to be applied to differential geome-  try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In doing so, we demonstrate that the language of tangent categories sheds light on the study of  classical mechanics on Lie algebroids and groupoids, as well as Mackenzie’s investigations into “Ehres-  mann doubles” of vector bundles and Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Overview  The first three chapters of this thesis build on previous work in the tangent category literature (for  example, Cockett and Cruttwell (2017, 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lucyshyn-Wright (2018)), providing tangent-categorical  sketches of differential-geometric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These structures follow Ehresmann’s original notion of  a sketch quite closely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' they are specified as graphs (a collection of objects and arrows in the category)  with a set of diagrams that must commute and cones that must be universal, except that the data may  now include the tangent functor T and the tangent natural transformations p, 0, +, ℓ, and c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Each  sketch is accompanied by a proof that its category of models in smooth manifolds (which we shall  often write SMan) is precisely the category of geometric structures it seeks to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first chapter reviews the basic theory of tangent categories, paying particular attention to the  category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first examples of “sketches” from the tangent categories literature  are covered, namely differential objects and affine connections, which model vector spaces and con-  nections respectively (Cockett and Cruttwell (2017, 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The chapter concludes with a study of tan-  gent submersions, which model submersions from classical differential geometry and are a useful ex-  ample of the sort of work that occurs in Chapters 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The second and third chapters develop tangent categorical sketches for vector bundles and Lie  algebroids respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Chapter 2 extends the observation due to Grabowski and Rotkiewicz (2009)  that the category of vector bundles is a full subcategory of multiplicative monoid actions by the non-  negative reals �+, and then applies the Euler vector field construction (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent  categorical sketch for a vector bundle, called a differential bundle, is then developed based on a mor-  phism λ : E → T E , and an isomorphism of categories between differential bundles in SMan and the  category of smooth vector bundles is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Chapter 3 introduces involution algebroids, which re-  place the bracket of a Lie algebroid with an involution map  σ : A̺×T πT A → A̺×T πT A  (where ̺ : A → T M is the anchor of the Lie algebroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Using Martinez’s presentation of the structure  equations for a Lie algebroid (Martínez (2001)), we are once again able to prove an isomorphism of  categories, this time that the category of Lie algebroids is isomorphic to that of involution algebroids  in SMan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This provides the initial bridge between differential geometric structures and tangent cate-  gorical sketches, making it possible to apply more sophisticated techniques in Chapters 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The fourth chapter constructs a syntactic tangent category for Lie algebroids, and demonstrates  that Lie algebroids are precisely generalized tangent bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We call this result the Weil nerve, as it  follows the same structure as Grothendieck’s original nerve theorem Segal (1974), in this case using the  categories presentation of tangent categories due to Leung (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This result has useful implications  for the study of generalized mechanics and geometric structures on Lie algebroids, as it introduces a  novel tangent structure on the category of Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This novel tangent structure corresponds to  Poincarè/Weinstein/Martinez’s characterization of classical mechanics on a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The fifth and final chapter introduces the enriched categories perspective on tangent categories  from Garner (2018), so that the work in Chapters 2 and 3 may be rephrased using the enriched sketches  2  of Kelly (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We construct a functor from the syntactic category of involution algebroids to the syn-  tactic category of a groupoid-in-a-tangent-category, thus giving a presentation of the Lie functor in  the spirit of Ehresmann’s sketch theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The syntactic version of the Lie functor is built by construct-  ing another novel tangent structure on the category of Lie groupoids (or more generally, groupoids in  a tangent category), which also agrees with previous investigations into classical mechanics on a Lie  groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' As a final result, we demonstrate that in a locally presentable tangent category we may use a  left Kan extension to construct a left adjoint to the Lie functor, which we call the the Lie realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3  Chapter 1  Tangent categories  Tangent categories are an example of convergent evolution in mathematics, in which two unrelated  lines of research with very different aims have arrived at a common endpoint, in this case the same  formal setting for abstract differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The older line of research has its roots in differential  geometry proper and, in particular, Weil’s algebraic characterization of the tangent bundle of a smooth  manifold (Weil (1953)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Weil’s work motivated Kock and Lawvere’s development of synthetic differen-  tial geometry, presented in the book of the same name Lawvere (1979);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kock (2006) as well as the Weil  functor formalism of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The second, more recent line of research has its foundations  in theoretical computer science following the publication of Linear Logic by Girard (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Ehrhard  and Regnier noticed that some models of linear logic have a notion of the “Taylor series approxima-  tion” of a proof;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this led to the development of differential linear logic by Ehrhard and Regnier (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Blute, Cockett, and Seely studied the categorical semantics of models of linear logic equipped with the  derivative operation - that is, they identified those categories whose internal language are were models  of differential linear logic - developing a categorical theory of differentiation in Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2006, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tangent categories arise naturally in each line of research: on the first path with the distillation  of synthetic differential geometry into abstract tangent functors in Rosický (1984), and more recently  when Cockett and Cruttwell refined abstract tangent functors following their investigations into the  manifold categoriesofcartesian differential restriction categoriesin Cockett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cockett and Cruttwell  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a sense, they are categories that axiomatize Weil’s characterization of the tangent bundle as  an endofunctor in a way that captures the combinatorics of higher-order derivatives when looking at  a certain class of internal commutative monoids (Cockett and Seely (2011)), as will be made precise  in Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tangent categories also pull Kock and Lawvere’s synthetic differential geometry into the  framework of enriched category theory, which is explored in Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Instances of tangent structure abound throughout mathematics and computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For exam-  ple, many categories of geometric spaces have natural tangent structure, such as the category of conve-  nient manifolds (the categoryofmanifoldsmodelled locallybyconvenient vectorspaces Kriegl and Michor  (1997)) and the category of schemes (see point (ii) in Example 2 of Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An example from  mathematical logic is the category of Köthe sequence spaces( Ehrhard (2002)), and categorical models  ofthe differential lambda-calculus(Cockett and Gallagher(2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' More recently, tangentand differen-  tial categorieshave found applicationsin differentiable programmingand machine learning(Wilson and Zanasi  (2021)), and to understanding Johnson and McCarthy’s functor calculus Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This thesis studies differential geometric structures using the language of tangent categories, fol-  lowing the tradition of synthetic differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' As such, this chapter will develop tangent cate-  gories with a focus on the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Extending the study of these formal structures  in the context of novel tangent categories is a significant endeavour and should be treated as a direc-  4  tion for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first section introduces Cartesian differential categories as the categorical  axiomatization of multivariable calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The second section introduces the category of smooth man-  ifolds and two characterizations of its tangent bundle (kinematic versus operational), while the third  section identifies the structure of the kinematic tangent bundle that characterizes abstract tangent  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The fourth section presents a pair of structures that allow for “local-coordinate calcula-  tions” in the tangent category of differential objects and connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The final section introduces  tangent submersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A submersion is a differentiable map between differentiable manifolds whose  differential is everywhere surjective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' as a preview of the work in Chapters 2 and 3, this section shows  that in the category of smooth manifolds, a tangent submersion is precisely a submersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  first appeared in MacAdam (2021), and is the only original work in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Differential calculus  As with most treatments of synthetic differential geometry, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kock (2006), it makes sense to begin  with the differential calculus - in this case, an introduction to the categorical theory of differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Categorical differentiation has recently gained quite a bit of attention due to its relationship with ma-  chine learning Cockett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2020), and applications to homotopy theory Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This sec-  tion will just consider the basic structures introduced in Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009), and the canonical example  of a cartesian differential category (the category of finite-dimensional real vector spaces and smooth  maps between them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' [Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009)] A cartesian left additive category is a cartesian  category1 � so that:  (i) Each hom-set �(A,B) is a commutative monoid with addition +AB and zero map 0AB : A → B (the  subscript AB will be suppressed when the context is clear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The composition operation ◦ preserves addition on the left:  (g + h) ◦ f = g ◦ f + h ◦ f  (iii) Projection is an additive map (preserves addition):  πi ◦ (f + g ) = (πi ◦ g ◦ f ) + (πi ◦ g )  Where πi denotes the projection from the i t h component of a product or pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' j  There are various examples of cartesian left additive categories - they all fit the same pattern of a  category where each object is equipped with a non-natural, but coherent, choice of linear structure:  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Any category with biproducts is a cartesian left additive category where every map is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The category of cartesian spacesCartSp, whose objects are finite-dimensional real vector spaces and  morphisms are smooth maps between them, is a cartesian left additive category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='" Clearly smooth  maps from A → B are closed under addition, projection is an additive map, and (g + h) ◦ f =  g ◦ f + h ◦ f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1We use the standard notation where 1 is the terminal object, × is product, and πi is the i t h projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  5  (iii) The category of topological vector spaces and continuous morphisms is a cartesian left additive  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In fact, cartesian left additive categories may equivalently be described as cartesian categories  where each object has a coherent choice of commutative monoid structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 (Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following are equivalent:  (i) � is a cartesian left additive category  (ii) � is a cartesian category so that each object has a chosen commutative monoid structure (A,+A,0A)  where the following coherence holds:  (A × B)2  A × B  A2 × B 2  +A×B  τ  +A×+B  (where τ = ((π0 ◦ π0,π0 ◦ π1),(π1 ◦ π0,π1 ◦ π1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) There is a category with biproducts �+ and a bijective-on-objects subcategory inclusion i : �+ �→ �  that creates products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Cartesian left additive categories provide an appropriate to define a differentiation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Re-  call that the usual derivative of a map f : � → � from elementary calculus can be written  ∂ f  ∂ x : � → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  More generally, for a map f : �n → �, one writes the Jacobian of f at x:  J [f ] : �n → (�n ⊸ �m) :=  \uf8ee  \uf8ef\uf8f0  ∂ f1  ∂ x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ∂ f1  ∂ xn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ∂ f1  ∂ x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ∂ f1  ∂ xn  \uf8f9  \uf8fa\uf8fb  The Jacobian, however, requires some notion of a “matrix”2 representing a linear map from �n to �m—  not every category that has a notion of differentiation supports that operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Instead, the directional  derivative:  D [f ](x,v) := lim  d→0  f (x + t · v)  t  gives an appropriately general notion of differentiation that extends to categories where the space of  linear maps A → B is not representable by an object in the category 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A cartesian differential category  axiomatizes the directional derivative as a combinator on a cartesian left-additive category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2This would be called an internal hom in the categorical logic literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3In the case of automatic differentiation, it is also worth noticing that computing the directional derivative of a map  �n → �m has complexity 2� (f ), while forming the Jacobian has complexity n� (f ), so the directional derivative is a more  appropriate primitive for purely practical computational reasons (see Section 5 of Hoffmann (2016) for a discussion of the  computational complexity of forward-mode automatic differentiation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  6  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' [Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 in Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009)] A cartesian differential category is a cartesian  left additive category equipped with a combinator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' a function on hom-sets)  A  f−→ B  A × A −−→  D[f ] B  satisfying the following axioms:  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1] Additive:  D [f + g ] = D[f ]+ D[g ]  D[0] = 0  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2] Additive in the second variable:  D[f ]◦ (g ,h + k) = D[f ]◦ (g ,h) + D[f ]◦ (g ,k)  D[f ]◦ (g ,0) = 0  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3] Projection is linear:  D [πi ] = πi ◦ π1  D[id ] = π1  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4] Pairing:  D[(f ,g )] = (D[f ],D[g ])  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5] Chain rule:  D[g ◦ f ] = D[g ]◦ (π0,D[f ])  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6] Linear in the second variable:  D[D [f ]]◦ ((a,0),(0,d )) = D[f ]◦ (a,d )  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7] Symmetry of partial differentiation:  D [D[f ]]◦ ((a,b ),(c ,d )) = D[D [f ]]◦ ((a,c ),(b,d ))  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of cartesian spaces (Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2(ii)), CartSp, is the canonical cartesian  differential category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let f : �n → �m, and consider its Jacobian at x ∈ �n, J [f ](x) ∈ �n×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define the  differential combinator:  D [f ]◦ (u,v) = J [f ](v) · u = lim  t →0  f (x + t · v)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2018), the authors construct a cartesian differential category based on the Abelian functor  calculus of Johnson and McCarthy (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In Wilson and Zanasi (2021), the authors consider a cartesian differential category whose objects are �2-  modules to apply gradient-based methods to learn the parameters of of Boolean circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Every cartesian differential category comes with a notion of linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This notion of linearity is  strictly stronger than additivity - there do exist examples of non-linear additive maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 of Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A map f : A → B is linear whenever D[f ] ◦  (0AB,id ) = f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We denote the category of linear maps in a cartesian differential category � as Lin(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category  Lin(�) will have biproducts, and will be a cartesian differential subcategory of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  7  Lemma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7(Corollary2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3in Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a cartesian differential category, and denote  its category of linear maps as Lin(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Linear maps preserve addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The category Lin(�) is a bijective-on-objects subcategory of � with biproducts, and the inclusion  Lin(�) �→ � creates products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Every category with biproducts is a cartesian differential category, where  D [f ] = f ◦ π1,  and this differential structure makes the inclusion Lin(�) �→ � preserve the left additive structure  and differential combinator (that is, it is a cartesian differential functor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  The category of smooth manifolds  Tangent categories axiomatize a more general structure than differential calculus, one in which spaces  are only “locally linear.” The category of smooth manifolds gives the historically canonical example,  and a good portion of this thesis relates to structures internal to that category, so it seems worthwhile to  set a working definition for that context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We follow Tu (2011), and allow for disconnected components  of a manifold to have different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' [Definitions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 in Tu (2011)] A chart on a topological space M is pair (Ui,φi :  Ui �→ �n), where Ui is an open subset Ui ⊆ M and φi : Ui → �n is a local homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An atlas is  a collection of charts {(Ui,φi : Ui → �n)|i ∈ I } (where n is fixed for each connected component of M ) so  that for each i, j ∈ I , the transition function ψi,j that completes the diagram  φ−1  i (Ui ∩Uj)  �n  Ui ∩Uj  Uj  φi  ⊆  φj  ψi,j  is a smooth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A smooth manifold is a topological space equipped with a (maximal4) atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A mor-  phism of smooth manifolds is a topological map f : M → N that is locally smooth - for each chart pair of  charts (Ui,φi) on M and (Vj ,θj), see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The map f is smooth whenever each  map fi,j that completes the diagram is smooth  Ui  Vj  M  N  f  fi,j  The category of smooth manifolds, SMan, is the category of smooth manifolds and their morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Chapter 2, some results will implicitly use partition of unity arguments, which require  that the underlying topological space for a manifold is Hausdorff and has a countable basis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' it is  second-countable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We will avoid any direct reference to these properties, and so we omit them from the  definition of a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4With respect to subset inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  8  φi  φ−1  i  φ−1  j  φj  M  Ui  Uj  φi(Ui)  �m  ψi j  φj(Uj)  �m  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1:  Overlapping charts in an atlas (Credit for this tex code belongs to user Cragfelt  https://tex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='stackexchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='com/a/388493/101171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Each vector space �n, for n ∈ �, has a canonical smooth manifold structure whose atlas is a single  chart (the identity map �n → �n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Most geometric shapes that do not have any singularities or sharp edges can be equipped with an  atlas without any issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For example, consider the circle: {(cos(x),sin(x)) : x ∈ [−π,π)}  For any appropriately small ε > 0, there are two charts from Iε = (−ε,π+ ε);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  φ0(t ) = (cos(t ),sin(t )), φ1(t ) = (cos(t − π),sin(t − π))  making the circle a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Category theory has not seen as many applications in differential geometry as it has in topology  or algebra, likely because the category of smooth manifolds (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1) is somewhat poorly be-  haved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The two main reasons that the category of smooth manifolds is “inconvenient” are as follows:  The set of smooth maps between two manifolds M ,N fails, in general, to form a smooth mani-  fold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus, the category is not cartesian closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of manifolds does not have quotients or arbitrary fibre products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  9  However, the category of smooth manifolds admits some limits, for example finite products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12 Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category SMan of smooth manifolds has finite products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given two manifolds M ,N , take the product of their underlying topological spaces together  with the product charts  (φi × ψj ) :Ui × Vj → �n × �m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of smooth manifolds—using our definition where disconnected components of a  manifold may have different dimensions—does have a class of (co)limits used throughout this paper,  namely idempotent splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 (Borceux and Dejean (1986)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An idempotent is an endomorphism e : E → E so that  e ◦e = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The splitting of an idempotent is given by a pair of maps e = s ◦r so that r ◦s = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The existence  of a splitting of an idempotent e is equivalent to asking that the following pair of parallel arrows has a  (co)equalizer:  E  E  e  The idempotent splitting � of a category (also known as the Cauchy completion of the category), is the  full subcategory of presheaves [�op ,Set] that are retracts of representable functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Any functor into a  category with idempotent splittings will factor through this inclusion of categories, so it is the free co-  completion of � under idempotent splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 (Lawvere (1989)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of smooth manifolds is the idempotent splitting of  the category whose objects are open subsets of Cartesian spaces and whose morphisms are smooth maps  f :U → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An idempotent in the idempotent splitting of a category is also an idempotent in the base category,  and thus admits a splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of smooth manifolds is closed to idempotent splittings: for every map  e : M → M so that e = e ◦ e there exists a pair of maps r :Q → M ,s : M →Q so that e = s ◦ r,r ◦ s = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The main construction of interest on the category of smooth manifolds, for tangent categories at  least, is the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given a smooth map f : M → N , restrict it to a morphism between coor-  dinate patches, so it may be regarded as a map f |U :U → V , where U ⊆ �m,V ⊆ �n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This gives a local  derivative operation (remembering that πi denotes the i t h projection from a product  �n Ai )5  D[f |U ] :U × �m → �n,  (f |U ◦ π0,D[f |U ]) :U × �m → V × �n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent bundle makes this construction global;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, there is a functor T : SMan → SMan giving  an assignment  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  −→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='N  that agrees with the local derivative on coordinate patches of M ,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  5The wording here was originally muddled, it has since been corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  10  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 (Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the algebra of smooth functions on a manifold as C ∞(M ) :=  SMan(M ,�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The set SMan(�,M )/ ∼= of tangent vectors on a smooth manifold M comprises the curves  � → M subject to the equivalence relation that for a pair of curves φ,θ : � → M , φ ∼= θ if and only if  φ(0) = θ(0) and for every f ∈ C ∞(M ),  ∂ f ◦ φ  ∂ x  (0) = ∂ f ◦ θ  ∂ x  (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The set SMan(�,M )/ ∼= has a naturally determined smooth manifold structure which we call the tangent  bundle over M , T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For a vector space, the space of linear paths crossing through a point v ∈ V is isomorphic to V , so  T V ∼= V × V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The tangent bundle above the circle is diffeomorphic to the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is follows from the classi-  cal result that a tangent vector on the circle must be perpendicular to its position vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent bundle lifts the “local derivative” into a globally defined construction, so the tangent  bundle construction is functorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent bundle is a product-preserving endofunctor on the category of smooth  manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Functoriality follows by showing that a morphism of smooth manifolds preserves the equiva-  lence relation on curves that defines a tangent vector:  ∀g ∈ C ∞(M ), ∂ g ◦ φ  ∂ x  (0) = ∂ g ◦ θ  ∂ x  (0)  Note that if f : N → M ∈ C ∞(N ), so that g ◦ f ∈ C ∞(M ), the chain rule ensures that  ∀g ∈ C ∞(N ), ∂ f ◦ g ◦ φ  ∂ x  (0) = ∂ f ◦ g ◦ θ  ∂ x  (0)  To show that T is product-preserving, it suffices to show that the equivalence classes of curves are  stable under pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, note that for any M and φ ∼= θ : � → M ,  (φ,id ) ∼= (θ,id ) : � → M × �  Given a pair of curves θM ,ψM : � → M , where θM ∼= ψM and similarly θN ∼= ψN for N , this implies that  f ◦ (φM ,φN )  = f ◦ (φM × id ) ◦ (id ,φN )  = f ◦ (φM × id ) ◦ (id ,θN )  = f ◦ (id ,θN ) ◦ (φM × id )  = f ◦ (id ,θN ) ◦ (θM × id )  = f ◦ (θM × θN )  so (φM ,φN ) ∼= (θM ,θN ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  11  The scalar action by � on tangent vectors and a partially defined addition additionally give the  tangent bundle the structure of a fibered �-module (that is, an �-module in the slice categorySMan/M  whose objects are morphisms into M , f : X → M , and morphisms are commuting triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent bundle over M is an �-module in SMan/M , as follows: let γ,ω be  tangent vectors on M , and define  p : T M → M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p(γ) = γ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  0 : M → T M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0(m) = [r �→ m] (the constant map � → M sending all r ∈ � to m ∈ M )  ·p : T M × � → T M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ ·p r = [x �→ γ(r · x)]  + : T M p ×p T M → T M := [γ],[ω] �→ [γ + ω] (where addition around γ(0) = ω(0) is defined using  local coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The second derivative is involved in the more nuanced axioms for a cartesian differential category,  namely linearity in the vector argument and the symmetry of mixed partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, set f |U  to be the restriction of f : M → N to a map between local coordinate patches U ⊆ M ,V ⊆ N , and then  define  f0 = f |U ◦ π0 ◦ π1,  ,  and  f1 = D[f ]◦ (π0 ◦ π0,π1 ◦ π0),  f2 = D[f ]◦ (π0 ◦ π0,π0 ◦ π1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The axioms [C D C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6],[C D C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7] then give:  (U × �n) × (�n × �n)  (V × �m) × (�m × �m)  T 2M  T 2N  T M  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='N  (U × �n)  (V × �m)  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  ((f0,f1),(f2,D[D[f |U ]]))  (f ,D[f |U ])  ℓ  ℓ  ((π0,0),(0,π1))  ((π0,0),(0,π1))  (U × �n) × (�n × �n)  (V × �m) × (�m × �m)  T 2M  T 2N  T 2M  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='N  (U × �n) × (�n × �n)  (V × �m) × (�m × �m)  c  ((π0◦π0,π0◦π1),(π1◦π0,π1◦π1)  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  c  ((π0◦π0,π0◦π1),(π1◦π0,π1◦π1)  ((f0,f1),(f2,D[D[f |U ]])  ((f0,f2),(f1,D[D[f |U ]])  The two natural transformations ℓ and c —the vertical lift and canonical flip—capture these coher-  ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Locally, ℓ is the map inserting zeros into the second and third coordinates, while c flips the  second and third arguments, leading to the coherences established in the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To capture  12  these coherences on the tangent bundle, first note that a tangent vector on T M is equivalent to an  equivalence class of surfaces on M ,  φ ∼= θ : �2 → M ⇐⇒ φ(0,0) = θ(0,0)  and ∀f ∈ C ∞(M ), ∂ f ◦ φ  ∂ xi  (0,0) = ∂ f ◦ θ  ∂ xi  (0,0),i = 0,1  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12 (Cockett and Cruttwell (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There are two natural transformations  ℓ : T M → T 2M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ([γ]) = [γ ◦ (π0 ·� π1)]  c : T 2M → T 2M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ([γ]) = [γ ◦ (π1,π0)]  satisfying the following coherences:  (i) ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ ℓ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ◦ ℓ6  (ii) The following maps are morphisms of fibred �-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  T M  T 2M  M  T M  ℓ  p  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  0  T M  T 2M  M  T M  ℓ  p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  0  (iii) c ◦ c = id  (iv) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (v) c ◦ ℓ = ℓ  (vi) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Equation (iv) is known as the Yang–Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is one of the coherences for  a symmetric monoidal category, and states that the twisting operation between two variables is coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We may regard the category of endofunctors on a category as a strict monoidal category and use string  diagram notation (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Selinger (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Interpreting the map c as twisting two strings, the coherence  becomes  =  The projection p : T M → M is locally trivial: for each connected component of M that is modeled  on �n, each point m lies in an open subset Um so that p −1(Um) ∼=Um ×�n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This local triviality property  leads to the following universality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following diagram is an equalizer:  T M  T 2M  T M  p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  0◦p◦p  ℓ  6Recall that we are using the 2-categorical notation described in the front-matter  13  Therefore the diagram in the following corollary is a pullback:  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the map µ : T M p ×p T M → T 2M to be T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' + ◦(ℓ× 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following diagram is a  pullback:  T M p ×p T M  T 2M  M  T M  0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  µ  ⌟  The five maps (p,0,+,c ,ℓ), along with their coherences and universal properties, characterize the  kinematic tangent bundle, axiomatized as a tangent structure in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, there is an  equivalent characterization of the tangent bundle for a finite-dimensional smooth manifold that will  be important throughout this thesis: the operational tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We first need to define the mod-  ule of vector fields on a manifold, where a vector field is essentially an ordinary differential equation  defined on a manifold rather than a cartesian space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A vector field on a manifold M is a section X : M → T M  of the projection p : T M → M so that p ◦ X = idM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The set of vector fields on a manifold M is written  χ(M ) and carries a C ∞(M )-module structure using the fibered �-module structure on p : T M → M :  X +χ(M ) Y := +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ◦ (X ,Y ),  0χ(M ) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ,  f ·��(M ) X (m) := f (m) ·T M X (m)  where X ,Y ∈ χ(M ), f ∈ C ∞(M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The module χ(M ) has an important universal property as a C ∞(M )-module—it is precisely the  module of derivations of C ∞(M ):  χ(M ) = {X : C ∞(M ) → C ∞(M ) : ∀f ,g ∈ C ∞(M ),X (f ·g ) = X (f ) ·g + f · X (g )}  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='17 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an isomorphism of C ∞(M ) modules:  Der(C ∞(M )) ∼= χ(M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Finally, we observe that there is a C ∞(M )-Lie algebra structure on χ(M ), with two equivalent def-  initions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, there is the kinematic definition of the bracket, which is induced using the universality  of the vertical lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given vector fields X ,Y on M , note that  X = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ X −T M c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ Y ),  0 = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ X −T M c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ Y )  so by Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='15, there is a unique map [X ,Y ] : M → T M so that  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ X −T M c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ Y ) = µ([X ,Y ],X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  Similarly, there is a bracket defined on Der(M ) using the anticommutator of derivations:  [X ,Y ](f ) = X (Y (f )) − Y (X (f )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2)  These brackets are equivalent for finite-dimensional smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' [Mackenzie(2013)]Recall that a Lie algebra overa ring R isan R-module A equipped  with a bilinear map  [−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='−] : A ⊗ A → A  that is alternating and satisfies the Jacobi identity:  [X ,[Y ,Z ]]+ [Z ,[X ,Y ]]+ [Y ,[Z ,X ]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The two brackets on χ(M ) (viewed as a Lie algebra) from Equations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  Tangent structures  This section develops the categorical framework to study more general categories of smooth manifolds  by axiomatizing the tangent bundle, tangent categories, which first appeared in Rosický (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An ar-  bitrary tangent category is significantly more general than smooth manifolds and captures examplesof  categorieswith “tangentbundle” from computerscience and logic, asdeveloped in Cockett and Cruttwell  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, observe that tangent categories forget the base ring from the previous section and only  consider the fibered commutative monoid structure of the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An additive bundle in a category � is a triple E  q−→ M ,+ : E q×qE → E ,ξ : M → E which  gives (q,+,ξ) the structure of a commutative monoid in the slice category �/M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If (q,ξ,+),(q ′,ξ′,+′) are  both additive bundles, a bundle morphism  E  E ′  M  M ′  q  f  q  f0  is additive if f ◦+ = +′ ◦(f ◦π0, f ◦π1) and f ◦ξ = ξ′ ◦ f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of additive bundles in a a category  � is given by additive bundles in � and additive bundle morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We will often write pullback powers of an additive bundle E q ×q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q ×q E as En, and use infix no-  tation to write addition so + ◦ (a,b ) becomes a + b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the category of smooth manifolds, the tangent  bundle functor gives a well-behaved functorial vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent category has a functorial ad-  ditive bundle and axiomatizes the coherences and universal properties of the tangent bundle from the  category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 (Rosický(1984);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cockett and Cruttwell (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent structure consists of a functor  T : � → � equipped with natural transformations  p : T ⇒ id , 0 : id ⇒ T,+ : T p ×p T ⇒ T  ℓ : T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  c : T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  satisfying the following axioms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' we call a category equipped with a tangent structure a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1] Additive bundle axioms:  (i) Pullback powers of p exist and are preserved by T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' write these Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Each triple (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M : T M → M ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M : M → T M ,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M : T2M → T M ) is an additive bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  T  T  T2  T  id  id  id  +  p◦πi  p  p  0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3)  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2] Symmetry axioms:  (i) Involution:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  c  c  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4)  15  (ii) Yang–Baxter: c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5)  (iii) Naturality equations:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  T2T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  c  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π0,c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  c  c  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  c  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6)  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3] Lift axioms:  (i) Naturality with addition:  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  id  T  T  T  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (ℓ◦π0,ℓ◦π1)  +  +  ℓ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ℓ  p  0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7)  (ii) Coassociativity:  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ℓ  ℓ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8)  (iii) Symmetric co-multiplication:  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ℓ  c  ℓ  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9)  (iv) Universality: for µ := T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' + ◦(0◦ π0,ℓ◦ π1), the following diagram is a pullback for all X :  T2X  T 2X  X  T X  µ  p◦πi  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  0  16  Definition1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3(Cockett and Cruttwell (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Atangent category ismonoidal whenever� isa monoidal  category, T is a monoidal functor, and +,p,c ,ℓ are monoidal natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent cate-  gory is cartesian whenever � is cartesian and is a strict monoidal tangent category for products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Throughout this thesis, two key pieces of notation apply:  Tn denotes pullback powers of p : T ⇒ id (and more generally En for q : E → M );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' iterated powers  of T are written T n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There will often be long strings of Tn pullbacks and functors F : A → B, so a 2-categorical notation  where functor composition is written with a period, Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ′  m, will often be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' While the  natural transformation c at Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M under the image of the functor T would often be written  T (cTn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ), in this thesis it will be written as  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M : T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tm ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tm  while the composition of 2-cells will be written using ◦ in applicative order, rather than the dia-  grammatic order typically used in the tangent category literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' That is, the composition  A  g−→ B  f−→ C  would be written f ◦ g rather than g f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Applying the results from section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, the category of smooth manifolds is a tangent  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that if M is a smooth manifold, there is a coordinate patch m ∈ U �→ M around each  point m ∈ M so that U ∼=U ′ ⊆ �n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' fibre above U ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' p −1(U ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' is locally isomorphic to U ′ ×�n and similarly  (p ◦ p)−1(U ) ∼=U ′ × (�n)3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' so that:  p :U × �n → U  (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x) ��→ m  0 :U →U × �n  m �→ (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0)  + :U × �n × R n →U × �n  (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' y ) �→ (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x + y )  ℓ :U × �n →U × �n × �n × �n  (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x) �→ (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x)  c :U × �n × �n × �n →U × �n × �n × �n  (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='z) �→ (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='z)  The study of tangent categories is closely related to Lawvere’ssynthetic differential geometry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' first in-  troduced in Lawvere (1979) and laterdeveloped in Kock (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lavendhomme(1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Moerdijk and Reyes  (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The setting of synthetic differential geometry is a topos � (for our purposes, we need only a com-  plete, cartesian closed category) equipped with a chosen ring object R that satisfies the Kock-Lawvere  axiom: given the object of nilpotent elements in R, D = [d : R|d 2 = 0], the following map is an isomor-  phism:  α : R × R → [D,R];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' α(a,b ) = (d �→ a + d b ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  One can find a class of objects that form a tangent category: the infinitesimally linear objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a model of synthetic differential geometry where the ring object is (R,·,1,+,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An object M in a model of synthetic differential geometry is infinitesimally linear when it satisfies the  following axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The following natural morphism must be an isomorphism:  [D (2),M ] ∼= [D,M ]0∗×0∗[D,M ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' where D(2) := [(d0,d1) ∈ D 2 : di ·d j = 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  17  (ii) M satisfies property W (credited by Kock (2006) to Gavin Wraith), namely that the following dia-  gram is a ternary equalizer:  [D,M ]  [D × D,M ]  [D,M ]  [0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='×D,M ]  [(0×0)◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',M ]  [D×0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',M ]  [·,M ]  (We will often use M as shorthand for idM in diagrams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An infinitesimal object generalizes the object D in the category of infinitesimally linear objects in  a model of synthetic differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The definition adopted in this thesis is a strict generaliza-  tion of the definition given in Cockett and Cruttwell (2014): here, we work with a symmetric monoidal  closed category rather than a cartesian closed category in order to capture some examples from logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 Cockett and Cruttwell (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An infinitesimal object in a symmetric  monoidal category  (�,⊗,I ,α,ρ,σ)  is a tuple (⊙ : D ⊗ D → D,0 : I → D,ε : D → I ,δ : D → D(2)) (where D(n) denotes pushout powers of 0)  so that:  [IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1] Pushout powers D(n) of 0 : I → D exist, and ε ◦ 0 = idI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2] ⊙ is a commutative semigroup with zero, so that the following diagrams commute:  D ⊗ D  D ⊗ D  D ⊗ D ⊗ D  D ⊗ D  D  D ⊗ D  D  D  D  D  ⊙  σ  ⊙  ⊙⊗D  ⊙  D⊗⊙  ⊙  (D,0)  ⊙  0D ◦ε  The third diagram shows that 0 is an absorbing element rather than a unit (think of 0 as being in  the commutative semigroup on the set [0,1) given by multiplication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3] The map δ : D → D(2) makes (0 : I → D,δ,ε) into a commutative comonoid in the coslice I /�:  D  D(2)  D  D(2)  D  D (2)  D  D (2)  D(3)  D(2)  δ  δ  D+I δ  δ+I D  δ  δ  (ι1|ι0)  (0◦ε+I D)  δ  [IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4] The following diagram commutes (⊙ is coadditive):  D ⊗ D  D  D ⊗ D(2)  D(2)  1×δ  (ι0◦⊙|ι1◦⊙)  δ  ⊙  The notation (f |g ) : A + B → C for pushouts/coproducts is dual to pairing (f ′,g ′) : C → A × B for  pullbacks/products, just as ιi is dual to projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Therefore, (f |g ) ◦ ι0 = f just as π0 ◦ (f ,g ) = f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  18  [IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5] The following diagram is a coequalizer:  D ⊗ I  D ⊗ D  D(2)  0◦ε⊗0  D⊗0  (⊙|I ε⊗D )◦(δ⊗id)  There are two tangent structures associated to an infinitesimal object in a symmetric monoidal  category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first relies on the exponentiability of the infinitesimal object, while the second tangent  structure is on the opposite category of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The enriched perspective on tangent structure in Section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 will clarify the relationship between these two tangent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (�,⊗,I ,α,ρ,σ) be a symmetric monoidal category, and  ⊙ : D ⊗ D → D,0 : I → D,ε : D → I ,δ : D → D(2)  define an infinitesimal object in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) There is a tangent structure on �op , where T = D ⊗ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) If � is also a symmetric monoidal closed category, then it is a tangent category with T = [D,−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The opposite category of � is a symmetric monoidal category:  (�op,⊗,I ,α−1,ρ−1,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent functor is D ⊗ (−), and the projection is  p = D ⊗ (−)  0op  −→ I ⊗ (−)  ρ−1  −→ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The zero map is given by  (−)  ρ−→ I ⊗ (−)  εop  −→ D ⊗ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Addition in �op is given by co-addition in �:  D(2) ⊗ (−)  δop ⊗(−)  −−−−→ D ⊗ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The lift is given by the semigroup structure (along with the monoidal coherences):  D ⊗ (−)  ⊙⊗(−)  −−−→ (D ⊗ D) ⊗ (−)  α−1  −→ D ⊗ (D ⊗ (−)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The flip is also given by monoidal coherences, combined with the symmetry on the monoidal  category:  D ⊗ (D ⊗ (−))  α−→ (D ⊗ D) ◦ (−)  σ⊗(−)  −−−→ (D ⊗ D ) ⊗ (−)  α−1  −→ D ⊗ (D ⊗ (−)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) First, we identify the following natural isomorphisms:  u : id ⇒ [I ,−],  b : [A,[B,−]] ⇒ [A ⊗ B,−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  19  The tangent functor is [D,−],  and the triple (0  :  I  →  D,δ  :  D  →  D(2),  ε : D → I ) gives the additive bundle structure  p : [D,−]  0∗  −→ [I ,−]  u−→ id ,  0 : id  u−→ [I ,−]  [ε,−]  −−→ [D,−]  + : [D (2),−]  δ∗  −→ [D,−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that by the continuity of [−,M ] : �op → �, we have [D(2),M ] ∼= [D,M ]p ×p [D,M ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The lift is given by  ℓ : [D,−]  ⊙∗  −→ [D ⊗ D,−]  b −1  −→ [D,[D,−]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The canonical flip is given by  c : [D,[D,−]]  b−→ [D ⊗ D,−]  σ∗  −→ [D ⊗ D,−]  b −1  −→ [D,[D,−]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The coherences and couniversality properties of an infinitesimal object, along with the continuity  of  [−,M ] : �op → �  then induce the coherences and universality properties for the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This completes  the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent structure on �op induced by the infinitesimal object is the dual tangent structure on  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This will be revisited in Chapters 4 and 5 when looking at tangent categories from the enriched  perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Returning to synthetic differential geometry, the co-universality conditions on an infinitesimal ob-  ject corresponds to infinitesimal linearity (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In some sense, the category of infinitesi-  mally linear objects is the largest subcategory of � for which D is an infinitesimal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a model of synthetic differential geometry (�,R), the object D = [d ∈ R|d 2 = 0] is an  infinitesimal object in the category of infinitesimally linear objects in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This thesis makes use of the 2-category of tangent categories (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 of Cockett and Cruttwell  (2014)), which formalizes the notion of a morphism of tangent structure and 2-cells between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This 2-categorical framework is a departure from the classical theory of synthetic differential geometry,  where the literature only really addresses morphisms of tangent structure in the form of fully faithful  embeddings SMan �→ Microl(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (�,�),(�,�′) be a pair of tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A pair (F : � → �,α : F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F ) is  a tangent functor if the following diagrams commute:  F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+  α2  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  α  α  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  α  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  α  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  20  A tangent functor is strong whenever α is a natural isomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' for a sub-(tangent category) inclusion,  α = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent functor (F,α) between cartesian tangent categories is a cartesian tangent functor if F  is an isomonoidal functor and α is a monoidal natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The coherences on the canonical flip c guarantee that (T : � → �,c : T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ) is a strong tangent  endofunctor on any tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given a pair of tangent functors (A,α) : � → �,(B,β) : � → �, the composition  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A : � → �,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A ◦ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α : B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T C  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  T E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  )  is a tangent functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) A model of synthetic differential geometry (�,R) is well-adapted whenever there is a fully faithful,  strict tangent functor fromSMan to the category of microlinear spaces of �,SMan �→ Microl(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  original development of well-adapted models for synthetic differential geometry may be found in  Dubuc (1981), and the reader may check Bunge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2018) for a recent account of the construction  of such models, or section 3 of Kock (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Atangentnatural transformation γ between tangent functors(F,α),(G ,β) isa natural  transformation so that the following diagram commutes:  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T (A)  G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T (A)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F (A)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G (A)  γT A  αA  βT A  T γA  If F,G are cartesian tangent functors, then γ is cartesian whenever it is an isomonoidal natural transfor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We will call the 2-category of tangent categories, tangent functors, and tangent nat-  ural transformations TangCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The 2-category of cartesian tangent categories is CartTangCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4  Local coordinates in a tangent category  This section develops some structures to facilitate reasoning about higher tangent bundles, using “lo-  cal coordinates.” In the case of a cartesian differential category, T n(A) =  �  2n A, whereas for an open  subset U ⊆ �m the tangent bundle decomposes as T nU ∼= U × (�m)2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This section introduces two  structures that allow for these arguments in an arbitrary tangent category: differential objects and con-  nections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 (Cockett and Cruttwell (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A differential object7 in a cartesian tangent category  is a commutative monoid (A,+A,0A) such that there is a section-retract pair A  λ−→ T A  ˆp−→ A which exhibits  T (A) as a biproduct in the category of commutative monoids:  A ⊕ A ∼= T A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Concretely, a differential object is a commutative monoid equipped with λ : A → T A, ˆp : T A → A, ˆp ◦λ =  id so that the following axioms hold:  7Not to be confused with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 from Barr (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  21  DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 Coherence between + and λ, ˆp:  T (A × A)  T A  A × A  A  T A × T A  T A × T A  A × A  A  T (A × A)  T A  ˆp  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+A  (T π0,T π1)  ˆp× ˆp  +A  +A  λ×λ  (T π0,T π1)−1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+A  λ  DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 Coherence between 0A and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A:  A  1  A  1  T A  A  T A  A  0  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  0A  ˆp  λ  p  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  0A  DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 Coherence between +A and +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A:  T2A  A × A  A × A  T2A  T A  A  A  T A  ˆp× ˆp  +A  +  ˆp  λ×λ  +A  λ  +  DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 Coherence between λ and ˆp with ℓ:  T A  T 2A  A  T A  A  T A  T A  T 2A  ℓ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆp  ˆp  ˆp  λ  λ  ℓ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  A map f : A → B between differential objects (A,λA, ˆpA,+A,0B) → (B,λB, ˆpB,+B,0B) is linear whenever  f preserves the lifts and projections  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ λA = λB ◦ f ) and ( ˆpB ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f = f ◦ ˆpA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Following the work in Section 3 of Cockett and Cruttwell (2018), it is sufficient to check that f pre-  serves λ or ˆp (each condition implies the other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) In the category of smooth manifolds, the real numbers are a differential object, as T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� = �[x]/x 2,  so the lift map in this case is  λ(a) = 0+ a · x,  ˆp(a + b · x) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  More generally, finite-dimensional real vector spaces, with their canonical smooth manifold struc-  ture, are exactly differential objects in the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The lift map is defined  using T � ∼= �[x]/x 2, so that  V  (0,1�)  −−−→ T (V × �)  T ·V  −→ T V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This is equivalent to the isomorphism T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V = R[x]/x 2 ⊗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  22  (ii) Every object in a cartesian differential category has a canonical differential object structure, as  T A := A × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Classically, the tangent space above each point of a smooth manifold is a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We have a  similar result for differential objects from Cockett and Cruttwell (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Suppose we have the following pullback in a cartesian tangent category �, and all powers  of T preserve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  E  T M  1  M  ι  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  p  m  There is a unique differential object structure (E ,λ, ˆp) so that ℓ◦ ι = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ι ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8  There are two natural classes of morphisms between differential objects, linear and “smooth” (that  is, arbitrary morphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 (Cockett and Cruttwell (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a cartesian tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We define the  following categories:  (i) Diff(�) is the category of differential objects and arbitrary morphisms, so for any differential objects  A,B, we have Diff(�)(A,B) := �(A,B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) DLin(�) is the category of differential objects and linear morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of differential objects and smooth maps is a cartesian differential category, exhibiting  differential calculus as a specialized logic in a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 of Cockett and Cruttwell (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a cartesian tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Then:  (i) Diff(�) is a cartesian differential category, where we define the differential combinator D to be  A  f−→ B  A × A  λ×0  −−→ T (A × A)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+A  −−→ T (A)  T (f )  −−→ T (B)  ˆpB  −→ B  (where λA,+A are the lift and addition for the differential object structure on A, and ˆpB is the pro-  jection map on the differential object structure on B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) There is an equality of categories, Lin(Diff(�)) = DLin(�), meaning that a morphism between differ-  ential objects in the cartesian tangent category � is linear if and only if it is linear in the cartesian  differential category Diff(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that if M is an open subset of �n, the second tangent bundle of an open subsetU ⊆ �n splits  as  T 2(U ) = T (U × �n) = (U × �n) × (�n × �n) = T3U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Every smooth manifold admits such a decomposition on its second tangent bundle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' these are known  as a affine connections9 and provide a way to reason about an object as though it has local coordinates  in an arbitrary tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  8Some diagrammatic notation had creeped into the original draft here, I have fixed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  9The prefix “affine” differentiates these from more general connections on differential bundles, which are introduced in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  23  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 (Cockett and Cruttwell (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a tangent category �, define the following:  (i) An affine vertical connection is a map κ : T 2M → T M so that  a) κ is a vertical descent, namely a section of the vertical lift ℓ : T M → T 2M , so κ ◦ ℓ = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  b) κ is compatible with both lifts on T 2M : T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='κ ◦ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='κ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ = ℓ◦ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) An affine horizontal connection is a map ∇ : T2 → T 2M so that  a) ∇ is a horizontal lift, namely a section to the horizontal descent (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p) : T 2 ⇒ T2, so that  (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p) ◦ ∇ = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  b) ∇  is  compatible  with  the  linear  structures  on  T 2,T2:T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='∇ (ℓ  ×  0)  = ℓ◦ ∇and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='∇(0× ℓ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ◦ ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) An affine connection is a pair (κ,∇) comprising a vertical and horizontal connection on M satisfy-  ing the compatibility conditions:  a) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' + ◦(+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ (ℓ◦ κ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ),∇(p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p)) = idT 2M ,  b) κ ◦ ∇ = 0◦ p ◦ πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An affine connection is torsion-free if κ ◦ c = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The data ofa vertical connection issufficienttodefine a full connection, asobserved in Lucyshyn-Wright  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7(Lucyshyn-Wright (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Afull connection isequivalent to a vertical connection in which  the following diagram is a fiber product:  T M  T 2M  T M  M  T M  p  p  p  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  κ  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Every differential object in a tangent category has a canonical vertical connection given by (p ◦  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T, ˆp ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A morphism of differential objects will preserve this vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Every smooth manifold has a non-natural choice of Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By the fundamental the-  orem of Riemannian geometry, there is a torsion-free connection associated with the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (See  any standard reference on Riemannian geometry, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' do Carmo (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  Connections allow for arguments on higher powers of the tangent bundle to be pushed down to  pullback powers of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Suppose M ,N each have connections (κ−,∇−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The map T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f : T 2M → T 2N can be  written using local coordinates �  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f : T3M → T3N as  T 2M  T 2N  T3M  T3N  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+◦(+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦(ℓ◦π2,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦π0),∇(π0,π1))  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p,κN )  24  where �  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f is given by  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ π0,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ π1,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ π2 +N ∇[f ](π0,π1))  with ∇[f ] := κN ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that in the case that f preserves the connections, ∇[f ] = 0, as  κN ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇M = T f ◦ κM ◦ ∇M = T f ◦ 0◦ p = 0◦ f ◦ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  Submersions  The category of smooth manifolds is incomplete: there are cospans10 X  f−→ M  g←− Y for which the  pullback fails to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Following Thom (1954), the pullback of a cospan exists and is preserved by T  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' it is a T-limit) whenever for each point f (x) = g (y ), the direct sum of the images of Tx f and Ty g is  the full vector space Tf (x)M , such cospans are called transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Submersions, then, form a convenient  class of maps, as any cospan where one map is a submersion will be transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' More precisely:  Definition  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  If  A  and  B  are  smooth  manifolds,  a  smooth  function  f  :  A → B is a submersion if and only if the derivative D f |a of f at every point a ∈ A is a surjective lin-  ear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  With this definition we have the following result:  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the category of smooth manifolds, let the class of submersions be denoted by � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Submersions are closed under the tangent functor: f ∈ � ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ∈ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Submersions are closed to pullback along arbitrary maps:  X  g ∗X  X  N  M  N  M  g  f ∈�  g ∗ f ∈�  g  f  ¯g  ⌟  This will often be referred to as T -stability under reindexing, as it induces a functor between slice  categories:  g ∗ : Submersions/M → Submersions/N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The properties of the class of submersions in the category of smooth manifolds were studied in  Cockett and Cruttwell (2018) and axiomatized as a tangent display system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent display system in a tangent category � is a class of maps � in � that is  stable under the tangent functor, d ∈ � ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='d ∈ �,  T -stable under reindexing (as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We call any tangent display system that is closed to retracts in the arrow category a retractive display  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If for all M , pM ∈ �, we call � a proper (retractive) display system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  10Following a general conventionincategory theory,where the prefix "co"-X means anX inthe opposite category,a cospan  in � is a span in the dual category of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  25  This section will show that the submersions in the category of smooth manifolds give a retractive  display system, yielding a general construction of retractive display systems from display systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The definition of a submersion may be rephrased as follows: f is a submersion if and only if for all  a ∈ A and all v ∈ T (B) such that f a = pv, there exists a w ∈ T (A) such that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦w = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is a weakly  universal cone over A  f−→ B  p←− T B: there exists at least one morphism into it for any other cone over  the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A commuting square is a weak pullback if for any x : X → A and y : X → B so that  f x = g y , there exists a map X → W making the following diagram commute:  X  W  A  B  C  x  y  ∃  a  b  f  g  If the above diagram is a weak pullback for each T n, then it is a weak T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Should the pullback of A  f−→ C  g←− B exist, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 is equivalent to asking that the  induced map (a,b ) : W → A f ×g B be a split epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let r be a retract of (a,b ) : W → A f ×g B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For any X  (x,y )  −−→ A f ×g B, the map r ◦ (x, y ) exhibits the  diagram as a weak pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For the converse, the unique map (a,b ) : W → A f ×g B will be a section of  any map A f ×g B → W induced by weak univerality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We now restate the submersion property for a map f using global elements (for all a ∈ A and all  v ∈ T (B) such that f a = pv, there exists a w ∈ T (A) such that T (f )w = v) using generalized elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An arrow f : A → B in a tangent category is a tangent submersion if and only if the  naturality diagram  T A  T B  A  B  T f  p  p  f  is a weak T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Following Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5, in the case that the pullback exists this is equivalent to asking for a section  h : A f ×p T B → T A of the horizontal descent (p,T f ) : T A → A f ×p T B (this section is sometimes called  a horizontal lift in differential geometry literature Cordero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1989)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In smooth manifolds, the T -  pullback along the projection p : T ⇒ id always exists, so to prove that every submersion is a tangent  submersion it suffices to show the existence of a horizontal lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the category of smooth manifolds, the tangent submersions are precisely the sub-  mersions (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an explicit construction of a horizontal lift for a classical smooth submersion in VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  of Cushman and Bates (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  26  It is possible to show that the T -stability properties for submersions in the category of smooth  manifolds follow from the general theory of weak pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We begin by showing that weak pullbacks  satisfy a weakened version of the pullback lemma and then show that the retract of a weak pullback is  a weak pullback (the second lemma is Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 of Adámek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 (Pullback lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider the diagram (A)  f  g (B)  (i) If f ,g are jointly monic and (A) + (B) is a weak pullback, then (A) is a weak pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) If (A),(B) are weak pullbacks then (A) + (B) is a weak pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) If (A)+(B) is a weak pullback, a map can be induced for a cone over (A) by concatenating it with (B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  the jointly monic condition on f ,g guarantees that the map induced for (A) + (B) will commute  for (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given a cone for (A) + (B) induce a map for (B), which then induces a cone for (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (Weak) pullbacks are closed to retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Suppose that S ′ is a weak pullback, and S is a retract of it in the category of commuting squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Consider the following diagram (suppressing the subscripts for s,r ):  Z  A  A′  A  B  B ′  B  C  C ′  C  D  D ′  D  x  y  s  x ′  y ′  r  x  y  w  s  w ′  r  w  z  s  z ′  r  z  s  r  Given a cone for S, there is a corresponding cone for S ′ which induces a map Z → A′ and postcompo-  sition with rA gives the desired map into A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using these lemmas, it is straightforward to prove that the following T -stability properties hold for  tangent submersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In any tangent category �,  (a) tangent submersions are closed to composition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (b) tangent submersions are closed to retracts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (c) any T -pullback of a tangent submersion is a tangent submersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  27  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (a) follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 while (b) follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It remains to prove (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Con-  sider a T -pullback, where u is a tangent submersion:  A  M  B  N  f  v  u  g  By naturality, the outer paths of the following two diagrams are equal:  T A  T M  M  T B  T N  N  T f  T v  p  T u  u  T g  p  =  T A  A  M  T B  B  N  p  T v  f  v  u  p  g  Note that the left diagram is a weak pullback by composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Therefore the outer perimeter of the right  diagram is a weak pullback, and the right square is a pullback, so the left square is a weak pullback by  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The class of tangent submersions is closed to retracts in the arrow category and is conditionally T -  stable under reindexing (if the T -pullback of a tangent submersion exists, it is a tangent submersion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This stability property leads to the following result:  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a tangent category that allows for reindexing of the class of tangent sub-  mersions �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then the class of tangent submersions is a display system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Any class of maps that is closed to reindexing is a tangent display system, and the class of sub-  mersions is closed to retracts in the arrow category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The class of submersions in the category of smooth manifolds is a proper retractive  display system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  28  Chapter 2  Differential bundles  Our principal aim in this thesis is to provide an abstract tangent-categorical axiomatization for Lie  algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To accomplish this, we must provide an axiomatization for Lie algebroids which is essen-  tially algebraic (in the sense of Freyd and Kelly (1972)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, in the category of smooth manifolds,  Lie algebroids are defined in terms of vector bundles and these are prima facie a highly non-algebraic  notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In addition to algebraic axioms which make it an �-module in the slice over its base M , a vector  bundle q : E → M satisfies a crucial topological requirement: it must be locally trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This means  that the projection q : E → M must be locally isomorphic to a projection π0 : U × �n → U for some  open subset U of M and natural number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is this property that permits calculations using local  coordinates, an approach deeply enshrined in the culture of differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Cockett and Cruttwell (2017) introduced the algebraicnotion ofa differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Evidence that  differential bundles are the appropriate generalization of vector bundles was provided by showing how  classical results for vector bundles could be generalized to differential bundles in any tangent category  Cockett and Cruttwell (2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, the precise relationship in the category of smooth man-  ifolds between vector bundles and differential bundles was left open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The main result of this chap-  ter (see MacAdam (2021)) is that vector bundles and differential bundles coincide in the category of  smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The axiomatization of differential bundles focuses on another important property of vector bun-  dles: given a vector bundle q : E → M and a vector v in the fibre Ex above x ∈ M , the tangent space  Tv(Ex) can be naturally identified with Ex .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This gives a lift map λ : E → T E which can be axiomatized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  While the lift map had long been noted in the differential geometry literature in the guise of the Euler  vector field (see 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11 of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993) and also Section 1 of Michor (1996) which explicitly uses the  term "lift"), it had not been adopted as the basis of an abstract axiomatization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  More recently, in the differential geometryliterature, Grabowski and Rotkiewicz (2009) and Bursztyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (2016) realized that the multiplicative �+-action on the total space E determines the vector bundle  structure of q : E → M , and conversely such a multiplicative action determines a vector bundle pre-  cisely when its Euler vector field (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8) satisfies an additional “non-singular” property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This  chapter extends these more recent observations on vector bundles to differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The chapter begins by reviewing vector bundles, describing the Euler vector field construction that  sends a vector bundle q : E → M to a "lift" map λ : E → T E from Grabowski and Rotkiewicz (2009),  whereas the rest of the chapter contains new results developed in collaboration with Matthew Burke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The second section establishes that these lifts are associative coalgebras for the weak comonad (T,ℓ),  and that there is a fully faithful functor from vector bundles into the category of lifts for smooth mani-  folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The third section identifies the universal property satisfied by the lift (or equivalently Euler vector  29  field), while the fourth section shows that a non-singular lift corresponds precisely to a differential bun-  dle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The fifth section proves the main theorem of the chapter: vector bundles are precisely differential  bundles for smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The final section contains some remarks on extending affine connec-  tions to arbitrary differential bundles, which will be useful in Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Vector bundles  A vector bundle over a manifold M axiomatizes the notion of a smoothly varying family of vector spaces  indexed by the points m ∈ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The driving example is that of the tangent bundle over a smooth man-  ifold M , where the fibre above each point m ∈ M is the tangent space TmM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The manifold structure  guarantees that the projection is locally trivial: given a chart U �→ M , the next pullback splits as a  product:  U × �n  T M  U  M  p  ⌟  The local triviality of the tangent bundle is essential for various constructions and is part of the defini-  tion of a vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A vector bundle is a tuple  (q : E → M ,ξ : M → E ,+ : E q ×q E → E ,· : � × E → E )  of morphisms in SMan so that  (i) the tuple (q,ξ,+,·) defines an �-module in SMan/M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) the map q : E → M is locally trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The fibred �-module structure means E is a family of vector spaces indexed by M , {Em|m ∈ M }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is important to note that the local triviality axiom guarantees that the projection of a vector bun-  dle is a submersion (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus pullback powers of q : E → M exist and are preserved by  the tangent functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider the cylinder, defined as the subset of �3 spanned by C = {(x, y,z)|x 2 + y 2 =  1,z ∈ �}:  �  Above each point i ∈ S 1 = {(x, y )|x 2 + y 2 = 1} the fibre over i is �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For each point i, we can choose a  sufficiently small ε and take the open set  Ui = {(x, y ) ∈ S 1|(ix − x)2 + (iy − y )2 ≤ ε},  which may be flattened to (−(1+ ε),1+ ε) × �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  30  The sections of a vector bundle also give rise to a C ∞(M )-module, generalizing that aspect of the  tangent bundle’s fibred �-module structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given a vector bundle q : E → M , write the set of sections of q as Γ(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' as, for example,  Γ(p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ) = χ(M ) (recall the notation from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The set Γ(q) has a C ∞(M )-module structure  in much the same way as χ(M ):  X +Γ(q) Y := + ◦ (X ,Y ), 0Γ(q) := ξ, (f ·Γ(q) X )(m) := f (m) · X (m)  There are also a variety of general constructions that yield vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The tangent bundle is a vector bundle: the construction in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 makes it clear that the pro-  jection p : T M → M is a locally trivial, fibred �-module over the base space M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) A trivial vector bundle over M with fibres in V is the product M × V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, every vector  space is a trivial vector bundle above the one-point space {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Each TkM will be locally trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' locally it lookslike the k-fold product of the tangent space p −1  k (U ) ∼=  U × (�n)k for an n-dimensional manifold M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' More generally, one can take the fibrewise pullback  Ek = E q ×q E q ×q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q ×q E and discover a vector bundle over M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The cotangent bundle of M , T ∗M , has the dual vector space of TmM above each point M : T ∗  m(M ) =  (TmM )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This space can be appropriately topologized to be smooth, and a set of sections of Γ(T ∗M )  is isomorphic to the set of morphisms T M → � that are linear in each fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This construction may  be applied to any vector bundle and is called the dual vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) Consider the space Λn(E ), the alternating tensor product of E ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The set of sections of this vector  bundle is equivalent to the alternating n-linear morphism En → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' when restricted to the tangent  bundle, this is the space of differential n-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There are two constructions on vector bundles that will be necessary to prove the main theorem of  this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,+q,·q) be a vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For any map f : N → M , the T -reindexing of q by f is a vector bundle:  f ∗E  E  N  M  q  f  f ∗q  ¯f  ⌟  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  (ii) Any retract of q in the space of arrows is a vector bundle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, given  F  E  F  N  M  N  q  r ′  r  π  s  s ′  π  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2)  if there is a vector bundle structure on q, then there is a vector bundle structure on π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  31  The category of vector bundles has “locally linear” bundle morphisms as its maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A morphism of vector bundles between q : E → M and π : F → N is a commuting  square  E  F  M  N  q  f  v  π  that is fibrewise linear, so that above each fibre  f |m : Em → Fv(m)  isa linearmorphism of vectorspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thismay equivalently be stated asa morphism of fibred �-modules,  so that the following diagrams commute:  E  F  E  F  M  N  M  N  E2  F2  � × E  � × F  E  F  E  F  q  f  v  π  ξ  ζ  f  v  +  +  f  f2  E  �×f  F  f  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For the pullback vector bundle in Diagram 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1, the pair ( ¯f , f ) is a linear bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For the section/retract vector bundle structure from Diagram 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, the section and retract are linear  morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that this is exactly the splitting of a linear idempotent on q : E → M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The lift on the tangent bundle was defined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 as  [γ]∼ �→ [γ ◦ ·�]∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Instead, consider the action of � on a tangent vector:  ([γ]∼,r ) �→ [γ ◦ (r · x)]∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='· gives the equation  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ·◦([ω ◦ (x, y )],[(a,b ) �→ a + b · x]) = (ω ◦ (a · x,a · b · y ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  so the lift map ℓ can be rederived as follows:  T M  T M × �  T (T M × �)  T 2M  [γ]  ([γ],1)  ([γ ◦ π0],[r �→ 1• r ])  [γ ◦ •�]  (id,1�)  0×λ  This general construction is known as the Euler vector field of a multiplicative action by �+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  32  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider a multiplicative monoid action a : �+ × E → E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Euler vector field1 of  the action is the morphism λ : E → T E constructed as follows:  λ := E  (id,1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  −−−−→ E × �  0×λ  −−→ T E × T � ∼= T (E × �)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='a  −→ T E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Local triviality for the tangent bundle is encoded by the universality of the vertical lift condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A  similar universality condition holds for vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let q : E → M be a vector bundle with corresponding Euler vector field λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then the  following diagram is a T -pullback:  E  T E  M  E × T M  (ξ,0)  q  λ  (p,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q)  Exploiting the fact that fibred �-modules have subtraction, the following result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following two diagrams are T-equalizers:  E2  T E  T M  T M p ×q E  T E  E  µE :=0◦π0+T q λ◦π1  νE :=T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π0+p λ◦π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q◦0◦p  p  p◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  (recall that E2 is the pullback of a submersion along a submersion and is therefore guaranteed to exist  and be preserved by the tangent functor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given v : X → T E so that  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q ◦ v = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q ◦ 0◦ p ◦ v  then  p ◦ (v −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q 0◦ p ◦ v) = p ◦ v −q p ◦ v = ξ ◦ q ◦ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  So there is a unique v ′ so that  λ ◦ v ′ = (v −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q 0◦ p ◦ v)  meaning that  v = 0◦ p ◦ v +T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q λ ◦ v ′ = µ(0◦ p ◦ v,v ′)  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The projection q is a submersion, so the pullback E2 is preserved by the tangent functor,  as is the pullback in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9, and the same calculation may be applied for each T n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The proof  for ν follows by the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that the class of submersions forms a retractive display system in the category of smooth  manifolds (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3), so they are stable under reindexing and closed to retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We may now  infer the following:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The projection for a vector bundle is a submersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This follows from the fact that π : T E → E is a submersion, so that q ◦ π0 : E q ×p T M → M is a  submersion, so the map q : E → M is a retract of the projection p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E : T E → E in the arrow category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1Somewhat confusingly, the Euler vector field is almost never a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  33  Preservation of the Euler vector field is also sufficient to guarantee that a morphism f : E → F  determines a vector bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let q : E → M ,π : F → N be a pair of vector bundles with Euler vector fields λE ,λF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Then a bundle morphism (f ,v) : q → π is a vector bundle morphism if and only if  λF ◦ f = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ λE  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the νF map from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10 is monic, and if f preserves the lift, it preserves ν:  νF ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v, f ) = + ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ζ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v,λF ◦ f ) = + ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='xi,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ λE ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ νE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Next, observe that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v × f is the unique map making the following diagram commute:  M  E  T M p ×q E  T E  T N p ×q F  T F  N  F  ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ν  ν  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  p  p  f  q◦π1  π◦π1  ζ  ξ  ⌟  ⌟  Now ν is a vector bundle morphism and monic, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f is a vector bundle morphism, so it follows  that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v × f is a vector bundle morphism and hence f is also a vector bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The reverse  implication is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  Lifts for the tangent weak comonad  The lift ℓ : T ⇒ T 2 gives rise to a weak comonad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Weak comonads were introduced in Wisbauer (2013)  and have a natural notion of an associative algebra2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An associative coalgebra of the weak comonad  (T,ℓ) is called a lift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this chapter will demonstrate that lifts provide the essential structure necessary to  formulate vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A weak comonad on a category � is an endofunctor S : � → � equipped with a coas-  sociative map δ : S ⇒ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S:  S  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S  δ  δ  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='δ  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S  An associative algebra of a weak comonad is an object E equipped with a map λ : E → SE so that  E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  λ  λ  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  2This is not strictly true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Wisbauer has a more nuanced hierarchy of almost-monads, and in his language (T,ℓ) would be  an endofunctor with an associative product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  34  A morphism of these algebras is a map f : (E ,λ) ⇒ (D,γ) so that  E  D  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D  f  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  λ  γ  Recall that for a full (co)monad, there is an adjunction between the base category and the category  of (co)algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' That result is weakened in this case:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every weak comonad on a category �, there is a free coalgebra functor  F : � → CoAlg(�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E �→ (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,δ : S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E )  an underlying object functor  U : CoAlg(�) → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(E ,λ) �→ E  and a natural transformation  λ : id ⇒ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  λ  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  λ  δ  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A lift in a tangent category � is an associative coalgebra of (T,ℓ), namely, a pair (E ,λ :  E → T E ) so that the following diagram commutes:  E  T E  T E  T 2E  λ  λ  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  A morphism of lifts is a coalgebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of lifts and lift morphisms in a tangent cate-  gory � is written Lift(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that the tangent bundle is not, in general, a comonad: while the tangent projection has the  correct type for a counit, p : T ⇒ id , it does not satisfy p ◦ℓ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In fact, if p were a counit, this would  force id = p ◦ ℓ = 0 ◦ p so that 0 = p −1, thus if (T,ℓ,p) is a comonad then T is naturally isomorphic to  the identity functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For every object M in a tangent category �, the pair (T M ,ℓ : T M → T 2M ) is a lift, called the free  lift on M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Every object M in a tangent category has a trivial lift, 0 : M → T M , where T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ 0 = ℓ◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Every differential object has a lift λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the coherence is equivalent to axiom [D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3] in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The Euler vector field of a multiplicative �+-action h : R + ×E → E in SMan is a lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that the  Euler vector field over the scalar action sM : T M × � → � induces the vertical lift on a manifold:  ℓ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='sM ◦ (0,λ� ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  35  In the category of smooth manifolds, T � ∼= �[x]/x 2, where λ(r ) = [x �→ r ·x] corresponds to the map  λ′(r ) = 0+r ·x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Similarly, there is an isomorphism �[x, y ]/(x 2, y 2) so that for the maps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tand T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T (a + r · x) ∼= a + r · x + 0y + 0x y,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0(a + r · x) ∼= a + 0x + r · y + 0x y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Since we know that (0 + x)(0 + y ) = (0 + x y ) in �[x, y ]/(x 2, y 2) (following 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2), we can use these  isomorphisms to see that  ℓ◦ λ� ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ λ� ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=') ·T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ λ� ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Consider a monoid action (�+,h) on a manifold E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Euler vector field of this action,  λ : E  (id,1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  −−−−→ E × �  (0,λ�)  −−−→ T E × T �  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h  −→ T E  will define an algebra if the induced scalar action on T E commutes with the natural scalar action:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ λ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ 0◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (0,λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ),0◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (T 0◦ 0,T 0◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ),0◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ 0,(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=') ·T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='))  = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (ℓ◦ 0,ℓ◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  = ℓ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (0,λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  = ℓ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12, morphisms preserve a monoid action if and only  if they preserve the associated Euler vector field of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This means the Euler vector field construc-  tion gives a fully faithful functor from monoid actions to lifts in the category of smooth manifolds, and  therefore from the category of vector bundles to the category of lifts in SMan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The following proposition gives a pair of constructions on lifts—closure under the tangent functor  and finite T -limits—that will be useful in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The tangent functor lifts to an endofunctor on the category of lifts in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given a diagram  D : � → Lift(�)  in the category of lifts of �, if the T -limit ofU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D exists in �, then limU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D has a natural lift λ′ asso-  ciated to it so that (limU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D,λ′) is the limit of D in Lifts(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (That is, T -limits of lifts are computed  pointwise in the base category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Simply check that  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  36  (ii) Concretely, a tangent terminal object will have a lift:  (1,1  0−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 ∼= 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Given (E ,λ) and (F,l ), if the tangent product E × F exists there is a lift  (E × F,E × F  λ×l  −−→ T E × T F ∼= T (E × F )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Given the T -equalizer of a fork f ,g : (E ,λ) → (F,l ), the equalizer has a lift induced as follows:  C  E  F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  f  g  k  λ  k  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  l  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of lifts is a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent functor sends  f : (E ,λ) → (F,l )  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f : (T E ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (T F,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To see that this is still an algebra morphism, compute  c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='l ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f = c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  The structure maps are the structure maps on the underlying object of the lift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the universality condi-  tions follow by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The following idempotent is key in the theory of lifts and will be used in defining non-singular  lifts (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1), and its splitting will present the projection and zero-section of a vector bundle  (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of lifts in a tangent category � has a natural idempotent:  e : id ⇒ id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e(E ,λ) : (E ,λ)  p◦λ  −−→ (E ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, we see that e = p ◦ λ is an idempotent:  p ◦ λ ◦ p ◦ λ = p ◦ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = p ◦ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ ℓ◦ λ = p ◦ 0◦ p ◦ λ = p ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Moreover, every f : (E ,λ) → (F,l ) preserves the idempotent:  f ◦ p ◦ λ = p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ λ = p ◦ l ◦ f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Finally, note that the idempotent is a lift morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' :  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = ℓ◦ λ = c ◦ ℓ◦ λ = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ  which implies that  λ ◦ e = λ ◦ p ◦ λ = p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = p ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  37  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  Non-singular lifts  Grabowski and Rotkiewicz (2009) introduced the notion of a non-singular lift as a means to axioma-  tize the Euler vector field of a vector bundle’s multiplicative �+-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' While it is not immediately  clear that our definition is the same as Grabowski’s, the results of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 will justify the use of this  language as they are necessarily the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A lift (E ,λ) in a tangent category � is non-singular whenever the following diagram is  a T -equalizer:  E  T E  T E  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  where e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E = p ◦ ℓ (the idempotent associated to the free lift on E ) and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ (the image of the  idempotent associated to (E ,λ) under the tangent functor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of non-singular lifts is written  NonSing(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The most prominent class of examples is given by the Euler vector field of the �-action on a vector  bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Euler vector field of a vector bundle is a non-singular lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,+,ξ,·) be a vector bundle with Euler vector field λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9, the  diagram  T M  E  T E  E  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  p◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q◦0◦p  is a T -limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The T -universality of this diagram will hold if and only if the diagram is universal after  each parallel pair of arrows is post-composed by a T -monic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A section is a T -monic, so the previous  diagram is T -universal if and only if the following diagram is T -universal:  E  T E  E  T E  T M  T E  p  ξ◦q◦p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  0◦q◦p  λ  0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ  Now simplify this diagram using the fact that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(ξ ◦ q) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ,0◦ p = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E :  T E  E  T E  T E  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  λ  38  Note that the the two pairs of parallel arrows have a common arrow, implying that they may be pulled  together into a single ternary equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' All that remains to check, then, is that for any x : X → T E ,  (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ x) ⇐⇒ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The forward implication is trivial, so it remains to prove the reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Suppose T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' then  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ x  giving the result, namely that the diagram  E  T E  T E  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  is a T -equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Every map f : E → F gives a map of free coalgebras T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f : (T E ,ℓ) → (T F,ℓ) and the idempotent e is  a coalgebra morphism by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8, so the following is immediate:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A non-singular lift is an equalizer in the category of lifts:  (E ,λ)  (T E ,ℓ)  (T E ,ℓ)  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  Observe that the category of non-singular lifts is closed under finite limits in the category of lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of non-singular lifts in � is closed under T -limits:  (i) The tangent functoron liftspreservesnon-singularlifts, so that if (E ,λ) isnon-singularthen (T E ,c ◦  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The trivial lift on an object, 0 : M → T M , is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) T -products of non-singular lifts are non-singular lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) T -equalizers of non-singular lifts are non-singular lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) This follows from the fact that the non-singularity condition is a T -limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The zero map splits the idempotent 0◦ p, so it is the equalizer of 0◦ p,p ◦ 0 = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) This follows by stability of limits under products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The following diagram commutes by naturality:  C  E  F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  λ  λ  f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  g  λ  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  Each horizontal diagram is a T -equalizer, and the two columns on the right are T -equalizers, so  the column on the left is a T -equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  39  Finally, when a tangent category has certain T -equalizers, there is an idempotent monad on the  category of lifts, whose algebras are non-singular lifts:  Theorem2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a tangent category with chosen T -equalizersof idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then the following  equalizer determines a left-exact idempotent monad on the category of lifts, whose algebras are non-  singular lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (F,l )  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,ℓ)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,ℓ)  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, take the equalizer in Lift(�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the functor sends (E ,λ) to the chosen limit (F,l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='The unit of  the monad is the unique morphism from (E ,λ) to the equalizer (F,l ) induced by universality:  (F,l )  (T E ,ℓ)  (T E ,ℓ)  (E ,λ)  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that non-singular lifts are closed under finite limits, so (F,l ) is a non-singular lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If (E ,λ) is a  nonsingular lift then λ : (E ,λ) → (T E ,ℓ) equalizes the diagram, so there is a unique isomorphism  (F,l ) ∼= (E ,λ), making the multiplication of the monad a natural isomorphism (and thus yielding an  idempotent monad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The functor is defined as a T -limit and therefore preserves all T -limits of lifts, so  it is left-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In the category of smooth manifolds, ℓ is the Euler vector field of an �+-action, this guarantees that  every λ is the Euler vector field of a multiplicative �+-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We note the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the category of multiplicative �+ actions in SMan, multiplication by 0 is equivalent  to the natural idempotent e in the fully faithful functor sending an �+-action to its Euler vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By non-singularity, the following diagram is an equalizer, giving E a multiplicative action by �+ whose  Euler vector field is λ:  (E ,·E )  (T E ,·T )  (T E ,·T )  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  λ  Moreover, λ is the Euler vector field of this lift, and the following diagram commutes:  E  T E  T E  �+ × E  �+ × T E  �+ × T E  T (�+ × E )  T (�+ × T E )  T (�+ × T E )  T E  T T E  T T E  T (�+×e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E )  T (�+×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e )  �+×λ  p  T ·p  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T λ  T ·E  (1�,id)  (1�,id)  (1�,id)  λ�×0  λ�×0  λ�×0  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  �+×e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  �+×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  λ  �+×λ  40  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4  Differential bundles  This section introduces (pre-)differential bundles, which provided the rest of the data for a vector bun-  dle: namely the projection, the zero section, and the addition map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The zero section and projection  data arise by splitting the natural idempotent e : id ⇒ id , and non-singularity will induce the addi-  tion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every differential bundle satisfies a pair of universality diagrams, linking this presentation  of differential bundles to the original definition in Cockett and Cruttwell (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) A pre-differential bundle is a lift λ : E → T E equipped with a chosen splitting of the natural idem-  potent e = p ◦ λ from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Pre-differential bundles are formally written (q : E →  M ,ξ,λ), where q : E → M is the retract, ξ : M → E the section, and λ : E → T E the lift (the types  are only necessary for the projection as the rest may be inferred, and will generally be suppressed to  save space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) A differential bundle is a pre-differential bundle (q : E → M ,ξ,λ) with the properties that λ is  non-singular and T -pullback powers of q exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Morphisms of (pre-)differential bundles are exactly morphisms of their underlying lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The categories  of (pre-)differential bundles are exactly (pre-)differential bundles and lift morphisms, and are written  Pre(�),DBun(�) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that as e is a natural idempotent in the category of lifts, any lift morphism will preserve e  and will consequently preserve its idempotent splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Preserving the idempotent means that every  differential bundle morphism is a bundle morphism, where the base map is given by  E  F  M  N  q  f  π  m:=π◦f ◦ξ  We now look at the limits of (pre-)differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The limit for a diagram of pre-differential bundles is the limit of the underlying lifts equipped with  a chosen splitting of p ◦ λ due to basic properties about idempotent splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The limit for a diagram of differential bundles is the limit in the category of pre-differential bun-  dles (the lift will be universal by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4), so long as T -pullback powers of the resulting  projection exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Because there is a projection associated to a (pre-)differential bundle, the T -reindexing operation  described in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 can now be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This gives a pullback differential bundle in a similar  way to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 (Cockett and Cruttwell (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ) be a (pre-) differential bundle in a  tangent category �, and consider a T -pullback in �:  u∗E  E  N  M  u  q  u∗q  ¯u  ⌟  41  Then the induced triple maps  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='u∗E  T E  N  u∗E  E  u∗E  E  N  M  N  M  T N  T M  u  q  u∗q  ¯u  ⌟  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ¯u  λ  0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='u  0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='u∗q  u∗λ  q  u  ⌟  ξ◦u  u∗ξ  induce a (pre-)differential bundle (u∗q : u∗E → N ,u∗ξ,u∗λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If λ is non-singular and T -pullback pow-  ers of u∗q exist, then (u∗q,u∗ξ,u∗λ) is a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note thata pre-differential bundle (q : E → M ,ξ,λ) in � mayalsobe regarded asa pre-differential  bundle (q : (E ,λ) → (M ,0),ξ,λ) in Lift(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Take the following pullback in Lift(�):  (u∗E ,u∗λ)  (E ,λ)  (N ,0)  (M ,0)  u  q  ⌟  It follows by construction that ι ◦ u∗λ = λ ◦ ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus the result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ) be a non-singular pre-differential bundle in a tangent category  �, so that T -pullback powers of q exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then there is an additive bundle structure (q,ξ,+q) so that the  differential bundle morphisms are additive:  (λ,ξ) : (q,ξ,+q) → (p,0,+)  (λ,0) : (q,ξ,+q) → (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore, every differential bundle morphism preserves addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Non-singularity forces the existence of an addition map:  E2  T2E  T2E  E  T E  T E  ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+q  +  λ×λ  λ  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  +  (e ×e ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  Note that this diagram commutes because T2E is a pre-differential bundle whose lift is ℓ×ℓ, and also +  is a linear morphism, so it commutes with the addition map e ◦a +e ◦b = e ◦(a ×b ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Post-composition  with λ ensuresthatξ isthe unitand thatassociativityholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Adifferential bundle morphism will induce  a morphism of the equalizer diagrams that induce each addition map to preserve addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, the Euler vector field for every vector bundle is a non-singular lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  If T M q ×q E exists, the map  νE : T M p ×q E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ×λ  −−−→ T2E  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  −→ T E  may be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Similarly, using the additive bundle structure from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4, the µ map may  be formed:  µE : E q ×q E  0×λ  −−→ T (E q ×q E )  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+q  −−→ T E  Note that any differential bundle morphism will preserve µ and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  42  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Differential bundle maps preserve µ(x, y ) := 0◦ x +T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q λ◦ y and ν(v, y ) := T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦v +p λ◦ y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following diagram demonstrates that lift maps preserve ν:  E q ×p T M  F q ′×p T N  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  ν  ν  f ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  g ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='n  Similarly, this diagram demonstrates that lift maps preserve µ:  T E  T F  T E2  T F2  E2  F2  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f )  (λ×0)  (λ′×0)  (f ×f )  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+′  T f  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider the full subcategory of differential bundles in � whose objects are differential  bundles (E ,λ) so that the forks  E2  T E  T M  T M p ×q E  T E  E  µE :=0◦π0+T q λ◦π1  νE :=T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π0+p λ◦π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q◦0◦p  p  p◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3)  are T -equalizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This subcategory is closed under T -equalizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Start with the T -equalizer of lifts:  C  E  F  f  g  k  Observe that k is a lift map, so by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5, the following diagram commutes:  C q c ×p T P  E q ×p T M  F q ′×p T N  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  ν  ν  f ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  k×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='k ′  ν  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='k  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='k  g ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='n  43  Next, since k is a morphism of pre-differential bundles, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 the following diagram com-  mutes:  C2  E2  F2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  µ  µ  f2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  g2  µ  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  The top row follows because maps satisfying Rosicky’s universality condition preserve µ, and the bot-  tom row by the naturality of e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C and the fact that linear maps preserve the natural idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, if  E ,F satisfy the universality diagrams in Diagram 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3, the equalizer C will as well, because T -equalizers  are closed to T -limits in the category of fork diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every differential bundle (q : E → M ,ξ,λ) in a tangent category, the diagram  E2  T E  T M  µE  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q◦0◦p  is a T -equalizer, and if T M p ×q E exists, then  E q ×p T M  T E  E  νE :=λ◦π0+p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π0  p  p◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  is a T -equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent bundle satisfies both universality conditions, and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6every differential  bundle will satisfy these conditions by the non-singularity of the lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a complete tangent category, the category of differential bundles is precisely the cat-  egory of algebras of the monad in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 on pre-differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The universality conditions in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 demonstrate that this definition of differ-  ential bundle agrees with that of Cockett and Cruttwell (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  The isomorphism of categories  It is now straightforward to show that the main theorem of this chapter holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, observe that for  every differential bundle (q : E → M ,ξ,λ) in the category of smooth manifolds, the T -pullback E p ×q  T M exists , as p is a submersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the universality of the two diagrams is equivalent:  E q ×p T M  T E  E  νE :=λ◦π0+p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π0  p  p◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  E q ×p T M  T E  M  E  νE  λ◦π0+p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π1  p  ξ  ⌟  Thus the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  44  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an isomorphism of categories between vector bundles and differential bundles  in smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 gives a fully faithful functor from the category of vector bundles to  differential bundles of smooth manifolds, as the lift associated to the vector bundle is non-singular and  the projection and zero section give the rest of the structure of a differential bundle: as remarked after  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1, local triviality guarantees that the projection is a submersion, so pullback powers of  the projection exist, yielding a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To see there is an isomorphism on objects, recall that every differential bundle is a fibred �-module  by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6, and that this identification is a bijective mapping (it recovers the original �-action  from the Euler vector field of the �-action, and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Last, note that the universality condition  E q ×p T M  T E  M  E  νE  λ◦π0+p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π1  p  ξ  ⌟  along with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 ensures the local triviality of q, so that the unique fibred �-module struc-  ture associated to a differential bundle is indeed a vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6  Connections on a differential bundle  The connections discussed in this section generalize the notion of an affine connection to a differential  bundle, giving a “local coordinates” presentation for T E similar to the presentation of T 2M as T3M  induced by an affine connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Chapter 3 makes extensive use of connections on vector bundles, so  it is useful to set out the basic definitions before embarking on algebroid theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 (Cockett and Cruttwell (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ) be a differential bundle in a tan-  gent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A vertical connection is a map κ : T E → E so that  (i) κ is a vertical descent, and hence a retract of the lift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus, κ ◦ λ = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) κ is compatible with both differential bundle structures on T E , so that the maps  κ : (T E ,ℓ) → (E ,λ)  κ : (T E ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (E ,λ)  are lift morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A horizontal connection is a map ∇ : E q ×p T M → T E so that  (i) ∇ is a horizontal lift, and so is a retract of (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q) : T E → E q×pT M 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus, (p,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q)◦∇ = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) ∇ is compatible with each pair of lifts, so that the maps  ∇ : (E q ×p T M ,λ × ξ) → (T E ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  ∇ : (E q ×p T M ,0× ℓ) → (T E ,ℓ)  are lift morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3That (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q) land in the pullback E q ×p T M is a consequence of naturality, as p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q = q ◦ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  45  A full connection is a pair (κ,∇) that satisfies the following compatibility relations:  (i) κ ◦ ∇ = ξ ◦ q ◦ π0,  (ii) ∇(p,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q) +T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q µ(p,κ) = id , so that there is an isomorphism E q ×p T M p ×q E ∼= T E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A notion that will be useful when dealing with classical differential geometry is that of a covariant  derivative, whose definition is equivalent to that of a vertical connection in the category of smooth  manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ,κ) be a vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 The covariant derivative associated  to (κ,∇) is the map  ∇(−)[=] : Γ(π) × Γ(p) → Γ(π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(A,X ) �→ (κ ◦ T A ◦ X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lucyshyn-Wright (2018) drastically simplified the notion of a full connection by showing that it is  exactly a vertical connection satisfying a universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 (Lucyshyn-Wright (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A connection on a differential bundle is equivalently spec-  ified by a vertical connection  κ : T E → E  so that the following diagram exhibits T E as a biproduct in the category of differential bundles over M :  E  T E  T M  M  E  p  T q  κ  q  q  p  There is also a notion of flatness for connections that extends to vertical connections on a differen-  tial bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A connection κ on a differential bundle (q : E → M ,ξ,λ) is flat whenever  κ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='κ ◦ c = κ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We can see that connections are closed under similar constructions to differential bundles, in par-  ticular idempotent splittings and the reindexing construction from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ) be a differential bundle equipped with a vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Any linear retract of (q,ξ,λ) will have a vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The pullback differential bundle induced by pulling back q along f : N → M will have a vertical  connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  If the vertical connection is flat or is part of a full connection (that is, it satisfies the universality condition  in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3), then the induced connection will be as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The universal property induces the vertical connection in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The construction will pre-  serve flatness as it is an equational condition, and it preserves effectiveness by the commutativity of  limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4The definition of a covariant derivative only uses a vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  46  There is no reason for every differential bundle in a tangent category to have a connection (for  example, the tangent bundle in the free tangent category Weil1 in Chapter 4 is a differential bundle  that does not have a connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, if the total space of a differential bundle has an affine  connection, this induces a compatible connection on the differential bundle and its base space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ) be a differential bundle in a tangent category in �, where E has a  (flat) vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then the total space M and the differential bundle (q,ξ,λ) each have a (flat)  vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If the connection is full (so there is a compatible horizontal connection), then the  induced connections are likewise full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By the strong universality condition for differential bundles, the differential bundle (q ◦ π0 :  E q×pT M → M ,(ξ,0),λ×ℓ) is the pullback differential bundle of p : T E → E along ξ : M → E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This gives  (q◦π0,(ξ,0),(λ,ℓ)),a (flat, effective) vertical connection byLemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5, and soityields(q : E → M ,ξ,λ)  and (p : T M → M ,0,ℓ) as (flat, effective) vertical connections by the idempotent splitting property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Every smooth manifold has an affine connection, thus inducing a connection on any vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every vector bundle has a connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  47  Chapter 3  Involution algebroids  This chapter accomplishes the first major goal of this thesis by providing a tangent-categorical axiom-  atization of Lie algebroids, namely involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Much like vector bundles, Lie algebroids are  a highly non-algebraic notion (in the sense of Freyd and Kelly (1972)), being vector bundles equipped  with a Lie algebra structure on the set of sections of the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Furthermore, the bracket on sec-  tions must satisfy a product rule with respect to �-valued functions on the base space (the Leibniz  law), introducing another piece of non-algebraic structure to the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent-categorical  definition of Lie algebroids will treat the tangent bundle as the “prototypical Lie algebroid” in which  the vertical lift ℓ : T ⇒ T 2 identifies the vector bundle structure and the canonical flip c : T 2 ⇒ T 2 plays  the role of the Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lie algebroidsare the natural many-objectanalogue toLie algebras, in the same waythatLie groupoids  are the many-object analogue of Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the single-object case, a Lie group is classically thought  of as a space of symmetries for some smooth manifold (one often identifies a group action G ×M → M ),  and a Lie algebra may similarly be thought of as a space of derivations (often identified as a sub-Lie-  algebra of χ(M ) for a manifold M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The extension of groups to groupoids is natural;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' in fact, Brandt’s  introduction of groupoids in Brandt (1927) predates MacLane and Eilenberg’s invention of category  theory in Eilenberg and MacLane (1945) by nearly two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The translation of Lie algebras to the  many-object case is not as straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first step is to replace the vector space underlying a Lie  algebra with a vector bundle (π : A → M ,ξ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The idea is to axiomatize this vector bundle so that each  section in Γ(π) corresponds to a derivation on C ∞(M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The anti-commutator operation on derivations  from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='18 suggests there should be a Lie bracket [−,−] : Γ(π) ⊗ Γ(π) → Γ(π) (similar to the  partially-defined multiplication for a groupoid), while the correspondence with derivations on C ∞(M )  suggests there be a vector bundle morphism ̺ : A → T M satisfying the Leibniz law:  [X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [X , f ] := ˆp ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ̺ ◦ X 1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  (the full definition of Lie algebroids may be found in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This “operational” definition of Lie alge-  broids makes it difficult to describe their morphisms, and furthermore it essentially fails to be an alge-  braic structure in the classical sense, as it axiomatizes structure on the set of sections of a map rather  than a morphism in the category itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Involution algebroids were introduced to provide a tangent-categorical presentation of Lie alge-  broids, similar to the relationship between differential bundles and vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Chapter 2 focused  on the Euler vector field construction on a vector bundle, showing that this induced a fully-faithful  functor from vector bundles to associative coalgebras (lifts) of the weak comonad (T,ℓ), and identified  vector bundles with a subcategory of Lift(SMan) satisfying a universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The corresponding  1Recall the notation from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  48  construction for Lie algebroids, then, is the canonical involution, which was identified by Eduardo  Martinez and his collaborators (a clearly written exposition may be found in Section 4 of de León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given a Lie algebroid (π : A → M ,̺ : A → T M ,[−,−] : Γ(π) ⊗ Γ(π) → Γ(π)), its canonical involu-  tion is a map  σ : A̺×T πT A → A̺×T πT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using this σ map, there is a straightforward characterization of Lie algebroid morphisms: a Lie alge-  broid morphism is precisely a vector bundle morphism (f ,m) : A → B that preserves the anchor and  involution maps:  A  B  A̺×T πT A  B ̺B ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πB T B  T M  T N  A̺×T πT A  B ̺B ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πB T B  ̺A  ̺B  f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  σB  f ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  σA  f ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  Furthermore, it is implicit in Martinez’s work (Martínez (2001)) that σ satisfies axioms corresponding  to the Lie algebroid axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, involutivity corresponds to antisymmetry of the Lie bracket  σ ◦ σ = id ⇐⇒ [X ,Y ]+ [Y ,X ] = 0,  while the Leibniz law holds if and only if the ̺ map sends the algebroid involution to the canonical flip  on M ,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ π1 ◦ σ = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ ̺ ⇐⇒ ∀f ∈ C ∞(M ),[X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y  (using the same definition as before for [X , f ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The idea of an involution algebroid, then, is to axiomatize the canonical involution directly, just as  differential bundles axiomatize the Euler vector field of a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An involution algebroid  is a differential bundle equipped with a pair of structure maps  ̺ : A → T M , σ : A̺×T πT A → A̺×T πT A  satisfying a collection of axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Some of them are straightforward translations of the structure equa-  tions for Lie algebroids given in Martínez (2001), for instance  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ λ = ℓ◦ ̺, σ ◦ σ = id , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ π1 ◦ σ = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  However, this requires a new coherence between the Euler vector field of the underlying vector bundle  and the involution map:  σ ◦ (ξ ◦ π,λ) = (ξ ◦ π,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The most striking new fact about this coherence is that the Jacobi identity on the bracket [−,−] corre-  sponds to the Yang–Baxter equation on σ:  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  σ×c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  σ×c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  1×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  σ×c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  1×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  This is both surprising (it is a new characterization of a central object of study in differential geom-  etry and mathematical physics) and yet in a way expected (the work in Cockett and Cruttwell (2015),  49  Mackenzie (2013) indicates that the Lie algebra structure on the set of vector fields over a manifold  follows from the Yang–Baxter equation on c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This vector-field-free presentation of the Jacobi identity  allows for a structural approach to Lie algebroids that drives the work presented in Chapters 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  As with Chapter 2, the first section is expository, and is concerned with introducing the category of  Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The second section introduces anchored bundles, together with the space of prolonga-  tions of an anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The relationship between anchored bundles and involution algebroids  is equivalent to that between reflexive graphs and groupoids (the subject of Chapter 5), the space of  prolongations of an anchored bundle being equivalent to the set of composable arrows for a groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This section is mostly a translation of Martinez’s prolongation construction to a general tangent cate-  gory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The rest of the chapter contains new results, developed in collaboration with Matthew Burke and  Richard Garner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Section 3 introduces involution algebroids, which are anchored bundles equipped with an involu-  tion map on their space of prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 4 considers an anchored bundle in a tangent cate-  gory with negatives that is equipped with a connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The connection gives an involution algebroid  a “local coordinates” presentation (in the sense of Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4) that is equivalent to the local character-  ization of Lie algebroids from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The final section of this chapter establishes the main result:  the category of Lie algebroids is isomorphic to that of involution algebroids in smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Lie algebroids  This section reviews the basic theory of Lie algebroids: their definition and that of their morphisms,  along with some introductory examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The classical definition will not appear elsewhere in this  chapter, however, as we quickly introduce Martinez’s structure equations for a Lie algebroid (Martínez  (2001)), then translate them into tangent-categorical terms using a connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Lie algebroid is a vector bundle π : A → M equipped with an anchor ̺ : A → T M  and a bracket [−,−] : Γ(π) ⊗ Γ(π) → Γ(π) satisfying the following axioms:  bilinear: [a X1 + b X2,Y ] = a[X1,Y ]+ b [X2,Y ] and [X ,a Y1 + b Y2] = a[X ,Y1]+ b [X ,Y2]  anti-symmetric: [X ,Y ]+ [Y ,X ] = 0  Jacobi: [X ,[Y ,Z ]] = [[X ,Y ],Z ]+ [Y ,[X ,Z ]]  Leibniz: [X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y  (where [X , f ] is defined as in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The canonical example of a Lie algebroid is, of course, the tangent bundle using the operational  tangent bundle from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) ALie algebra isa Lie algebroid overthe terminal object: fora groupG , the bundle of source-constant  tangent vectors is the usual Lie functor from Lie groups to Lie algebras, because a groupoid is a one-  object group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  50  (iii) The bundle of source-constant tangent vectors s,t : G → M of a Lie groupoid forms a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This bundle is defined by the pullback  A  T G  T M  M  T M ×G  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='s,p)  (0,e )  π  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='t  ̺  ⌟  where the projection is π and the target is given by ̺ in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an injective �-module  morphism from sections of π, Γ(π) to vector fields on G , χ(G ), and the Lie bracket on G is closed  over the image of this lift, putting a Lie bracket on Γ(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, we can see that T M is the  bundle of source-constant tangent vectors for the pair groupoid on a manifold M :  T M  T (M × M )  M  (M × M ) × T M  (p,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π0)  (∆,0)  p  (id,0◦p)  ⌟  Given (u,v) : X → T (M × M ) above (m,m) : X → T M with u = 0 ◦ m, it follows that u = 0 ◦ p ◦ u,  so v is the unique map induced into T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) Every group is a groupoid over a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Lie algebroid associated with a group G , then, is  the usual Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) From the Hamiltonian formalism of mechanics, every Poisson manifold has an associated Lie alge-  broid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Poisson manifold is a manifold M equipped with a Poisson algebra structure on C ∞(M ),  namely a Lie algebra that is also a derivation:  [f ·g ,h] = f ·[g ,h]+ [f ,h]·g  where · is the multiplication in the algebra C ∞(M ) as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The cotangent bundle over  a Poisson manifold M is canonically a Lie algebroid, called a Poisson Lie algebroid Courant (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (vi) Any Lie algebra bundle—that is, a vectorbundle equipped with a Lie bracket on itsspace of sections—  is a Lie algebroid, with anchor map ξ ◦ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Morphisms of Lie algebroids are notoriously difficult to work with, and have an involved definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A,B be a pair of anchored bundles over M ,N , and Φ : A → B an anchored bundle  morphism over a map φ : M → N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Φ-decomposition of X ∈ Γ(πA) is a set of Xi ∈ Γ(πB) and fi ∈ C ∞(M )  so that  Φ ◦ X =  �  i  fi · Xi ◦ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An anchor-preserving vector bundle morphism Φ is a Lie algebroid morphism if and only if for any X ,Y ∈  Γ(πA) and Φ-decompositions {Xi , fi},{Yj,g j} of X ,Y , the following equation holds:  Φ ◦ [X ,Y ] =  �  fi ·g j ·([Xi ,Yj]◦ φ) +  �  [X , fi ]·(Xi ◦ φ) −  �  [Y ,g j](Yj ◦ φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  51  The equation defininga Lie algebroid morphism holdsindependentlyofthe choice ofΦ-decomposition  (see Higgins and Mackenzie (1990) for a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) If A,B are Lie algebroids over a base manifold M , Φ : A → B is a Lie algebroid morphism if and only  if it preserves the anchor and Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given two Lie algebroid morphisms f : A → B,g : B → C , their composition g ◦ f is also a Lie  algebroid morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) A Poisson Sigma model (see Bojowald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2005)) is a morphism of Lie algebroids  φ : T Σ → T ∗M  for which Σ is a 2-dimensional manifold and T Σ denotes its tangent Lie algebroid, while M is a  Poisson manifold and T ∗M denotes the Lie algebroid structure on its cotangent bundle .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lie algebroids are a natural generalization of Lie algebras to the “multi-object” setting, but they  are ill-suited for a functorial presentation of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A step in this direction is to consider the  coordinate-based presentation of the Lie bracket and its coherences due to Martínez (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A  be an anchored bundle over M equipped with a bilinear bracket on its space of sections, and choose a  pair of bases for Γ(π) and χ(M ): for Γ(π) write {eα}, and for χ(M ) write { ∂  ∂ x i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The anchor and bracket  then have a presentation in local coordinates:  ̺(eα) =  �  i  ̺i  α  ∂  ∂ x i  [eα,eβ] =  �  γ  C γ  αβ eγ  (from here on out, we use Einstein summation notation to simplify our calculations, so instead write  ̺(eα) = ̺i  α  ∂  ∂ x i  [eα,eβ] = C γ  αβ eγ  with  �  suppressed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following characterization of the Lie algebroid axioms uses Martinez’s struc-  ture equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' [Martínez (2001)] An anchored bundle A over M equipped with a bracket is a Lie  algebroid if and only if ̺ and [−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='−] satisfy the following structure equations:  (i) Alternating:  C ν  αβ + C ν  βα = 0  (ii) Leibniz:  ̺ j  α  ∂ ̺i  β  ∂ x j = ̺i  γC γ  αβ + ̺ j  β  ∂ ̺i  α  ∂ x j  (iii) Bianchi:  0 = ̺i  α  ∂ C ν  βγ  ∂ x i + ̺i  β  ∂ C ν  γα  ∂ x i + ̺i  γ  ∂ C ν  αβ  ∂ x i + C µ  βγC ν  αµ + C µ  γαC ν  βµ + C µ  αβC ν  γµ  52  This proposition is a straightforward translation of the Lie algebroid axioms into local coordinates  using a covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, recall that by the smooth Serre–Swan theorem (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='33 Nestruev  (2003)), the bilinearity of the bracket  [−,−] : Γ(π) × Γ(π) → Γ(π)  as a morphism of C ∞(M )-modules guarantees that it corresponds to a bilinear morphism A2 → A of  vector bundles, meaning that there exists a globally defined bilinear map A2 → A that is equal to the  Lie bracket when applied to sections of the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We record this as a lemma:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every Lie algebroid � , there is a bilinear morphism  〈−,−〉 : A2 → A  so that for any sections X ,Y ∈ Γ(π), 〈X ,Y 〉 = [X ,Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There are two structure maps derived from 〈−,−〉 that encode the coherences of a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first map measures the extent to which the anchor maps fail to preserve the chosen connections  on the vector bundle π : A → M and the tangent bundle p : T M → M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be a Lie algebroid, and for a chosen horizontal connection ∇ on A and vertical  connection κ′ on T M , set  {v, x}κ′,∇ := T M p ×q A  ∇  −→ T A  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  −→ T 2M  κ′  −→ T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  When the choice of connection is evident by context, we will suppress the subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The above parentheses bracket corresponds to the symbol  {−,−} = ̺ j  β  ∂ ̺i  α  ∂ x j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The Leibniz coherence may be rewritten as follows:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be an anchored bundle with a bilinear bracket (inducing an involution σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Choose  connections (κ,∇),(κ′,∇′) on A and T M respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The bracket and anchor map satisfy the Leibniz  law if and only if  ̺ ◦ 〈x, y 〉(κ,∇) + {̺x, y }(κ′,∇) = {̺y, x}(κ′,∇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The condition is equivalent to the identity  ̺ j  α  ∂ ̺i  β  ∂ x j = ̺i  γC γ  αβ + ̺ j  β  ∂ ̺i  α  ∂ x j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The Bianchi axiom measures the failure of the Jacobi identity in local coordinates, and states that  it must be corrected for by the curvature of the brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We see that  C ν  αµC µ  βγ = [eα,[eβ,eγ]]  while  ∂ C ν  βγ  ∂ x i eν = κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ ( ∂  ∂ x i ,eβ,eγ)  53  determines a trilinear map  { ∂  ∂ x i ,eβ,eγ}(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ ( ∂  ∂ x i ,eβ,eγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This is the second derived map used in the structure equations for a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle equipped with a bilinear map  〈−,−〉 : A2 → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The derived ternary bracket {−,−,−} : T M p ×πAπ×πA → A is defined as  {v, x, y }(κ,∇) := T M p ×πAπ×πA  ∇[A2]  −−−→ T A2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='〈−,−〉  −−−−→ T A  κ−→ A  where ∇[A2] is the pairing (∇(π0,π1),∇(π0,π2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The ternary bracket corresponds to the following symbol:  {−,−,−}(κ,∇) := ̺i  α  ∂ C ν  βγ  ∂ x i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be an anchored bundle over M with a bilinear bracket and denote the induced  involution by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Choose a pair of connections and write the derived maps 〈−,−〉,{−,−,−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then  (i) the bracket is antisymmetric if and only if its globalization is;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, 〈eα,eβ〉 + 〈eβ,eα〉 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) the bracket satisfies the Jacobi identity if and only if it is alternating in the last two arguments and  0 =  �  γ∈Cy(3)  〈xγ0,〈xγ1, xγ2〉〉 +  �  γ∈Cy(3)  {̺xγ0, xγ1, xγ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) This is equivalent to C ν  αβ + C ν  βα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) This is equivalent to  0 = C µ  βγC ν  αµ + C µ  γαC ν  βµ + C µ  αβC ν  γµ + ̺i  α  ∂ C ν  βγ  ∂ x i + ̺i  β  ∂ C ν  γα  ∂ x i + ̺i  γ  ∂ C ν  αβ  ∂ x i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Lie algebroid is exactly an anchored vector bundle (π : A → M ,ξ,λ,̺) with a  bilinear, alternating map  〈−,−〉 : A2 → A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  〈x, y 〉 + 〈y, x〉 = 0  so that for any connection (∇,κ) on A, the maps  {v, x}(κ,∇) := κ′ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ ∇(v, x),  {v, x, y }(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ (v, x, y )  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2)  satisfy the equations  (i) ̺ ◦ 〈x, y 〉 + {̺x, y }(κ′,∇) = {̺y, x}(κ′,∇),  54  (ii)  �  γ∈Cy(3)〈xγ0,〈xγ1, xγ2〉〉 +  �  γ∈Cy(3){̺xγ0, xγ1, xγ2}(κ,∇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There is also a local coordinates presentation of morphisms as in Section 2 of Martínez (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An  anchored bundle morphism A → B is a Lie algebroid morphism whenever  f β  γ Aγ  αδ + ̺i  δ  ∂ f β  α  ∂ x i = B β  θσ f θ  α f σ  δ + ̺i  α  ∂ f β  δ  ∂ x i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The A and B arguments are understood as the brackets, so this condition can be rewritten as  f β  γ Aγ  αδ = f ◦ 〈α,δ〉,  B β  θσ f θ  α f σ  δ = 〈f ◦ α, f ◦ δ〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Set the following notation for maps between vector bundles with connection:  f : A → B  (κA,∇A,A,λA)  (κB,∇B,B,λB)  ∇[f ] : A2 → B := κB ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇A  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3)  The ̺ terms are understood to be the torsion, so that  ̺i  δ  ∂ f β  α  ∂ x i = κ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇(̺eδ,eα) = ∇[f ](̺eδ,eα),  ̺i  α  ∂ f β  δ  ∂ x i = κ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇(̺α,δ) = ∇[f ](̺eα,eδ),  using the notation set up in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The notion of a Lie algebroid morphism, then, has the fol-  lowing presentation:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14 (Martínez (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,̺A,〈−,−〉A),(q : B → N ,̺B,〈−,−〉B) be a pair of  Lie algebroids with chosen connections (κ−,∇−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An anchor-preserving vector bundle morphism f : A →  B is a Lie algebroid morphism if and only if  f ◦ 〈eα,eδ〉 + ∇[f ]◦ (̺eδ,eα) = 〈f ◦ α, f ◦ δ〉 + ∇[f ]◦ (̺eα,eδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  Anchored bundles  Anchored bundlesare toLie algebroidswhatreflexive graphsare togroupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Each theoryhas(mostly)  the same structure, but while a reflexive graph is missing a groupoid’s composition operation, an an-  chored bundle lacks a Lie algebroid’s bracket operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This section reviews the basic theory of an-  chored bundles and their prolongations (see Mackenzie (2005) for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An anchored bundle in a tangent category isa differential bundle (A  π−→ M ,ξ,λ) equipped  with a linear morphism  A  T M  M  ̺  π  p  A morphism of anchored bundles is a linear bundle morphism (f ,v) that preserves the anchors  T A  T B  A  B  A  B  T M  T N  λ  l  f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  ρ  ̺  f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v  The category of anchored bundles and anchored bundle morphisms in a tangent category � is written  Anc(�), and a generic anchored bundle is written (A  π−→ M ,ξ,λ,̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  55  There are two pullbacks that are associated with every anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These play the role of the  spaces of composable arrows G2 := G t ×s G ,G3 := G t ×sG t ×sG for a reflexive graph s,t : G → M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (A  π−→ M ,ξ,λ,̺) be an anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Its first and second prolongations are  given by the limits  � (A)  T A  � 2(A)  T 2A  A  T M  T A  T 2M  A  T M  ̺  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  ⌟  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  ̺  ⌟  (The notation for the fibre product is slightly non-standard, as it is not technically a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=') Through-  out this chapter, it will always be assumed that the first and second prolongations of an anchored bundle  exist (although no choice of prolongation is explicitly made).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is not strictly necessary that the prolongations of an anchored bundle exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this con-  dition is primarily a matter of convenience when discussing involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every result in this  section, that does not explicitly mention prolongations, holds for an anchored bundle independently of  their existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For any object M , id : T M → T M is an anchor for the tangent differential bundle, and every  f : M → N yields a morphism of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Moreover, for any anchored bundle over M the  anchor is itself a morphism of anchored bundles (A,π,ξ,λ,̺) → (T M ,p,0,ℓ,id ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This induces a  fully faithful functor:  � �→ Anc(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This inclusion has a left adjoint, which sends an anchored bundle (π : A → M ,ξ,λ,̺) to its base  space M (the unit is the anchor map ̺), so that � is a reflective subcategory of Anc(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For any differential bundle (A,π,ξ,λ), the map 0 ◦ π : A → T M is an anchor and every morphism  f : A → B of differential bundles again yields a morphism of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The naturality of  0 ensures that every differential bundle morphism will preserve this trivial anchor map, giving a  fully faithful functor  DBun(�) �→ Anc(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This functor has a right adjoint that replaces the anchor map with the trivial anchor map  (π : A → M ,ξ,λ,̺) �→ (π : A → M ,ξ,λ,0◦ π)  where the counit is given by the natural idempotent e = ξ ◦ π : (A,λ) → (A,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is trivial to check  that the anchor map is preserved by the bundle morphism (e ,id ):  ̺ ◦ ξ ◦ π = 0◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='id ◦ π  and so the following diagram commutes:  T M  T M  A  A  0◦π  p◦λ  ̺  56  This means that differential bundles are a coreflective subcategory of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Any reflexive graph (s,t : C → M ,e : M → C ) in a tangent category has an anchored bundle (when  sufficient limits exist), constructed as  C ∂  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e s  (where e s : C → C = e ◦ s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Construct a lift on C ∂ :  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C ∂  T 2C1  T 2C1  C ∂  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C1  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e s  ℓ  ℓ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e s  λ  This lift will be non-singular by the commutativity of T -limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The pre-differential bundle data is  given by the projection  C ∂ �→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e s  −−→ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The section is induced by  M  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e )  −−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  while post-composition with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='t gives the anchor map:  C ∂ �→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='t  −→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The diagram is a pullback by composition, and the outer perimeter defines the pullback � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note  that any reflexive graph morphism will give rise to an anchored bundle morphism by naturality,  making the construction of an anchored bundle from a reflexive graph functorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) For any object M in a tangent category, cM : T 2M → T 2M is an anchor on (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ), and  every map f : M → N gives a morphism of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) For any anchored bundle (q : E → M ,ξ,λ,̺), the differential bundle (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,  c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) has an anchor  ̺T : T E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  −→ T 2M  c−→ T 2M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first prolongation of an anchored bundle � (A) behaves similarly to the second tangent bundle,  except that it does not have a canonical flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the definition of an affine connection, the tangent  bundle played a similar role to the “arities” of a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a lift map that makes this connection  stronger:  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We define a  generalized lift:  ˆλ : A → A̺×T πT A := (ξ ◦ π,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This generalized lift satisfies the same coherences as the lift on the second tangent bundle:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It follows that  (i) (Coassociativity of ˆλ) (ˆλ × ℓ) ◦ ˆλ = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ) ◦ ˆλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  57  (ii) (Universality) the following diagram is a T -pullback:  A2  A̺×T πT A  M  A  ˆµ  π0  π◦πi  ξ  ⌟  where ˆµ := (ξ ◦ π◦ π0,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Compute  (ˆλ × ℓ) ◦ ˆλ = (ξ ◦ π◦ ξ ◦ π,λ ◦ ξ ◦ π,ℓ◦ λ)  = (ξ ◦ π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (ξ ◦ π) ◦ λ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ)  = (π0,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (ξ ◦ π) ◦ π1,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ π1) ◦ (ξ ◦ π,λ)  = (id × T (ˆλ)) ◦ ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Use the pullback lemma to observe that the following diagram is universal for any anchored bun-  dle:  A2  � (A)  T A  M  A  T M  ξ  ̺  0  T π  π0  π◦π1  ˆµ  π1  µ  The top triangle of the diagram commutes by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The right square and outer perimeter  are pullbacks by definition, and the bottom triangle also commutes by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The pullback  lemma ensures that the left square is a pullback, so for every anchored bundle, the general lift is  universal for � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now post-compose with the involution:  A2  � (A)  � (A)  M  A  A  ξ  π0  π◦π1  ˆµ  id  pπ1  σ  ˆν  It suffices to check that the top triangle commutes, so σ ◦ ˆµ = ν:  σ ◦ ˆµ ◦ (a,b ) = σ ◦ ((ξ ◦ π,0) ◦ a +π0 (ξ ◦ π,λ) ◦ b ) = (id ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ ◦ ̺ ◦ a) +pπ1 (ξ ◦ π,λ) ◦ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus, the lift (ξπ,λ) involution algebroid is universal for � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For T (M ) = T (M ), the space of prolongations is T (M )id ×T p T 2M = T 2M , and the second prolon-  gation is given by T 2M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  58  (ii) In a tangent category with a tangent display system, if π ∈ � then the prolongations for (π,ξ,λ,̺)  automatically exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, for every anchored bundle in the category of smooth manifolds,  all prolongations exist because the projection is a submersion (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) For any differential bundle with the anchor 0 ◦ π, it follows that � (A) ∼= Aπ×π◦πi (A2) ∼= A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  universality of the vertical lift factors b into (a1,a2), as the following diagram is a pullback:  A3  Aπ×πA  T A  A  M  T M  π0  (π1,π2)  µ  ππi  T π  π  0  (iv) Returning to the anchored bundle constructed from a graph, the space of prolongations � (A) em-  beds into the second tangent bundle of the space of composable arrows:  A̺×T πT A �→ T 2(G t ×sG ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of anchored bundles is, in a sense, “tangent monadic” over the category of differential  bundles: the forgetful functor from anchored bundles to differential bundles “creates” T -limits and  the tangent structure (this is all made precise in Chapter 5 using an enriched perspective on tangent  categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A limit of anchored bundles is the limit of the underlying differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Be-  cause the anchor is preserved by every map in the diagram, this induces a natural transformation for  any D : � → Anc(C ):  �  Anc(�)  Anc  DBun(�)  D  S  U  ̺  Thus, limU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D computes the limit of the underlying objects, while limU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D computes the limit of the  underlying differential bundles in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The anchor/unit map, then, induces a differential bun-  dle map:  lim̺i : (limAi ,limλi ) → (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(lim  i Mi),ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (lim  i Mi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This is the limit in the category of anchored bundles (so long as pullback powers of the limit projection  and the two prolongations of the limit anchor bundle exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent structure on lifts defined in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 lifts to a tangent structure on anchored  bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of anchored bundles in a tangent category has a tangent structure that maps  objects as follows:  (π : A → M ,ξ,λ,̺) �→ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π : T A → T M ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺)  where the structure maps are all defined using the pointwise structure maps in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  59  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given an anchored bundle (q : A → M ,ξ,λ,̺), there is an anchor on the differential bundle  (T q,T ξ,c ◦ T λ) given by c ◦ T ̺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the diagram commutes by naturality and the coherences on c ,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  T 2A  T 3M  T 3M  T A  T 2M  T 2M  T M  T M  T M  T q  T p  c ◦T λ  c ◦T ℓ  ℓ  c  p  T ̺  T 2̺  T c  The tangent structure maps and universality properties all follow from the forgetful property of the  functor from anchored bundles to differential bundles as a consequence of Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For any anchored bundle A, there are two differential bundles associated to � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first is the  usual pullback differential bundle given by pulling back T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π along ̺ as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3:  A̺×T πT A  T A  A  T M  π1  π0  T π  ̺  This gives the differential bundle structure  (A̺×T πT A  π0  −→ A,A  (id,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦̺)  −−−−−−→ A̺×T πT A,A̺×T πT A  (0,c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  −−−−−→ T (A̺×T πT A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Taking the fibre product in the category of anchored bundles as in Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 yields the second  differential bundle structure:  (A,π,ξ,λ)  (̺,id)  −−−→ (T M ,p,0,ℓ)  (T π,π)  ←−−− (T A,p,0,ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The two lifts behave similarly to the pair (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ) on the second tangent bundle, and (ℓ,λT ) on the  tangent bundle of a pre-differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a tangent category and Anc(�) the category of anchored bundles in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If T -  pullback powers of p ◦π0,π1 : A̺×T πT A → A exist, then there are two differential bundles on � (A), with  lifts induced as  (� (A),λ × ℓ)  (T A,ℓ)  (� (A),0× (c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ))  (T A,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  (A,λ)  (T M ,ℓ)  (A,0)  (T M ,0)  ̺  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  ⌟  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  ̺  ⌟  with structure maps  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (� (A)  p◦π1  −−→ A,A  (ξπ,0)  −−−→ � (A),� (A)  λ×ℓ  −−→ T ◦ � (A)),  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (� (A)  π0  −→ A,A  (id,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦̺)  −−−−−−→ � (A),� (A)  0×c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  −−−−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore, the two lifts λ� and λ̺ commute:  c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(λ × ℓ) ◦ (0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) ◦ (λ × ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  60  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The two differential bundles exist as a consequence of Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The commutativity of limits follows by postcomposition, as both c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ,ℓ and λ,0 commute  by the differential bundle axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The above lemma determines a functor, which we denote � , that sends an anchored bundle in �  to an anchored bundle in Lift(�) (equipped with the tangent structure from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a functor � from anchored bundles in � to anchored bundles in the cate-  gory of lifts Lift(�), that sends an anchored bundle (π : A → M ,ξ,λ,̺) to the tuple in Lift(�)  π� := (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  p◦π1  −−→ (A,0),  ξ� := (A,0)  (ξ◦π,0)  −−−→ (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  λ� := (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  (λ×ℓ)  −−→ (T (� (A)),c ◦ T (0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ),  ̺� := (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  π1  −→ (T A,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  (note that this functor lands in anchored bundles of non-singular lifts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Morphisms  of anchored bundles  (f ,m) : (π : A → M ,ξ,λ,̺) → (q : E → N ,ζ,l ,δ)  are sent to  � (f ,m) := f Tm ×Tm T f : A̺×T πT A → E δ×T q T E  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, note that the tuple  (π� : � (A) → A,ξ� ,λ� ,̺� )  is an anchored bundle, and that each morphism is a lift morphism:  π� : (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (A,0) follows because  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1 ◦ (λ × ℓ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ ℓ◦ π1 = 0◦ p ◦ π1  ξ� : (A,0) → (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) follows since  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0) ◦ 0 = (0◦ ξ ◦ π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ 0) = (0◦ ξ ◦ π,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ 0) = (0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) ◦ (ξ ◦ π,0)  λ� : (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)) follows by the commutativity of 0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  and λ × ℓ, and  ̺� : (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (T A,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) is a lift by definition of ̺� = π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This gives an anchored bundle in the category of non-singular lifts in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Next, check that the mapping is functorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To see that � (f ,m) preserves the lifts, note that (f ,m)  gives a morphism of diagrams for each pullback in Lift(�) defining the two lifts, so the induced map  � (f ,m) preserves each lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To see that � (f ,m) preserves the anchor, check that  ̺� B ◦ � (f ,m) = π1 ◦ (f × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ π1 = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ̺� A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  61  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a tangent category � where pullbacks along differential bundle projections exist,  such as a tangent category equipped with a proper retractive display system (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3) like the  category of smooth manifolds, there is an endofunctor on the category of anchored bundles in �,  � ′ : Anc(�) → Anc(�) =U Lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' T -Limits in Anc(Lift(�)) are computed as pointwise limits in � by Observations  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, and � is constructed as a limit, so it follows that � preserves T -limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The prolongation endofunctor � ′ : Anc(�) → Anc(�) has natural transformations  p ′ : � ′ ⇒ id  0′ : id ⇒ � ′  +′ : � ′p ′×p ′� ′ ⇒ � ′  ℓ′ : � ′ ⇒ � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′  satisfying all of the axioms of a tangent category that do not incvolve the canonical flip (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (Sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the full argument is given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3, but it is not difficult to sketch it out here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For the projection, zero map, and addition, use the differential bundle structure induced by Corollary  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To see the lift axiom, note that up to a choice of pullback, we have:  � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′(π : A → M ,ξ,λ,̺) =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  (p ◦ π1,p ◦ π2) :  � 2(A) → � (A)  (ξ ◦ π◦ π0,0◦ π1,0◦ π2) :  � (A) → � 2(A)  (λ × ℓ× ℓ) :  � 2(A) → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� 2(A)  (π1,π2) :  � 2(A) → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  The “lift map” is then  ℓ′ : � ′ ⇒ � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � (A)  A×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ  −−−→ � 2(A)  (where we recall that ˆλ = (ξ ◦ π,λ) : A → � (A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The structure described in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11 leads to the theory of double vector bun-  dles, developed by MacKenzie and his collaborators F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' and Mackenzie (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Mackenzie (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A dou-  ble vector bundle is a commuting square  D  B  A  M  q D  A  q D  B  q A  q B  where each projection is a vector bundle projection, and “vertical” and “horizontal” orientations of the  square are each vector bundle homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It was observed by Grabowski and Rotkiewicz (2009)  that this is equivalent to a pair of commuting �+-actions on the total space E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' following the development  in Chapter 2, this is a commuting pair of non-singular lifts on E ,  λA,λB : E → T E , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λB ◦ λA = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λA ◦ λB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A proper exposition of the so-called “Ehresmann doubles” (Mackenzie (2011)) for the structures in Lie  theory would substantially expand the scope of this thesis, and so it has been relegated to the margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  62  Finally, observe that a connection (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6) on an anchored bundle’s underlying differential  bundle behaves similarly to an affine connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle with � (A) existing in a tangent category  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If (π,ξ,λ) has a connection, then define  ˆκ := κπ1,  ˆ∇ := ∇(π0,̺π1)  and note that  (i) ˆκ : � (A) → A is a retract of (ξπ,λ), and ˆκ : (� (A),l ) → (A,λ) is linear for both l = (λ×ℓ),(0×c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) ˆ∇ : A2 → � (A) is a section of (π0,p ◦ π1) and is bilinear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) there is an isomorphism � (A) ∼= A3:  ∇(p ◦ π1,̺ ◦ π0) +(π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π) ˆµ(p ◦ π1, ˆκ) = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every vector bundle in the category of smooth manifolds has a connection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' it follows  that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16 holds for every anchored bundle in the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of anchored bundles in a tangent category is almost a tangent category,  except that it lacks a symmetry map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The “differential objects” in such a category will act like a cartesian  differential category, except that the symmetry of mixed partial derivatives fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There has been some  interest in settings for differentiable programming where the symmetry of mixed partial derivatives need  not hold (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 along with the discussion at the end of Section 6 in Cruttwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2021));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  the category of anchored bundles in a tangent category appears to be a source of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  Involution algebroids  Involution algebroids are a tangent-categorical axiomatization of Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that a Lie al-  gebroid has almost all of the structure of the operational tangent bundle, in particular a Lie bracket  on sections that satisfies the Leibniz law so that it gives a directional derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14,  it was demonstrated that the category of anchored bundles have almost all of the same maps as the  tangent bundle, the only structure map missing being the canonical flip  c : T 2M ⇒ T 2M  which, we may recall from the chapter introduction (and Cockett and Cruttwell (2015), Mackenzie  (2013)), is used in constructing the Lie bracket of vector fields in a tangent category with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus, a natural next step is to add an involution map to an anchored bundle and require that it satis-  fies the same coherences as c from the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An involution algebroid is an anchored bundle (A,π,ξ,λ,̺) equipped with a map  σ : � (A) → � (A) satisfying the following axioms:  (i) Involution:  � (A)  � (A)  � (A)  σ  σ  63  (ii) Double linearity:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  � (A)  � (A)  σ  0×c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  λ×ℓ  (iii) Symmetry of lift:  A  � (A)  � (A)  ˆλ=(ξ◦π,λ)  σ  ˆλ  (iv) Target:  � (A)  T A  T 2M  � (A)  T A  T 2M  σ  π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  c  (v) Yang–Baxter:  � 2(A)  � 2(A)  � 2(A)  � 2(A)  � 2(A)  � 2(A)  σ×c  σ×c  σ×c  A×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  A×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  A×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  A is an almost-involution algebroid if the involution does not satisfy the Yang–Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A mor-  phism of involution algebroids is a morphism of anchored bundles, so that � (f )◦σA = σB ◦� (f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (Note  that because σ is an isomorphism and σ = σ−1,  σ : (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (� (A),λ × ℓ)  is linear as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=') Write the category of involution algebroids and involution algebroid morphisms in �  is written Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is not immediately clear that the Yang–Baxter equation is well-typed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This follows  from the target axiom (iv) and the double linearity axiom (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Starting with (u,v,w ) : A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  T 2A, we see that σ × c is well-typed if and only if  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ π1 ◦ σ(u,v) = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ c ◦ w = c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ w = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Similarly, 1 × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ is well-typed if T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ u = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π ◦ π0 ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ(v,w );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' then use the double linearity axiom to  compute  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ π0 ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ(v,w ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1(v,w ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ w  = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ w = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ v = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  64  This perspective on Lie algebroids has already appearad in the work of Martinez and his collabora-  tors in de León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2005), where a “canonical involution” was derived on space of prolongations of a  Lie algebroid using the formula  σ : � (A) → � (A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ(x, y,z) = σ(y, x,z + 〈x, y 〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The structure of this map has been largely unexplored;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' helpfully, involution algebroids succeed in  reverse-engineering axioms for an involution map that will induce a Lie bracket on the sections of  the projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The bracket from the original Lie algebroid is induced using the same formula as  for the bracket of vector fields on the tangent bundle:  λ ◦ [X ,Y ]∗ =  �  (π1 ◦ σ ◦ (id ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺) ◦ Y −p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ X ◦ ̺) −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π 0Y  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore, a morphism of anchored bundles is a Lie algebroid morphism if and only if it preserves  the derived involution map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For any M in �, (T M ,p,0,id ,c ) is an involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Furthermore, for any involution alge-  broid anchored on M , (̺,id ) is a morphism of involution algebroids (by the target axiom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This  defines a fully faithful functor � �→ Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The same construction as the anchored bundle case in  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4(i) exhibits � as a reflective subcategory of Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For any differential bundle, the trivial anchored bundle (π : A → M ,ξ,λ,0◦ π) has an involution  using the isomorphism � (A) ∼= A3, given by σ := (π1,π0,π2) (proving this map satisfies the involu-  tion algebroid axioms is just an exercise in combinatorics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It follows that every differential bundle  morphism gives rise to a morphism of these trivial involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The same construction  from the anchored bundle case (Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4(ii)) exhibits Diff(�) as a coreflective subcategory of  involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Consider a groupoid  s,t : G → M , e : M → G , (−)−1 : G → G , m : G2 → G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The underlying reflexive graph has an associated anchored bundle, constructed in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4,  and the space of prolongations of this anchored bundle includes into (T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that there is a  well-formed involution map:  σ(u,v) = c ◦ ((0◦ p ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v)  The direct proof of this involves a more conceptual construction, which is the focus of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An involution algebroid resembles a generalized tangent bundle, and so the lift (ξ◦π,λ) : A → � (A)  satisfies the same universality conditions as ℓ : T ⇒ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The double linearity condition is equivalent to  the naturality condition for c ,ℓ in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2:  � (A)  � (A)  T 2  T 2  � 2(A)  � 2(A)  � 2(A)  T 3  T 3  T 3  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  c  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  id×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ  σ  σ×c  id×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  ˆλ×ℓ  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4)  65  which, using string diagrams for monoidal categories (Selinger (2010)), is the equation  =  where the circle denotes the lift and c is the crossing of two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle, and suppose that  σ : � (A) → � (A)  satisfies the involution axiom (i) and the symmetry of lift axiom (iii),and furthermore that the involution  “exchanges” the idempotents associated to the two lifts (λ×ℓ) and (0×c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) on � (A) (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8)  σ ◦ ((p ◦ λ) × (p ◦ ℓ)) = (p ◦ 0) × (p ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) = id × (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  Then the double linearity axiom is equivalent to the left-hand commuting diagram in Diagram 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4:  (ˆλ × ℓ) ◦ σ = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ × c ) ◦ (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Starting with the left-hand side of the equation,  (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ × c ) ◦ (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ) ◦ (u,v)  = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ × c ) ◦ (u,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ v,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λv)  = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ ◦ (u,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ ◦ ̺ ◦ u),T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λv)  = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (ξ ◦ π◦ u,0◦ ̺ ◦ u,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λv)  = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (ξ ◦ π◦ u,0◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ v,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λv)  = (ξ ◦ π◦ u,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ ˆλ◦ v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For the right-hand side:  (ˆλ,ℓ) ◦ σ ◦ (u,v)  = (ξ ◦ π◦ π0 ◦ σ ◦ (u,v),(λ × ℓ) ◦ σ ◦ (u,v))  = (ξ ◦ π◦ p ◦ v,(λ × ℓ) ◦ σ ◦ (u,v))  = (ξ ◦ p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ v,(λ × ℓ) ◦ σ ◦ (u,v))  = (ξ ◦ p ◦ ̺ ◦ u,(λ × ℓ) ◦ σ ◦ (u,v))  = (ξ ◦ π◦ u,(λ × ℓ) ◦ σ ◦ (u,v))  = (ξ ◦ π◦ u,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) ◦ (u,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  So the naturality equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 is equivalent to  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) = (λ × ℓ) ◦ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of involution algebroids is “tangent monadic” over the category of anchored bundles,  in the same sense that anchored bundles are tangent monadic over the category of differential bundles,  or internal categories over reflexive graphs in a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The “tangent monadicity” leads to a similar  observation about T -limits of involution algebroids as in Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  66  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The forgetful functor from involution algebroids to anchored bundles creates limits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  that is to say, the T -limits of the underlying anchored bundles give the limits of involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that by Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13, the limit � (limAi ) = lim� (Ai ) in the category of anchored bundles,  and this induces a map between objects in �  limσi : lim� (Ai ) → lim� (Ai ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The axioms for an involution algebroid are induced by universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that a point-wise tangent structure may be defined following the anchored bundles example  from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For a tangent category �, the category of involution algebroids has a “point-wise”  tangent structure that maps objects as follows:  (π : A → M ,ξ,λ,̺,σ) �→ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺,σT )  where σT is defined as  σT := (1×c c ) ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (1×c c ) : � (AT ) → � (AT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent structure on involution algebroids is inherited from the functor Inv(�) → Anc(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus, it suffices to give the involution map for the tangent involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that given (π : A → M ,ξ,λ,̺,σ), the space of prolongations on (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,  c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ,c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺) is T Ac ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2πT 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We can construct an isomorphism between the objects � (T A) and  T (� (A)) in � using the cospan isomorphism  T A  T 2M  T 2A  T A  T 2M  T 2A  c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  id  c  c  thus inducing a map  T Ac ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2πT 2A  id×c  −−→ T (A̺×T πT A)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  −→ T (A̺×T πT A)  id×c  −−→ T Ac ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2πT 2A  which we call σT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The linearity and involution axioms follow by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now check the rest of the axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For the unit:  (1×c c ) ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (1×c c ) ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(ξ ◦ π),c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  = (1×c c ) ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (T (ξπ),T λ) = (1×c c ) ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(ξ ◦ π),T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) = (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(ξ ◦ π),c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For the anchor:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺T ◦ π1 ◦ σT = T (c ◦ T ̺) ◦ π1 ◦ (1×c c ) ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (1×c c )  = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1 ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (1×c c ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1 ◦ (1×c c )  = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c ◦ T 2̺ ◦ c ◦ π1 = c ◦ T c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ π1 = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺T ◦ π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The Yang–Baxter equation is straightforward to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent bundle is a canonical involution algebroid on every object in a tangent category, and  the anchor induces a morphism from an involution algebroid to the tangent involution algebroid on  its base space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The anchor acts as a reflector from involution algebroids in � to � itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  67  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Any tangent category � is a reflective subcategory of the category of involution alge-  broids in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, observe that the inclusion of � into Inv(�) (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1) is fully faithful because the  anchor on the tangent involution algebroid is id : T M → T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This means that the only involution  algebroid morphisms T A → T B are pairs (T f , f ), f : A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now consider the functor Inv(�) → � that  sends (π : A → M ,ξ,λ,̺,σ) to M : this gives an endofunctor S : Inv(�) → Inv(�) along with a natural  transformation ̺ : id ⇒ S, so that S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ = ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S = id , given by the anchor map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, the category � is  the category of algebras for an idempotent monad on Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let  �A = (π : A → M ,ξ,λ,̺,σ)  be an involution algebroid in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The morphism  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π,π) : T A → T M  is an involution algebroid morphism from the tangent involution algebroid on A to the tangent  involution algebroid on M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The morphism  (̺,id ) : �A → T M  is an involution algebroid morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  If pullback powers of p ◦ π1 : � (A) → A exist, then  (p ◦ π1 : � (A) → A,(ξ ◦ π,0),(λ × ℓ))  is a differential bundle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' note that π0 acts as an anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If the prolongations exist, then  (p ◦ π1 : � (A) → A,(ξ ◦ π,0),(λ × ℓ),π0,σ′)  is an involution algebroid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this follows from computing the pullback in the category of involution alge-  broids (σ′ is induced as in Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The above corollary puts an involution algebroid structure on the differential bundle (� (A),λ × ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that the map σ gives an isomorphism of differential bundles  (� (A),λ × ℓ) → (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Martinez observed that the canonical involution σ puts a unique Lie algebroid structure on (� (A),0×  c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ), and σ isa uniquelydetermined isomorphism ofinvolution algebroids(see de León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2005)):  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For an involution algebroid (π : A → M ,ξ,λ,̺,σ), the isomorphism of differential  bundles  σ : (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (� (A),λ × ℓ)  induces a second involution algebroid structure on � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  68  Recall that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14 sketched out a proof that the category of anchored bundles in � has  an endofunctor � ′ and natural transformations p ′,0′,+′,ℓ′ satisfying the axioms of a tangent structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  this endofunctor and the natural transformations all lift to Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The involution map σ is the missing  piece that gives a tangent structure on Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The construction may be spelled out here at a big-picture  level, but the actual proof brings up tricky coherence issues that make up the bulk of Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of involution algebroids in a tangent category � has a second tangent  structure, where the tangent functor is given by  � ′ : Inv(�) → Inv(�)  and the tangent natural transformations are given as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14, with the canonical flip  σ′ : � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′ → � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′ := � 2(A)  1×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  −−−→ � 2(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Starting with an involution algebroid, � ′(A) and � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′(A) are given by  � ′(A)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  π′ :  � (A)  p◦π1  −−→ A  ξ′ :  A  (ξ◦π,0)  −−−→ � (A)  λ′ :  � (A)  λ×ℓ  −−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  ̺′ :  � (A)  π1  −→ T A  σ′ :  � 2(A)  σ×c  −−→ � 2(A)  � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′(A)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  π′′  � 2(A)  (p◦π1,p◦π2):  −−−−−−−→ � (A)  ξ′′ :  � (A)  (ξ◦π◦π0,0◦π1,0◦π2)  −−−−−−−−−−−→ � 2(A)  λ′′ :  � 2(A)  (λ×ℓ×ℓ)  −−−−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� 2(A)  ̺′′ :  � 2(A)  (π1,π2)  −−−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  σ′′ :  � 3(A)  (σ×c ×c )  −−−−−→ � 3(A)  (where � 3(A) = A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (A̺×T πT A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent natural transformations are given by  p : � ′ ⇒ id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � (A)  π0  −→ A  0 : id ⇒ � ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A  (id,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π)  −−−−−−−→ � ′(A)  + : � ′  2 ⇒ � ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A̺×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦πi T2A  id×+  −−−→ � (A)  ℓ : � ′(A) ⇒ � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � (A)  1×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (ξ◦π,λ)  −−−−−−→ � 2(A)  ̺′ : � ′ ⇒ T  c : � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′(A) ⇒ � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � 2(A)  1×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  −−−→ � 2(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Deferred to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4  Connections on an involution algebroid  In this section we take an involution algebroid with a chosen linear connection on its underlying an-  chored bundle (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1), and rederive Martinez’s structure equations for a Lie algebroid  (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In a tangent category � with negatives, there is a natural transformation  n : T ⇒ T  69  making each fibred commutative monoid (p : T M ⇒ M ,0,+,n) a fibred abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Because the  additive bundle structure on differential bundles is induced via universality (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4), in a  tangent category with negatives the additive bundle structure for differential bundles will likewise have  negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We adopt the following notation for the “fibred linear algebra” used in this section, as there  are a significant number of computations done on bundles with multiple choices of additive bundle  structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the second tangent bundle of a differential bundle has three additive bundles structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let E be an object with multiple differential bundle structures (πi : E → M i ,ξi,λi)  in a tangent category with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Addition over a specific differential bundle is written using infix  notation, where a subscript is added to the symbol denoting the projection of the differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Letting x, y : X → E denote a pair of generalized elements for which the πi -addition operation is well-  defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' we set  x +π[i] y = X  (x,y )  −−→ E π[i]×π[i]E  +i  −→ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Similar notation is used for subtraction:  x −π[i] y = X  (x,y )  −−→ E π[i]×π[i]E  id×n[i]  −−−−→ E π[i]×π[i]E  +i  −→ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Throughout this section, we work in a tangent category � with negatives and a chosen anchored  bundle (π : A → M ,ξ,λ,̺), equipped with a connection (κ,∇), whose base object has a torsion-free  connection (κ′,∇′) and a morphism  σ : � (A) → � (A)  that exchanges the two projection maps p ◦ π1,π0, meaning that the following diagram commutes:  � (A)  A  A  � (A)  π0  p◦π1  σ  p◦π1  π0  The following notation will be useful when working with local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, recall the ∇-notation for morphisms of differential bundle where each morphism  has a choice of connection from Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3:  f : A → B  (κA,∇A,A,λA)  (κB,∇B,B,λB)  ∇[f ] := κB ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇A : A2 → B  Let  (π : A → M ,ξ,λ,̺),(q : B → N ,ζ,l ,ρ)  be a pair of anchored bundles equipped with connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' “Hatting” a map f : � (A) → � (B) refers to  the map:  �f : A3  ˆν(π0,π2)+ ˆ∇(π0,π1)  −−−−−−−−−−→ � (A)  f−→ � (B)  (π0,p◦π1,κ◦π1)  −−−−−−−−→ B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Similarly, for f : T A → T B,  �f : T M p ×πAπ×πA  ν(π0,π2)+∇(π0,π1)  −−−−−−−−−−→ T A  f−→ T B  (π0,p◦π1,κ◦π1)  −−−−−−−−→ T M p ×q B π×q B  70  Clearly, �  f ◦ g = �f ◦ �g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Similarly, a map A3 → B3 may be “barred” to form a map � (A) → � (B):  g : � (A)  (π0,p◦π1,κ◦π1)  −−−−−−−−→ A3  g−→ B3  ˆν(π0,π2)+ ˆ∇(π0,π1)  −−−−−−−−−−→ � (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is straightforward to see that �f = f , �g = g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' σ : � (A) → � (A) induces a bracket on Γ(π):  λ ◦ [X ,Y ] = (σ ◦ (id ,T X ◦ ̺) ◦ Y −T π (id ,T Y ◦ ̺) ◦ X ) − 0◦ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let X ,Y ∈ Γ(π) and compute:  p ◦ π1 ◦ (σ ◦ (id ,T X ◦ ̺) ◦ Y −π0 (id ,T Y ◦ ̺) ◦ X )  = p ◦ (π1 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −T π T Y ◦ ̺ ◦ X )  = p ◦ π1 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −π p ◦ T Y ◦ ̺ ◦ X  = π0 ◦ (id ,T X ◦ ̺) ◦ Y − Y ◦ p ◦ ̺ ◦ X  = Y − Y = ξ,  π0 ◦ (σ ◦ (id ,T X ◦ ̺) ◦ Y −π0 (id ,T Y ◦ ̺) ◦ X )  = π0 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −π π0 ◦ (id ,T Y ◦ ̺) ◦ X  = p ◦ π1 ◦ (id ,T X ◦ ̺) ◦ Y −π X  = p ◦ T X ◦ ̺ ◦ Y −π X = X −π X = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The universality of the lift induces a new section [X ,Y ] so that  λ ◦ [X ,Y ] = (σ ◦ (id ,T X ◦ ̺) ◦ Y −T π (id ,T Y ◦ ̺) ◦ X ) − 0Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The map σ : � (A) → � (A) is linear whenever the two bundle morphisms  σ : (� (A),λ × ℓ) → (� (A),0× c ◦ T λ) σ : (� (A),0× c ◦ T λ) → (� (A),λ × ℓ)  are linear, and cosymmetric if  σ ◦ �λ = �λ = (ξπ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that whenever σ is linear and cosymmetric, σ ◦ ˆµ(u,v) = ˆν(u,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Linearity and unit axioms, along with the connection on the differential bundle, force the existence  of a bilinear bracket 〈−,−〉 as in the definition of a Lie algebroid in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For an anchored bundle (π : A → M ,ξ,λ,̺) with connection (κ,∇), a cosymmetric  and bilinear σ is equivalent to a bilinear 〈−,−〉 : A2 → A, with the correspondence given by  〈−,−〉 : κ ◦ σ ◦ ∇−π κ ◦ ∇  σ : � (A) −−−−−−−−−−→  (π1,π0,π2+〈π0,π1〉)  � (A)  71  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Derive  �σ ◦ (x, y,z) = (y, x,κ ◦ σ ◦ (∇(̺x, y ) +p◦π1 µ(y,z)))  = (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π κ ◦ σ ◦ µ(y,z))  = (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π κ ◦ ν ◦ (y,z))  = (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π z)  =: (y, x,〈x, y 〉 +π z)  where 〈−,−〉 is certainly bilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The converse is immediate: take  �σ(u,v,w ) := (v,u,w + 〈u,v〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is easy to see that σ is linear and cosymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The linear bracket must be involutive for the bilinear bracket to be alternating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If σ is cosymmetric and linear, then the bilinear bracket 〈−,−〉 is alternating if and  only if σ ◦ σ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, note that σ ◦ σ = id if and only if �σ ◦ �σ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then check that  �σ ◦ �σ(u,v,w ) = �σ(v,u,w + 〈u,v〉) = (u,v,w + 〈u,v〉 + 〈v,u〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By the cancellativity of addition on A,  w = w + 〈u,v〉 + 〈v,u〉 ⇐⇒ 0 = 〈u,v〉 + 〈v,u〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A bilinear 〈−,−〉 on an anchored bundle with a connection induces the maps  {v, x}(κ,∇) := κ′ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ ∇(v, x),  {v, x, y }(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ (v, x, y )  from Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let σ be cosymmetric and bilinear, with the associated bracket 〈−,−〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then  T ̺ ◦ π1 ◦ σ = c ◦ T ̺ ◦ π1  if and only if the Leibniz equation is satisfied:  ̺ ◦ 〈u,v〉 + {̺v,u} = {̺u,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='}  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Following the given notation and using the hypothesis that the connection is torsion-free on  M ,  �  T ̺ ◦ π1(u,v,w ) = (̺u,̺v,w + {u,v})  �c (x, y,z) = (y, x,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Computing each side,  �  T ̺ ◦ �  π1 ◦ �σ ◦ (u,v,w ) = �  T ̺ ◦ �  π1 ◦ (v,u,w + 〈u,v〉)  = �  T ̺ ◦ (̺v,u,w + 〈u,v〉)  = (̺v,̺u,̺w + ̺〈u,v〉 + D[̺](̺v,u))  = (̺v,̺u,̺w + ̺〈u,v〉 + {v,u}),  �c ◦ �  T ̺ ◦ �  π1(u,v,w ) = �c (̺u,̺v,̺w + {u,v})  = (̺v,̺u,̺w + {u,v}),  so it follows that the two terms are equal if and only if the desired equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  72  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let σ be linear and cosymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then  (i) T (〈−,−〉) : T A2 → T A satisfies  �  T (〈−,−〉)(ax,uy ,uz,ux y ,ux z)  = (ax ,〈uy ,uz〉,{ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ satisfies  �  (id × T (�σ))(ux ,uy ,ux y ,uz,ux z,uy z,ux y z)  = (ux ,uz,ux z,uy ,ux y ,uy z + 〈uy ,uz〉,  ux y z + {ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) By the universality of the vertical lift and bilinearity of 〈−,−〉, the outer squares below are pull-  backs:  A4  A2  T (A)  T (A)  T M  T M  M  M  µA2  (〈π0,π1〉,〈π0,π3〉+〈π1,π2〉)  µA  T 〈,〉  T π◦T πi  T π  0  0  so that  T (〈,〉)(0,uy ,uz,ux y ,ux z) = µA(〈uy ,uz〉,〈uy ,ux z〉 + 〈ux y ,uz〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now we compute  T (〈,〉)(µA2((uy , yz),(ux y ,ux z)) +p ∇A2(ax,(uy ,uz)))  = T (〈,〉)µA2((uy ,uz),(ux y ,ux z)) +p T (〈,〉) ◦ ∇(ax,(uy ,uz))  = µA(〈uy ,uz〉,〈uy ,ux z〉 + 〈ux y ,uz〉) +p T (〈,〉) ◦ ∇(ax,(uy ,uz))  and then postcompose this with (T π,p,κ) to obtain  (0,〈uy ,uz〉, 〈uy ,ux z〉 + 〈ux y ,uz〉) +p (ax,〈uy ,uz〉,{ax,uy , yz})  = (ax,〈uy ,uz〉,{ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Consider the following diagram:  T M p ×πA3π×πA3  T M p ×πA3π×πA3  T (A3)  T (A3)  T ◦ � (A)  T ◦ � (A)  �  T �σ  ∇A3+p µA3  T �σ  T (π1,π0,π2+〈π0,π1〉)  T (∇′+π0µ′))  (T (π3),p,κA3)  T σ  T (π0,p◦π1,κ◦π1)  73  We want to find �  T �σ =  �  T (π1,π0,π2 + 〈π0,π1〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that  T (π1,π0,π2 +π 〈π0,π1〉) = T (π1,π0,π2) +T π T (ξππi ,π0,〈π0,π1〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The left term is straightforward:  �  T (π0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π2)(ax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z))  = (ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z))  and for the right term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' use part (i) of this lemma:  �  T 〈−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='−〉◦ �  T (π0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1))(ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z))  = �  T 〈−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='−〉(ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z)  = (ax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='〈uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz〉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='{ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' yz} + 〈uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z〉 + 〈ux y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz〉)  Then compute  �  T �σ(ax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z)  = (ax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy z + 〈uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz〉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z + q)  where q = {ax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' yz} + 〈uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z〉 + 〈ux y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz〉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' giving the desired equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let σ be cosymmetric, doubly linear, and involutive, and satisfy the target axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Then σ satisfies the Yang–Baxter equation if and only if 〈−,−〉 and {−,−,−} satisfy the Bianchi identity:  0 =  �  γ∈Cy(3)  〈xγ0,〈xγ1, xγ2〉〉 +  �  γ∈Cy(3)  {̺xγ0, xγ1, xγ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='}  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We expand �  id × T σ and �  σ × c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Start with T (id × σ), which was derived in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9:  σ1(u) =  �  (id × T (�σ))(ux,uy ,ux y ,uz,ux z,uy z,ux y z)  = (ux ,uz,ux z,uy ,ux y ,uy z + 〈uy ,uz〉,  ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using the fact that κ′ is torsion free, so ˆc (ux z,uy z,ux z y ) = (uy z,ux z,ux y z), it follows that  σ2(ux,uy ,ux y ,uz,ux z,uy z,ux y z) = (uy ,ux,ux y + 〈ux,uy 〉,uz,uy z,ux z,ux y z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Then compute  σ2σ1σ2(u)  =  �  uz,uy ,uy z + 〈uy ,uz〉,ux,ux z + 〈ux,uz〉,ux y + 〈ux ,uy 〉,z1  �  ,  σ1σ2σ1(u)  =  �  uz,uy ,uy z + 〈uy ,uz〉,ux,ux z + 〈ux,uz〉,ux y + 〈ux ,uy 〉,z2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  74  Note that the first five terms are equal, so it suffices to check z1 = z2 for z1 = π6σ1σ2σ1(u),z2 =  π6σ2σ1σ2(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  z1 = ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }  + 〈ux,uy z + 〈uy ,uz〉〉 + 〈ux z + 〈ux ,uz〉,uy 〉  = ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }  + 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈ux z,uy 〉 + 〈〈ux ,uz〉,uy 〉  = ux y z + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }  + 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉,  z2 = ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y + 〈ux ,uy 〉,uz〉  = ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y ,uz〉 + 〈〈ux,uy 〉,uz〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  So z1 = z2 is equivalent to requiring  ux y z + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }  + 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉  = ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y ,uz〉 + 〈〈ux,uy 〉,uz〉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  cancelling alike terms, this is equivalent to  〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }  = {̺ ◦ uy ,ux,uz} + 〈〈ux,uy 〉,uz〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using the fact that 〈−,−〉 and {−,−,−} are alternating in the last two arguments, this is equivalent to  0 = 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉 + 〈uz,〈ux,uy 〉〉  + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } + {̺ ◦ uy ,uz,ux}  = 〈ux,〈uy ,uz〉〉 + 〈uy ,〈uz,ux〉〉 + 〈uz,〈ux,uy 〉〉  + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } + {̺ ◦ uy ,uz,ux}  giving the desired identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category with negatives,  with anchored connection (∇,κ) on A and torsion-free affine connection (∇′,κ′) on M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An involution  algebroid structure on A is equivalent to a bilinear map  〈−,−〉 : A2 → A  with derived maps  {−,−} : Aπ×p T M → T M := κ′ ◦ T ̺ ◦ (π0,π1),  {−,−,−} : T M p ×π◦πi A2 → A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='{a,u1,u2} := κ ◦ T (〈−,−〉) ◦ (∇(a,u1),∇(a,u2))  satisfying  (i) 〈−,−〉 is linear and cosymmetric,  (ii) 〈−,−〉 is alternating,  75  (iii) 〈−,−〉 and {−,−} satisfy the Leibniz equation, Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) 〈−,−〉,{−,−}, and {−,−,−} satisfy the Bianchi identity, Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Morphisms of involution algebroids may also be characterized by preservation of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A,B be a pair of involution algebroids with chosen connections in a tangent  category with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then an anchored bundle morphism f : A → B is an involution algebroid  morphism if and only if (recalling the notation from Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3)  ∇[f ](x, y ) + 〈f ◦ x, f ◦ y 〉 = ∇[f ](y, x) + f ◦ 〈x, y 〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that  (σ ◦ � (f ) = � (f ) ◦ σ)  ⇐⇒ (π1,π0,π2 + 〈π0,π1〉) ◦ (f ◦ x, f ◦ y, f ◦ z + ∇[f ](x, y ))  = (f , f , f ◦ π2 + ∇[f ](π0,π1)) ◦ (y, x,z + 〈x, y 〉)  while the second condition reduces to  ∇[f ](x, y ) + 〈f ◦ x, f ◦ y 〉 = ∇[f ](y, x) + f ◦ 〈x, y 〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  The isomorphism of Lie and involution algebroid categories  Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 have made the relationship between involution algebroids and Lie algebroids clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is important to note that while the proofs used connections as a tool to identify the local coherences  satisfied by involution and Lie algebroids, the construction of a Lie algebroid from an involution alge-  broid (or vice versa) is independent of the choice of connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an isomorphism of categories between Lie algebroids and involution algebroids  in smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For the equivalence of categories, note that by Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11,  and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12 there is an isomorphism of categories between involution algebroids with a  choice of connection and Lie algebroids with a choice of connection (morphisms are not restricted to  connection preserving morphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This allows us to chain together isomorphisms  Inv(SMan) ∼= Inv(SMan)ChosenConn ∼= LieAlgdChosenConn ∼= LieAlgd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To complete the proof, we must show that the assignment that sends an involution algebroid to a  Lie algebroid whose bracket is given by  λ ◦ [X ,Y ]∗ =  �  (π1 ◦ σ ◦ (id ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺) ◦ Y −p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ ̺◦) −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π 0◦ Y  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8)  is a bijection on objects, which brings up some subtleties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, while an involution map  σ : A̺×T πT A → A̺×T πT A  76  is defined with respect to a particular choice of pullback A̺ ×T πT A, the category of involution alge-  broids does not distinguish between different choices of this pullback (and therefore different repre-  sentations of the map σ), and it is not part of the data of an involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is immediate by  universality that the left-hand-side of Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 is independent of the choice of pullback A̺×T πT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now, recall thatthe canonical involution ofa Lie algebroid isuniquelybyTheorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7ofde León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (2005) (this was also mentioned in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Once we make a choice of prolongation A̺×T πT A,  we have made a choice of pullback � (A) in LieAlgd, which uniquely determines the canonical involu-  tion  σ : A̺×T πT A → A̺×T πT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  While the exact map σ depends on the choice of pullback A̺ ×T π T A, they all determine the same  involution algebroid, thus proving the bijection correspondence of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  77  Chapter 4  The Weil nerve of an algebroid  The first three chapters of this thesis demonstrated that tangent categories allow for an essentially alge-  braic description of Lie algebroids by axiomatizing the behaviour of the tangent bundle, and showing  that a Lie algebroid over a manifold M is a “generalized tangent bundle”, namely an involution al-  gebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This chapter will make precise the sense in which an involution algebroid is a generalized  tangent bundle, by showing that the category of involution algebroids in a tangent category � is equiv-  alent to a certain tangent-functor category from the free tangent category over a single object to �, or  more generally that there is a fully faithful functor  Inv(�) �→ TangLax(FreeTangCat(∗),�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This functor, the Weil nerve of an involution algebroid, builds a functor from the free tangent category  over a single object to � using a span construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This chapter primarily builds on two pieces of work:  Leung’s construction of the free tangent category Weil1 (Leung (2017)) , and Grothendieck’s original  nerve construction (first published in Segal (1974)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To understand Leung’s construction of the free tangent category, and more generally his actegory-  theoretic presentation of tangent categories (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2), we first look at Weil’s original insight relating  the kinematic and operational descriptions of the tangent bundle in SMan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The definition of a tangent  vector on a manifold M as an equivalence class of curves (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8) puts a bijective correspon-  dence between tangent vectors and �-algebra homomorphisms from the ring of smooth functions  C ∞(M ) to the ring of dual numbers:  C ∞(M ) → �[x]/x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The hom-set �Alg(C ∞(M ),�[x]/x 2) is precisely the set of derivations on C ∞(M ), which defines the  operational tangent bundle discussed in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16: there is a natural smooth manifold structure  on this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Weil functor formalism, most notably developed in Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993), extends this ob-  servation to a general class of endofunctors on SMan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For example, the fibre product T2M corresponds  to �-algebra morphisms,  C ∞(M ) → R[x, y ]/(x 2, y 2, x y ),  while the second tangent bundle corresponds to �-algebra morphisms into the tensor product,  C ∞(M ) → R[x]/(x 2) ⊗ R[y ]/(y 2) = R[x, y ]/(x 2, y 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By applying Milnor’s exercise (Problem 1-C Milnor and Stasheff (1974)), which states that the C ∞ func-  tor  SMan → �Algop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' M �→ C ∞(M ) = SMan(M ,�)  78  is fully faithful, the structure maps occur as natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For example, the tangent projec-  tion is induced by the morphism  p : �[x]/x 2 a+b x�→a  −−−−−→ �,  so that  T M  p−→ M = [C ∞(M ),�[x]/x 2]  (p)∗  −→ [C ∞(M ),�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The zero map and addition are similarly induced by  0 : �  a�→a+0x  −−−−−→ R[x]/x 2 and + : R[x, y ]/(x 2, y 2, x y )  a+b x+c y  −−−−−−−→  �→a+(b+c )x R[x]/x 2,  respectively, while the lift and flip are induced by the morphisms  ℓ : �[x]/x 2  a+b x  −−−−−→  �→a+b x y �[x, y ]/(x 2, y 2)  and  c : �[x, y ]/(x 2, y 2)  a+b x+c y +d x y  −−−−−−−−−−−→  �→a+c x+b y +d x y �[x, y ]/(x 2, y 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  More generally, there is a monoidal category of Weil algebras (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1) which has a monoidal  action on the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Weil functor formalism, then, studies differential ge-  ometric structures from the perspective of the endofunctors and natural transformations induced by  this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Leung’s insight is that there is an analogous category of commutative rigs1 built by replac-  ing �[x]/x 2 with �[x]/x 2, called Weil1 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' a tangent structure is precisely a monoidal  action by Weil1 satisfying some universal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, this category Weil1 is precisely the  free tangent category over a point, FreeTang(∗), so that every object A in a tangent category � deter-  mines a strict tangent functor  T (−)A : Weil1 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V �→ T V A  and morphisms f : A → B are in bijective correspondence with tangent-natural transformationsT (−)A ⇒  T (−)(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The axioms of an involution algebroid in a tangent category � correspond bijectively with those  of the tangent bundle - this suggests that an involution algebroid should determine a tangent functor  from Weil1 to �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A first guess would lead one to think that p : �[x]/x 2 → � is sent to π : A → M , 0 to  ξ, and + to +A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' As the space of prolongations � (A) = A̺ ×T πT A plays the role of the second tangent  bundle, we can see that  ℓ : N [x]/x 2 → N [x, y ]/(x 2, y 2) �→ (ξ ◦ π,λ) : A → � (A),  and  c : N [x, y ]/(x 2, y 2) → N [x, y ]/(x 2, y 2) �→ σ� (A) → � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This pattern may be neatly summed up using span composition - we will construct a functor that sends  �[x]/x 2 to the span  A  M  T M  ̺  π  1A rig is a ring without negatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' a commutative monoid equipped with a bilinear multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  79  and the tensor product �[x]/x 2 ⊗ �[x]/x 2 to the span composition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the pullback)  � (A)  A  T A  M  T M  T 2M  ̺  π  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  ⌟  which is the space of prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This leads to the first major result of this chapter, the Weil Nerve  (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9), which states there is a fully faithful embedding  NWeil : Inv(�) �→ [Weil1,�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This bears a strong similarity to Grothendieck’s original nerve theorem, which takes an internal cate-  gory s,t : C → M and constructs a functor ∆op → � (where ∆op is the monoidal theory of an internal  monoid) by sending tensor to span composition, and the composition and unit maps given span mor-  phisms  C2  M  M  M  M  M  C  C  s◦π0  t ◦π1  s  t  m  s  t  e  while the unit and associativity axioms for a category are exactly the unit and associativity laws for  a monoid in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Segal conditions identify exactly the functors C : ∆op → � that lie in  the image of the nerve functor N as those whose [n]t h object is sent to the wide pullback C ([n]) =  C [2]t ×s C [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' t ×s C [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The corresponding result for involution algebroids is found in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8,  which states that a tangent functor (A,α) : Weil1 → � is the nerve of an involution algebroid if and  only if A preserves tangent limits and α is a T -cartesian natural transformation (this forces A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (W ⊗n) =  A̺×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T n−1̺×T nπT nA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The similarity between the Weil nerve and Grothendieck/Segal’s nerve  runs deep, and in Chapter 5 we demonstrate that the enriched perspective on tangent categories puts  these both into the same formal framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Sections4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2give a detailed introduction tothe Weil functorformalism (Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993),Bertram and Souvay  (2014)) and Leung’s unification of Weil functors with tangent categories Leung (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The rest of the  chapter contains contains new results developed by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 proves the embedding  part of the Weil nerve theorem, that the category of involution algebroids embeds into the category  of tangent functors and tangent natural transformations [Weil1,�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 identifies exactly those  tangent functors (A,α) : Weil1 → � that are the nerve of an involution algebroid, completing the proof  of the Weil nerve theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 uses the Weil nerve to develop a novel tangent structure on the  category of involution algebroids in a tangent category (in particular, the category of Lie algebroids will  have this novel tangent structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Weil algebras and tangent structure  This section gives a more thorough introduction to the Weil functor formalism of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993),  and in particular how the structure maps of a tangent category may be teased out of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We begin by  80  introducing Weil algebras, and the prolongation of a smooth manifold by a Weil algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (The relation-  ship with prolongations of involution algebroids from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 will be made clear in Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An �-Weil algebra2 is a finite-dimensional �-algebra V so that V = � ⊕ ˙V as �-  modules and ˙V is a nilpotent ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category �Weil is the full subcategory of �Alg spanned by the  �-Weil algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The prolongation of a manifold by a Weil algebra V is given by the manifold  T V M := �Alg(C ∞(M ,�),V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Eck (1986) showed that every product-preserving endofunctor on the category of smooth mani-  folds is constructed as the Weil prolongation by some Weil algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consequently, �-algebra homo-  morphisms induce natural transformations between these product-preserving endofunctors on the  category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider the following �-Weil algebras and their associated prolongation functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Prolongation by � induces the identity functor, and the tangent bundle is given by �[x]/x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  tangent projection, then, is equivalent to the �-algebra morphism  p : �[x]/x 2 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' p(a + b x) = a  while the 0-map induces the zero vector field:  0 : � → �[x]/x 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 0(a) = a + 0x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The algebra �[xi ]1≤i≤n/(xi x j )1≤i≤j ≤n = (�[x]/x 2)n is the wide pullback TnM = T M p×pT M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p×p  T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, prolongation by �[x, y ]/(x 2, y 2, x y ) gives the bundle T2M = T M p×pT M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  �-algebra morphism  + : �[x, y ]/(x 2, y 2, x y ) → R[x]/x 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' +(a0 + a1x + a2y ) = a0 + (a1 + a2)x  corresponds to the addition of tangent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The algebra �[x, y ]/(x 2, y 2) = (�[x]/x 2) ⊗ (�[x]/x 2) is the second tangent bundle T 2M = T T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The vertical lift T → T 2 is induced by the morphism  ℓ : �[x]/x 2 → �[x, y ]/(x 2, y 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ℓ(a + b x) = a + b x y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The monoidal symmetry map induces c : T 2 ⇒ T 2, as follows:  c : (�[x]/x 2) ⊗ (�[y ]/y 2) → (�[y ]/y 2) ⊗ (�[x]/x 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  c (a + b1x + b2y + b3x y ) = a + b2x + b1y + b3x y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) For n ≥ 2, the algebra �[x]/x n gives the n-jet bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that this is the equalizer of ⊗n�[x]/x 2  by the symmetry actions of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2Not to be confused with the normal usage of “Weil algebra” in Lie theory, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Meinrenken and Pike (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  81  Further examples may be found in the monograph Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tangent categories bridge  the gap between the Weil functor approach to studying the differential geometry of smooth manifolds  and the synthetic differential geometry approach of axiomatizing a tangent bundle using nilpotent in-  finitesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The main structure axiomatized here is that of monoidal action of a symmetric monoidal  category on a category M × � → �, or equivalently, a lift from a category to the category of complexes  � → [M ,�], which involves translating a bit of classical category theory to the 2-categorical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 leads to the classical theorem that the category of smooth manifolds has an action by  the category of �-Weil algebras that preserves all connected limits that exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These “natural” universal  properties (in the sense of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993)) is foundational to synthetic differential geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' see,  for example, Chapter Two of Lavendhomme (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Unfortunately, Weil algebras are not an ideal syn-  tactic presentation: they are not a finitely presentable category, and it is not immediately clear when  a diagram is a connected limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 Moving from �-algebras to commutative rigs and restricting to an  appropriate subcategory solves this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 Leung (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category Weil1 is defined to be the full subcategory  of commutative rigs, CRig, generated by the rig of dual numbers W := �[x]/x 2, constructed as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Start with finite product powers of W in CRig, and make a strict choice of presentation:  W0 = �, Wn := �[xi ]/(xi x j )i≤j ,0 ≤ i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then take the closure of Wn under coproduct of commutative rigs, written ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Again, make a strict  choice of presentation:  Wn(0) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='⊗ Wn(m−1) := �[xi,j ]/(xi j xik)j ≤k,0 < i < m,0 < j < n(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that we will often suppress the tensor product ⊗ and simply write  U V :=U ⊗ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 Leung (2017) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The category Weil1 is a symmetric strict monoidal category with unit � and coproduct ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) � is a terminal object in Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that there is a forgetful functor  Weil1 → (CMon/�) → CMon  that reflects connected limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This gives the following class of limits, identified in Leung (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We say the following pullback diagrams in Weil1 are transverse:  W  W  W 2  W  W 2  W W  W  W  W  �  �  W  id  id  id  id  ⌟  π0  π1  p  p  ⌟  p◦πi  µ  0  pW  where µ(a +a1x +a2y ) = a +a1x +a2x y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The ⊗-closure of these three pullbacks is the set of transverse  squares, and they are also pullback squares by Leung (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3It should be noted that Nishimura and Osoekawa (2007) made progress applying techniques from computer algebra to  latter problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  82  To see that each transverse square in the ⊗-closure is a pullback diagram, take the two non-identity  squares and rewrite them in CMon:  � × (� × �)  � × �  � × (� × �)  � × (� × � × �)  � × �  �  �  � × �  �×π0  �×π1  π0  π0  ⌟  π0  �×(π0,0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',π1)  (id,0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  �×π1  ⌟  The coproduct of Weil algebras is the tensor product of the underlying commutative monoids, which  are finite-dimensional and free, so these limits are closed under ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 (Leung (2017) Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category Weil1 is a tangent category, where the  tangent functor is  T := W ⊗ (_) : Weil1 → Weil1  and the natural transformations are given by  p : W ⊗ (_)  p⊗(_)  −−→ (_), 0 : (_)  0⊗(_)  −−→ W ⊗ (_), + : W2 ⊗ (_)  +⊗(_)  −−→ W ⊗ (_),  ℓ : W ⊗ (_)  ℓ⊗(_)  −−→ W ⊗ W ⊗ (_),  c : W ⊗ W ⊗ (_)  p⊗(_)  −−→ W ⊗ W ⊗ (_)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category Weil1 is, in some sense, a finitely presented theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is precisely the free tangent  category on a single object:  Proposition4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7(Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5, Leung(2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The categoryWeil1 isgenerated by the maps{p,0,+,ℓ,c }  from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, closed under composition, tensor, and maps induced by transverse limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category Weil1 is the free tangent category over a single object: every object C in a  tangent category � determines a strict tangent functor T−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C : Weil1 → �, mapping  V = W n1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='⊗ W n(k) �→ Tn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C = T V C  so that there isan isomorphism of categoriesbetween � and the category of strict tangent functorsWeil1 →  � with tangent natural transformations as morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Throughout this section, the notation T V C will refer to the action of the Weil algebra  V on an object C in a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, we will make use of the isomorphism T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V C =  T U V C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  Tangent structures as monoidal actions  The presentation of Weil1 as the free tangent category situates the formal theory of tangent categories  as an instance of more general categorical machinery, namely monoidal actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that in a sym-  metric monoidal category (�,⊗,I ), an internal monoid (C ,•,i) determines a monad:  (C ⊗ _ : � → �,µ : � ⊗ (� ⊗ _)  ⊗_  −→ � ⊗ _,η : _  ρ−→ I ⊗ _  e ⊗_  −−→ C ⊗ _).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  83  The category of algebras for this monad is exactly the category of C -modules, objects with an associa-  tive and unital action by C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A morphism will be a map on the base object that preserves the action:  C  M ⊗ C  M ⊗ M ⊗ C  M ⊗ C  C  M ⊗ C  C  M ⊗ C  M ⊗ D  C  D  M ⊗∝  ⊗M  ∝  ∝  (u,C )  ∝  f  ∝C  ∝D  M ⊗f  Strict actegories are the 2-categorical generalization of modules over a monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The coherences for a 2-  monad follow from the coherences from a strict monoidal category in the 2-category of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  following proposition relies on a few facts from enriched category theory (treating the cartesian closed  category Cat as a Cat-enriched category, per Kelly (2005)) but a more general treatment of non-strict  actegories may be found in Janelidze and Kelly (2001):  A 2-functor and 2-natural transformations are exactly a functor and natural transformations  that satisfy extra coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These coherences follow for free by constructing the monad and  comonad � × _,[� ,_].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An algebra of the underlying 1-monad is exactly an algebra of the 2-monad (the same result holds  for comonads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  When working with algebras of a 2-monad, four different notions of morphisms can come into play  (Lack and Power (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These arise through using the 2-categorical data to weaken the notion of a  morphism:  (i) Strict: this is exactly a morphism of the underlying algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the 2-category of strict � -  actegories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Strong: the morphisms preserve the action to an isomorphism:  � × �  � × �  �  �  � ×F  ∝C  ∝D  F  α  (iii) Lax: the 2-cell is no longer an isomorphism:  � × �  � × �  �  �  � ×F  ∝C  ∝D  F  α  (iv) Oplax: the 2-cell travels in the opposite direction (these will not figure into this account).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2-cells between actegory morphisms must satisfy a coherence between the natural transformation  parts of the actegory morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  84  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the case of strict, strong, and lax tangent functors, the same notion of a 2-cell applies:  a natural transformation γ : F ⇒ G satisfying the following coherences with the natural transformations  α and β:  � × �  � × �  � × �  � × �  �  �  �  �  � ×F  ∝�  ∝�  F  α  G  � ×G  � ×F  ∝�  ∝�  G  β  γ  � ×γ  =  We call these actegory natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that for strict actegory morphisms, this condition holds for any natural transformation γ : F ⇒  G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now consider the following three 2-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (� ,⊗,I ) be a strict monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define the following three 2-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �Actstrict: the 2-category of strict � -actegories, strict actegory morphisms, and natural transfor-  mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �Actstrong: the 2-category of strict � -actegories, strong actegory morphisms, and actegory nat-  ural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �Actlax: the 2-category of strict � -actegories, lax actegory morphisms, and actegory natural  transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that the inclusions of these 2-categories are locally fully faithful, so only the 1-cells differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The case where the action preserves certain limits in the monoidal category is of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A small category equipped with a class of chosen limits is known as a sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The previous correspon-  dence restricts to the class of limit-preserving actions in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A sketch is a small category with a class of chosen limits, and a sketch morphism is a  functor sending chosen limits to the chosen limits in the domain (up to isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of  models of a sketch � in a category �, Mod(� ,�), is the full subcategory [� ,�] whose functors preserve  the chosen limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A monoidal sketch, then, is a sketch (� ,� ) equipped with a symmetric monoidal  category structure on � so that _ ⊗ _ preserves limits in each argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now use the fact that the category Weil1 is a monoidal sketch, since it is a small, strict monoidal  category equipped with a class of limits stable under the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 (Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1, Leung (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following are equivalent:  (i) A tangent structure on �,  (ii) A sketch action ∝: Weil1 × � → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a coalgebraic perspective on tangent categories, coming from the equiva-  lence between algebrasof the 2-monad (Weil1×(_),⊗,I : 1 → Weil1) and the 2-comonad ([Weil1,_],[⊗,_],[I ,_]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For any category �, there is a free tangent category given by  Weil1 × �  85  and this agrees with the free Weil1-actegory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, for the cofree tangent category, take  Mod(Weil1,�),  the category of transverse-limit-preserving functors Weil1 → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We can use Leung’s theorem to induce a monoidal functor Weil1 → � when a tangent structure is  induced by a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (�,⊗,I ) be a strict monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If an additive bundle (p : A → I ,+ : A2 →  A,0 : I → A) equipped with morphisms  A ⊗ A  c−→ A ⊗ A  A  ℓ−→ A ⊗ A  determines a tangent structure on � using the endofunctor A⊗(−), then A determines a strict, transverse-  limit-preserving, monoidal functor  A(−) : Weil1 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn[1] ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='⊗ Wn[k] �→ An[1] ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='⊗ An[k] = Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='I  Note that this allows for a more conceptual description of representable tangent structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a symmetric monoidal closed category, an infinitesimal object is exactly a strict  symmetric monoidal functor D : Weil1 → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This presentation of an infinitesimal object makes it tautological that �op has a tangent structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given a strict symmetric monoidal functor  D : Weil1 → �op  there is a strict action of Weil1 on �op given by  Weil1 × �op D op ⊗�  −−−−→ �op × �op  ⊗−→ �op .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There is a clear correspondence between the notions of a (strict, strong, lax) tangent functor and  a (strict, strong, lax) actegory morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This proposition extends to the following equivalence of 2-  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9 (Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 Leung (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following pairs of 2-categories are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The 2-category of tangent categoriesand strict tangent functorsisthe full sub-2-category ofWeil1ActStrict  spanned by sketch actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The 2-category of tangent categoriesand strong tangent functorsisthe full sub-2-category ofWeil1Actstrong  spanned by sketch actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The 2-category of tangent categoriesand lax tangent functorsisthe full sub-2-category ofWeil1ActLax  spanned by sketch actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  86  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  The Weil nerve of an involution algebroid  The construction in this section is analagous to the nerve of an internal category—hence the “Weil  nerve” construction—and deals with similar technical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, the construction in this  section will mimic the nerve construction for internal categories by replacing the tensor product of  Weil1 with span composition in the domain category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that every anchored bundle or internal  category has a canonical span associated with it:  A  C  M  T M  M  M  π  ̺  s  t  In any category �, there is a category of spans in � as well as span composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A span from A to B in a category � is a diagram of the form  X  A  B  l  r  There is a notion of span composition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' so given a span X : A → B and Y : B → C ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' then the composition  of X and Y is the pullback (if it exists):  Z  X  Y  A  B  C  l  r  l ′  r ′  ⌟  l ′′  r ′′  A morphism of spans is a commuting diagram of the form  A  X  B  C  Y  D  l  r  l ′  fl  fr  r ′  fc  Note that if f and g are span morphisms with fr = gl ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' then the horizontal composition may be formed  if each respective span composition exists: X  W  A  B  E  C  D  F  Y  Z l  r  l ′  fl  fr =gl  r ′  fc  gr  gc  ⌟  ⌟  87  When discussing span composition in a tangent category,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' it is assumed that the pullback is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the category of spans in � is a functor category, so that limits are computed  pointwise in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This also means that the horizontal composition operation, when it exists, preserves  limits in either argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  These span constructions can be helpful in constructing functors from a monoidal category into  a non-monoidal category �, by forming a monoidal category from � using spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the case of an  internal category over M , one takes the slice category �/(M × M ) where the tensor product is span  composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An internal category s,t : C → M is a monoid in this category of spans over M , so that it  determines a monoidal functor  C : ∆op → �/(M × M ),  remembering that ∆op is the monoidal theory for monoid (every monoid in a monoidal category �  determines a monoidal functor ∆ → �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The construction of the corresponding monoidal category for  spans is more nuanced, as the category Weil1 is not �-indexed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Observe that the prolongation of an  anchored bundle is constructed as a span composition:  � (A)  A  T A  M  T M  T 2M  π  ̺  T π  T ̺  π0  π1  ⌟  The third prolongation is given by span composition as well:  � 2(A)  A  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  M  T M  T 2M  π  ̺  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦π0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺◦π1  π0  π1  ⌟  This horizontal composition will play the same role as the tensor product in �/(M × M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a tangent category �, consider a pair of spans  X : M → T U M , Y : M → T V M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Define X ⊠ Y to be the horizontal composition (when it exists):  Z  X  T U Y  M  T U M  T U V M  lX  rY  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='lY  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='r  ⌟  88  (recall that we will often suppress the ⊗ in Weil1 to save space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' So the span composition is  M  X−→ T U M  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y  −−−→ T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V M  M  X  T U M  M  Y  T U M  M  A  T V  M  B  T V ′  lX  lY  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  rX  rY  f  rA  rB  ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  g  lB  lA  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  The horizontal composition f ⊠ g is defined as f ×θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g :  Z  X  Y  M  T U M  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V M  M  T U ′M  T U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V ′M  A  T U ′B  C  lX  lA  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  rX  rA  f  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='rY  T U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='rB  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='lY  T U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='lB  ⌟  ⌟  In any tangent category with a tangent display system (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3), the category of spans on  M whose maps are of the form given by Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 with l ∈ � is a monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Any anchored  bundle in a tangent category gives rise to a monoidal category after a strict choice of T -pullbacks (as-  suming those T -pullbacks exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the span  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn := (M  π◦πi  ←−− An  ̺n  −→ TnM ),  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� := (M = M = M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A choice of prolongations for (π,ξ,λ,̺) is a strict choice of horizontal composition for each V ∈ Weil1:  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V = �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (Wn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn[k]) := �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn[1] ⊠ ··· ⊠ �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We will write the span as follows:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  M  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  πV  ̺V  (notice that the apex is not hatted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given a choice of prolongations for an anchored bundle (π,ξ,λ,̺),  the category Span(π,ξ,λ,̺) is defined as follows:  89  Objects are �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V for V ∈ Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Morphisms are given by pairs  (f ,φ) : �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V → �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  where f : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U and φ : V →U determine a span morphism of the form  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  πV  ̺V  φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  πV  ̺U  f  as discussed in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tensor structure: The tensor product is defined using the horizontal composition ⊠ as defined in  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The idea is to show that an involution algebroid structure on an anchored bundle induces a tangent  structure on the monoidal categoryofprolongations, and then toapplyLeung’stheorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following  two lemmas will simplify this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle with chosen prolongations in a tangent  category �, and identify the monoidal category Span(π,ξ,λ,̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) There is a functor  U ̺ : Span(π,ξ,λ,̺) → �  constructed by sending a span morphism to the morphism between the objects at its apex:  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  πV  ̺V  φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  πV  ̺U  f  f  (ii) Suppose we have a square  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Z  (f ,φ)  (g ,ψ)  (l ,α)  (r,β)  whose image under U ̺ is a T -pullback in �, and so that the square in Weil1 is a transverse T -  pullback:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y  U  Y  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Z  X  Z  f  g  l  r  φ  ψ  β  α  ⌟  ⌟  Then U ̺ reflects the limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, the original square in Span(π,ξ,λ,̺) is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) T -pullbacks of the form described in (ii) are closed under ⊠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  90  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The functor in in (i) is straightforward to construct, as it simply forgets the left and right legs  of the spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For (ii), note that because the Weil1 part of the diagram is a transverse T -pullback, then  given a pair of maps  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Z  (f ,φ)  (g ,ψ)  (l ,α)  (r,β)  (x,ω)  (y,γ)  a unique span morphism �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V → �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U may be induced using the apex map from �, and the unique  map induced in Weil1 by the universality of transverse squares (this square is also universal in �), so  the span morphism diagram will commute by universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For (iii), T -pullback squares of the form in (ii) are closed under ⊠ as transverse squares in Weil1  are closed under ⊗, so the result follows by the commutativity of limits and by applying part (ii) of this  lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It will be useful to have a “flat” presentation of the prolongation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The higher prolongations of an anchored bundle may be concretely described as the T -pullback of the  zig-zag below:  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(k)  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(1)  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(2)  Tn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='n(k−1) ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(k)  Tn(1)M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺n(2)  Tn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='n(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (π◦π0)  ̺n(1)  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (π◦π0)  so that the prolongation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W n[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W n[k] may be written concretely as  (u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',uk) : An[1]̺′×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π′Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='An[2]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺′×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π′Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='n[k−1]An[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore, the choice of prolongation identifies the following limits:  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  ̺U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV  ⌟  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V  T U ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U M  T U M  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  ⌟  ̺U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV  so that  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ idM ⊠ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  where idM is the span M = M = M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that a ̺ sends the involution algebroid structure map to its corresponding tangent structure  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Each of the structure maps, then, gives a span morphism where ⊠ is well defined:  91  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,+q,λ,̺,σ) be an involution algebroid in � with chosen prolonga-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then define the following maps in Span(π,ξ,λ,̺):  The projection p : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W → ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�,  M  A  T M  M  M  M  ̺  π  p  π  The zero map 0 : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� → ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W ,  M  M  M  M  A  T M  0  π  ̺  ξ  The addition map + : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W2 → ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W ,  M  A2  T2M  M  A  T M  ̺2  π◦πi  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  π  ̺  +  The lift map ℓ : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W → ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W ,  M  A  T M  M  A̺×T πT A  T 2M  π  ̺  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  π◦π0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺◦π1  (ξπ,λ)  The flip map c : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W → ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W ,  M  A̺×T πT A  T 2M  M  A̺×T πT A  T 2M  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  π◦π0  ̺◦π1  π◦π0  ̺◦π1  σ  The idea is to show that the monoidal category of chosen prolongations Span(π,ξ,λ,̺) for an in-  volution algebroid has a tangent structure generated by the structure maps in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 and the  endofunctor �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W ⊠(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Using the flat presentation, we can then show thatU ̺ will determine a tangent  functor in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following lemma about ̺ will be useful in constructing the natural transformation  part of a tangent functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle with chosen prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that  by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4, the right leg of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U is written ̺, so it induces a span map:  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  M  T U M  T U M  ̺U  ̺U  πU  pU  92  This map has a flat presentation as  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ̺U ×T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U M id ×T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  ̺U ⊠A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  πU ×T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  We write the map  ̺U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V := ̺U ⊠ ( �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V )  which corresponds to the following span morphism:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  M  T U M  T U V M  T U M  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  ̺U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV  ̺U  πU  ⌟  pU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9 (The Weil Nerve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a fully faithful functor  NWeil : Inv(�) → [Weil1,�]  that sends an involution algebroid to the transverse-limit-preserving tangent functor:  ( �A,α) : Weil1 → �  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For the first step of this proof, we show that an involution algebroid structure on an anchored  bundle (π : A → M ,ξ,λ,̺) determines a tangent category structure on the monoidal category Span(π :  A → M ,ξ,λ,̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We check that the endofunctor ˆA ⊠ (−) determines a tangent structure, with the structure maps  given by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7:  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1] Additive bundle axioms:  (i) Use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 to see that  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W2 = ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W p ×p ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W Aπ×πA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  this is preserved by ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V ⊠ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The triple ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+, ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p, ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0) = (+q ,π,ξ) is an additive bundle induced by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4,  and ⊠ preserves pullbacks (and therefore additive bundles), so the additive bundle axioms  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  93  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2] Symmetry axioms:  (i) ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = id follows from the involution axiom σ ◦ σ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For Yang–Baxter, note that  ( ˆA ⊠ c ) ◦ (c ⊠ ˆA) ◦ ( ˆA ⊠ c ) = (c ⊠ ˆA) ◦ ( ˆA ⊠ c ) ◦ (c ⊠ ˆA)  follows from the Yang–Baxter equation on an involution algebroid  (σ × c ) ◦ (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ × c ) = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ × c ) ◦ (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ),  since  σ × c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A = ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ) ⊠ ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W ) and id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ = ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W ) ⊠ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) For the naturality conditions:  (a) The interchanges of +,0,p all follow from the fact that  σ : (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W,λ × ℓ) → (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W,id × c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  is linear, and so is an additive bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (b) The axiom  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ c = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  is equivalent to the equation  (σ × c ) ◦ (1× T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (ˆλ× ℓ) = (1× T ˆλ) ◦ σ  which is equivalent to the double linearity axiom on σ by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3] The lift axioms:  (i) The additive bundle equations are a consequence of λ being a lift and + being the addition  induced by the non-singularity of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The coassociativity axiom  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ ℓ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ◦ ℓ  is equivalent to  (ˆλ× ℓ) ◦ ˆλ = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ) ◦ ˆλ  proved in (i) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The symmetry of comultiplication, c ◦ℓ = ℓ, is given by the unique equation for an involu-  tion algebroid, so that σ ◦ (ξ ◦ π,λ) = (ξ ◦ π,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The universality of the lift follows from part (ii) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 ensures  that for any V ∈ Weil1, �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V ⊠ µ and µ ⊠ �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V are universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This lemma puts a tangent structure on Span(π,ξ,λ,̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now consider the functor sending spans to  the apex map,  U ̺ : Span(π,ξ,λ,̺) → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The family of maps  {̺U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V : ̺U ⊠ �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V |U ,V ∈ Weil1}  94  gives a family of natural transformations  ̺U : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U ⇒ T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A,  so that the following pair constitute a tangent functor  (U ̺,̺) : Span(π,ξ,λ,̺) → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Because the universality conditions on Span(π,ξ,λ,̺) followed by reflecting limits in � using Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5, it follows that (U ̺,̺) will preserve the tangent-natural limits in Span(π,ξ,λ,̺) corresponding to  transverse limits in Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By Leung’s Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 (by way of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6), the tangent structure on Span(π,ξ,λ,̺) in-  duces a strict, monoidal, transverse-limit-preserving functor  ¯A : Weil1 → Span(π,ξ,λ,̺)  that sends the tensor product ⊗ to the span composition ⊠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By composing the strict tangent functor  ( ¯A,id ) and (U ̺,̺), we have a lax, transverse-limit-preserving, tangent functor:  (A,̺) : Weil1 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V �→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  Now, check the bijection on morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Starting with an involution algebroid morphism (f ,m) :  A → B, note that this gives a span morphism ˆf :  M  A  T M  N  B  T N  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  m  f  This gives a natural definition of ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V using the horizontal composition of span morphism, so that  ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (U V ) = ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V and ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� = m,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2)  giving a family of maps { ˆfU : U ∈ objects(Weil1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Because f will commute with the structure maps  {π,ξ,+,(ξ ◦ π,λ),σ}, it follow immediately that ˆf is a natural transformation, because the following  calculation holds for each θ : X → Y ∈ {p,0,+,ℓ,c }:  ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U Y V ◦ ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ θ ⊠ ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V )  = ( ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ⊠ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V ) ◦ ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ θ ⊠ ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V )  = ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ ( ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ θ) ⊠ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  = ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ (θ ◦ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ) ⊠ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  = ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ θ ⊠ ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V ) ◦ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U X V  Tangent naturality will follow by the preservation of the anchor map by f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The equality, for any Weil  algebra U , of the diagrams  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  N  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U N  M  T U M  T U M  N  T U N  T U N  N  T V N  T V N  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  m  πU  ̺U  ̺U  ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  ̺U  pU  πU  pU  ̺U  pU  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  m  ̺U  =  95  is precisely the tangent-naturality condition from Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For the inverse of this mapping, consider a tangent natural transformation (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12)  γ : (A,α) → (B,β),  A ◦ T (A)  B ◦ T (A)  T ◦ A(A)  T ◦ B(A)  γT A  αA  βT A  T γA  where (A,α) and (B,β) are tangent functors Weil1 → � built out of involution algebroids with chosen  prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For any U ,V , the map γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V decomposes as γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V :  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A  �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �B  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T =π0  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T =π1  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T =π0  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T =π1  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T T  Applying this relationship inductively, it is clear that the base maps γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� determine the entire  morphism γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V :  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  N  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U N  M  An[1]  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  N  Bn[1]  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='N  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  ̺n[1]  π◦πi  π◦πi  ̺n[1]  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn[1]  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (π◦πi)  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (π◦πi)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  πU  πU  ̺U  ̺U  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  =  Thus, every tangent-natural transformation is constructed out of a pair  (γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� : M → N ,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W : A → B)  using the ⊠ construction from Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' All that remains to show is that this pair is an involution  algebroid morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tangent naturality gives the following two coherences:  ̺B ◦ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ◦ ̺A and σB ◦ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W ◦ σA  since ̺B = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W,̺A = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W,σB = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c , and σA = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following diagram proves  that γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W preserves the lifts, so that (γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�) is an involution algebroid morphism:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (ξπ,λA)  π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  m  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (ξπ,λB )  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  λA  λB  96  Thus, a tangent natural transformation ( �A,α) → ( �B,β) is exactly a morphism of involution algebroids  A → B, proving the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now, the projection for a Lie algebroid is a submersion, as we may make a choice of prolongations  for each U ∈ Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These prolongations lead to a new observation about Lie algebroids: they embed  into a category of functors into smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Using the Weil nerve construction, the category of Lie algebroids embeds into the  tangent-functor category:  LieAlgd �→ [Weil1,SMan].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4  Identifying involution algebroids  This section identifies those tangent functors  (A,α) : Weil1 → �  that are involution algebroids as precisely those where A preserves transverse limits and α is a T -  cartesian natural transformation (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These conditions will force each A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V to be the  V -prolongation of the underlying anchored pre-differential bundle:  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T → A, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 : A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  −→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  −→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T, α : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A)  (these conditions also ensure that this tuple is an anchored differential bundle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Initially, it is only clear that α is T -cartesian for the projection p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Indeed, recall that the prolonga-  tion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V is defined to be the T -pullback of the cospan:  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  αU  −→ T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  ←−−−−− T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  Then consider the following diagram:  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U W V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  ⌟  ⌟  This means that every naturality square of α for p is a T -pullback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' natural transformations satisfying  this property for every map in the domain category are called T -cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A natural transformation γ : F ⇒ G is cartesian whenever each naturality square  F C  G C  F D  G D  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  γ  γ  G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  ⌟  is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A natural transformation between functors into a tangent category is T -cartesian when-  ever each component square is a T -pullback (we will generally suppress the T when the context is clear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  97  Now, recall that the Weil complex determined by an involution algebroid has A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V determined  by the following T -pullback squares:  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  ⌟  Then it is not difficult to show that the T -cartesian condition on a Weil complex forces it to be an  involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We first need:  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A T -cartesian Weil complex in � is a tangent functor  (A,α) : Weil1 → �  for which A sends transverse limits to T -limits and α is a T -cartesian natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first condition to check is that a T -cartesian Weil complex gives a natural anchored bundle ˆA  whose Weil prolongations coincide with the functor assignments on objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (A,α) be a T -cartesian Weil complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then we have an anchored bundle  (M := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�,  ˆA := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W, π := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π, ξ := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ, λ := α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore,  � ( ˆA) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W,  � 2( ˆA) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Suppose we have a tangent functor (F,α) : � → � and a differential bundle (π,ξ,λ) in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If F pre-  serves T -pullbacks of π, it preserves the additive bundle structure on (π,ξ,+), so to show (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,α ◦  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) is universal it suffices to show that the following diagram is a T -pullback in �:  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π2  µα◦F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  Expand this to  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='µλ  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  In this case, it restricts to the diagram  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='µ  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  α  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  ⌟  98  Each square is a T -pullback by hypothesis, so the universality of the lift follows by the T -pullback  lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Because the complex is T -cartesian, the assignment A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V gives a coherent choice of prolonga-  tions by the T -pullback  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  αU  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  αU  There is, of course, a natural candidate for the involution map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be the anchored bundle induced by a T -cartesian Weil complex  in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then we have an involution map  σ : � (A)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  −→ � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The equations for an involution algebroid should follow immediately by functoriality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' one need  only ensure that the maps take the correct form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be a T -cartesian Weil complex in a tangent category �, with (π : A → M ,ξ,λ,σ) its  underlying anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then we have:  (i) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = σ × c ,  (ii) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = 1× T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ,  (iii) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = ˆλ × ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A,  (iv) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ = id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Consider the diagram  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  αT T  αT T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ⌟  ⌟  Observe that this forces A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c × c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = σ × c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  99  (ii) Likewise, the diagram  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T T  αT  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  αT  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  ⌟  ⌟  forces A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The diagram  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  αT T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  α  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  αT T  ⌟  ⌟  ⌟  ⌟  forces A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ× ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A = ˆλ × ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) As α is T -cartesian, the following diagram is a T -pullback:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  ⌟  Using previous results, this means that A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T is the unique map making the following diagram  commute:  A̺×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT A  A̺×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  T A  T AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  (π1,π2)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ  π1  which we can see is id × ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Pulling together this lemma and the previous proposition, the following is now clear:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A T -cartesian Weil complex determines an involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  However, we have not yet exhibited an isomorphism of categories between the image of the Weil  nerve functor and T -cartesian Weil complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' At first glance, the Weil nerve construction only gives  a Weil complex that is T -cartesian for the tangent projection p ∈ Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Being T -cartesian for p is,  100  however, sufficient: a Weil complex that is T -cartesian for tangent projections will be T -cartesian for  every map in Weil1 (a similar result appears in the context of differentiable programming languages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  see Cruttwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A lax tranverse-limit-preserving tangent functor (F,α) : Weil1 → � for which F pre-  serves pullback powers of each T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p is T -cartesian if and only if each  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  αA  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  αB  is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We only check the converse since the forward implication is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We make use of the T -  pullback lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) c is an isomorphism, so its naturality square is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For projections T2 → T , the retract of a T -pullback diagram is a T -pullback, so the following  diagram is universal:  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  α  α  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πi  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πi  α  α  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πi  (iii) For 0, observe that the following two diagrams are equal:  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  α  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  α  α  α  α  ⌟  ⌟  The right diagram is a T -pullback, and the right square of the left diagram is a T -pullback by  hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By the T -pullback lemma, the left square of the left diagram is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) For ℓ, observe that  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ⌟  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α  ⌟  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ⌟  The outer perimeter of the right diagram is a T -pullback (left square by hypothesis, right square  by (ii)), as is the right square of the left diagram (by hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By the T -pullback lemma, the  left square of the left diagram is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  101  (v) For +, observe that  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ⌟  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πi  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πi  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ⌟  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  α  ⌟  The outer diagram on the right is a T -pullback by composition, and the right square on the left  diagram is a T -pullback by hypothesis, so the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To check that the naturality square is a T -pullback for every map in Weil1, we once again use Leung’s  characterization of maps in Weil1 from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Inductively, the set of maps generated by  {p,0,+,ℓ,c } closed under ⊗ and ◦ follows as T -pullback squares are closed to composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For maps  induced by a tranverse limit in Weil1, F preserves transverse limits so this follows by the commutativity  of limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For any tangent category �, the replete image of the Weil nerve functor  Inv(�) �→ [Weil1,�]  is precisely the category of T -cartesian Weil complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' That α : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A is T -cartesian is equivalent to requiring that the tangent functor  (A,α) : Weil1 → �  restricts to an anchored bundle  (π : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  −→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�, ξ : A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  −→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T, λ : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  −→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T T  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  −→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T, ̺ : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A)  and each A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V is the V -prolongation of this anchor bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The condition in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9 is analogous to the Segal conditions identifying those  simplicial complexes  ∆ → �  that are internal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that every simplicial object has an underlying reflexive graph  tr1(X ) := (s,t : X ([1]) → X ([0]),i : X ([0]) → X ([1]))  where X ([n]) is isomorphic to the object of n-composable arrows for the underlying reflexive graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Notably, being T -cartesian for p is enough to force that a natural transformation is  T -cartesian for the other tangent-structural natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This has consequences when one  uses partial maps to combine topological notions with tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In this context, a partial map  N → X with domain M �→ N is a span  M  N  X  f  m  whose right leg is monic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The intuition is that the map f is defined on the subobject M of N , which  introduces a new problem: what is the proper notion of a subobject in a tangent category?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Such a notion  102  should give rise to a stable class of monics: one that is closed under horizontal span composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' One  answer is the notion of etale monics: a morphism is etale whenever the naturality square for p is a T -  pullbacks:  T M  T N  M  T N  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  p  p  ⌟  Geometrically, this means that the morphism is a local diffeomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' for example, an etale subobject  of �n in the Dubuc topos is precisely an open subset in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An endofunctor lifts to the partial  map category whenever it preserves the class of monics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A natural transformation lifts to endofunctors  on the partial map category whenever it is T -cartesian for the class of monics, and the same proof will  show that this property holds for etale monics Cruttwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  The prolongation tangent structure  One of the most important consequences of the Weil Nerve Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9 is that the category of in-  volution algebroids (with chosen prolongations) may be equipped with two tangent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  first tangent structure is the pointwise tangent structure described in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent  functor sends  (A,α) �→ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A : Weil1 → �,c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α : T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A)  (recall the composition of tangent functors given in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11 (ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The structure morphisms will  be given by whiskering, so in this case θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A,θ ∈ {p,0,+,ℓ,c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The restriction to tangent functors that  preserve transverse limits along with the fact that the natural part α is T -cartesian, however, ensures  that precomposition with the tangent functor  (A,α) �→ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T : Weil1 → �,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T )  returns an involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The structure maps are once again given by whiskering, with the pre-  composition tangent structure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='θ,θ ∈ {p,0,+,ℓ,c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Preservation of transverse limits guarantees that  this tangent structure will satisfy the necessary universality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of involution algebroids with chosen prolongations  in a tangent category � has a second tangent structure, where the action by Weil1 is given by  (A,α) �→ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T : Weil1 → �,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The proposition statement means that the structure morphisms for this new involution alge-  broid are given by  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ) ∼=  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = ̺′ :  � (A)  π1  −→ T A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p = π′ :  � (A)  p◦π1  −−→ A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 = ξ′ :  A  (ξ◦π,0)  −−−→ � (A)  ̺′ ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ = λ′ :  � (A)  λ×ℓ  −−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = σ′ :  � 2(A)  σ×c  −−→ � 2(A)  103  Similarly, we can see that  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c )  =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = ̺′′ :  � 2(A)  (π1,π2)  −−−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p = π′′ :  � 2(A)  (p◦π1,p◦π2):  −−−−−−−→ � (A)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 = ξ′′ :  � (A)  (ξ◦π◦π0,0◦π1,0◦π2)  −−−−−−−−−−−→ � 2(A)  ̺′′ ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ = λ′′ :  � 2(A)  (λ×ℓ×ℓ)  −−−−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� 2(A)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = σ′′ :  � 3(A)  (σ×c ×c )  −−−−−→ � 3(A)  These coincide with the involution algebroids � ′(A),� ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′(A) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10: the second tan-  gent structure follows from the fact that the natural transformations for the tangent structure there are  given by  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='φ : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⇒ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V,φ :U → V ∈ {p,0,+,ℓ,c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The result follows as a corollary of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The Jacobi identity for involution algebroids  Classically, the theory of Lie algebroids uses the algebra of sections Γ(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' One key observation is that  when using the Lie tangent structure (Inv(�),� ), sections of π are in bijective correspondence with  χ� (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This observation allows for different statements about Lie algebroids to be translated into for-  mal statements about the tangent bundle in (Inv(�),� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be an involution algebroid in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a bijection between the sections of π  in � and the vector fields on A in Inv(�):  X ∈ Γ(π) �→ ((id ,T X ◦ ̺),X ) : A → TL(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ˆX ∈ χ� (A) �→ ˆXR : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='R → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall the coherence for tangent natural transformations γ : (H ,φ) ⇒ (G ,ψ):  H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='H  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='K  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  φ  ψ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ  We specify this to a morphism ˆX : (A,α) ⇒ TL(A,α) = (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ) at �,W :  A  A̺×T πT A  T M  T A  X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ̺  π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X  and so infer that, if we set X := XR , we have π1 ◦ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = T (X ) ◦ ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Furthermore, the condition that  pL ◦ X = id forces id = π0 ◦ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus, we can see that every section X of pL is given by a morphism of  the form ((id ,T X ◦ ̺),X )) on the underlying involution algebroids, where π◦ X = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We now show that every X ∈ Γ(π) gives rise to a section of πL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Observe that the following is a mor-  phism of involution algebroids:  A  A̺×T πT A  M  A  π  (id,T X ◦̺)  p◦π1  X  104  Note that it is well typed, as T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺ = ̺ ◦ id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Check that it is a bundle morphism:  p ◦ π1 ◦ (id ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺) = p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺ = X ◦ p ◦ ̺ = X ◦ π  and that it is linear:  (λ ◦ π0,ℓ◦ π1) ◦ (id ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺) = (λ,ℓ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺)  = (λ,T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ℓ◦ ̺) = (λ,T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ λ) = T (id ,T X ) ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Then check that it preserves the anchor:  π1 ◦ (id ,T X ◦ ̺) = T X ◦ ̺  and the involution:  (π0,π1,T 2X ◦ T ̺π1) ◦ σ = (σ,T 2X ◦ T ̺ ◦ π1σ)  = (σ,T 2X ◦ c T ̺ ◦ π1) = (σ(π0,π1),c π3) ◦ (id ,T 2X ◦ T ̺ ◦ π1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus we have that (id ,T X ◦̺π1) is a morphism of involution algebroids, inducing a morphism of  T -cartesian Weil complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lastly, we check that it is a section of p L  A , but this is clear, since  π0 ◦ (id ,T X ◦ ̺) = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  thus we have the desired bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that given an involution on an anchored bundle, there is a bracket on its set of sections (see  the explicit construction in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given an X ,Y ∈ Γ(π), there is a bracket defined as follows:  ˆλ ◦ [X ,Y ]A +1 (ξπ,0) ◦ Y = ((σ ◦ (id ,T Y ◦ ̺) ◦ X −2 (id ,T X ◦ ̺) ◦ Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A direct proof of the Jacobi identity is a detailed calculation (see the original preprint on involution al-  gebroids Burke and MacAdam (2019)) and still relies on Cockett and Cruttwell’s result for an arbitrary  tangent category with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' As a result of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, we can instead use Cockett and Crut-  twell’s result directly:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be a complete involution algebroid in a tangent category � with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There  is a Lie bracket defined on Γ(π), [−,−] induced by  ˆλ ◦ [X ,Y ]A +1 (ξπ,0) ◦ Y = ((σ ◦ (id ,T Y ◦ ̺) ◦ X −2 (id ,T X ◦ ̺) ◦ Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The bracket is induced by Rosicky’s universality diagram, as  0 = p ◦ ((σ ◦ (id ,T Y ◦ ̺) ◦ X − (id ,T X ◦ ̺) ◦ Y )) − 0Y  = T p ◦ ((σ ◦ (id ,T Y ◦ ̺) ◦ X − (id ,T X ◦ ̺) ◦ Y )) − 0Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We look at the Lie tangent structure for Inv∗(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this is precisely the vector field induced by  e vR([ ˆX , ˆY ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We complete the proof by using the result that for any A in a tangent category with negatives, the  bracket on χ(A) that is defined by  ℓ◦ [X ,Y ] = (c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ Y − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ X ) − 0X  satisfies the Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  105  Identifying categories of involution algebroids  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 identified whenever a functor Weil1 → � is an involution algebroid, whereas this section  identifies tangent categories � that embed into the category of involution algebroids in some tangent  category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We call this structure an abstract category of involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This notion involves  some 2-category theory, using a modified notion of codescent (see Bourke (2010) for a development of  codescent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that for any tangent category �, the category of involution algebroids has � as a reflective  subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Furthermore, because limits of involution algebroids are computed pointwise, this reflec-  tor is left-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This left-exact reflection is the main structure we axiomatize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An abstract category of involution algebroids is a tangent category � with a left-exact  T -cartesian tangent localization (Z ,̺) : � → �, where L satisfies a codescent condition:  TangCatStrict(Weil1,�) �→ TangCatLax(Weil1,�)  L∗  −→ TangCatLax(Weil1,�)  (where L∗ denotes post-composition by L) is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of involution algebroids in any tangent category � is an abstract category  of involution algebroids using (Inv(C ),� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The reflector is the functor sending an involution algebroid  to its base space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the T -cartesian natural transformation is the anchor map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Any tangent subcategory of  Inv(�) that contains � as a full subcategory will give rise to an abstract category of involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � �→ � be an abstract category of involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then there is an embed-  ding � �→ Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The proof follows by treating objects in � as strict tangent functors Weil1 → � and morphisms  as tangent natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Weil1  �  �  Z  � (−,A)  � (−,B)  � (−,f )  The natural part of Z is T -cartesian, and the functor part preserves limits, so Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (−,A) =: Z [A] de-  termines an involution algebroid in �, and f a morphism of involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The embedding is  guaranteed by the codescent condition so that the post-composition functor is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An abstract category of involution algebroids � �→ � is exactly a full subcategory � �→  � �→ Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  106  Chapter 5  The infinitesimal nerve and its realization  The main thrust of Chapters 2, 3, and 4 has been that the tangent categories framework allows for Lie  algebroids to be regarded as tangent functors  Weil1 → SMan  which satisfy certain universality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This chapter, which is more experimental than the pre-  vious four chapters and represents work still in progress, puts Lie algebroids into the framework of  enriched functorial semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This new perspective on algebroids uses Garner’s enriched perspective  on tangent categories (Garner (2018)) and the enriched theories paradigm from Bourke and Garner  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The functorial-semantics presentation of the Lie functor will generalize the Cartan–Lie the-  orem (that the category of Lie algebras is a coreflective subcategory of Lie groups) into a statement  within the general theory of functorial semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The goal is to show that the infinitesimal approximation of a groupoid, as discussed in Example  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, may be constructed as a nerve, just like Kan’s original simplical approximation of a topological  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The nerve of a functor K : � → � approximates objects and morphisms in � by � -presheaves,  so it sends an object in � to the � -presheaf  NK : � → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C �→ �(K −,C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus there will be an infinitesimal object,  ∂ : Weilop  1  → Gpd(� )  where Gpd(� ) denotes groupoids in the category � of Weil spaces (formally defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The nerve of ∂ has a left adjoint, the Lie realization, given by the left Kan extension (just as Kan’s geo-  metric approximation of a simplicial set is, in Kan (1958)):  Weilop  1  Gpd(� )  [Weil1,� ]  �  ∂  Lan��� ∂  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13 (The Lie Realization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a tangent adjunction between the category of involution  algebroids and groupoids in � , where each functor preserves products and the base spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Gpd(� )  Inv(� )  N∂  |−|∂  ⊣  107  Note, however, that the left Kan extension in Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 does not immediately give the desired  adjunction of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13, as there is no guarantee that the nerve functor N∂ lands in the category  of algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To prove this we must revisit the work in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 presenting involution algebroids as  those functors A : Weil1 → � for which each A(V ) is the prolongation of its underlying anchored bundle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  this leads naturally to the formalism for enriched theories developed in Bourke and Garner (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The presentation of algebroids as models of an enriched theory requires situating the categories  of differential bundles, anchored bundles, and involution algebroids in � as full subcategories of � -  presheaf categories on small � -categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 reviews the work in Garner (2018) characteriz-  ing tangent categories as categories enriched in � = Mod(Weil1,Set) (the cofree tangent category on  Set, by Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 reconfigures the content of Chapter 2 using the enriched perspective, so that lifts are � -  functors from the � -monoid D + 1, while differential bundles are a reflective subcategory of functors  from its idempotent splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Similarly, the category of anchored bundles are a reflective subcategory  of the category [Weil1  1,�], where Weil1  1 is the category of 1-truncated Weil algebras (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 reviews the basic idea of an enriched nerve/approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A particularly important ex-  ample is the linear approximation of a reflexive graph, the functor introduced in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7, which  is the nerve of a functor  ∂ : (Weil1  1)op → Gph(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 appliesthe enriched theoriesframework introduced in Bourke and Garner(2019), where  a dense subcategory of a locally presentable � �→ � forms the “arities”, and a bijective-on-objects func-  tor � → � is the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of models is the pullback of (enriched) categories:  ��  [� ,� ]  �  [� op,� ]  The first step is to freely complete the � -category Weil1  1 of truncated Weil algebras (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8)  so that its base anchored bundle has all prolongations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' call this � -category � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then, in every tangent  category�, the categoryofanchored bundleswith chosen prolongations(Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4) in � embeds  fully and faithfully into the functor category:  Anc� (�) �→ [� ,�]  (here, Anc� (�) denotes the category of anchored bundles with chosen prolongations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular,  the tangent bundle on � in Weil1 determines a bijective-on-objects functor  � → Weil1  so that the category of involution algebroids in any tangent category � is the pullback in � Cat:  Inv∗(�)  [Weil1,�]  Anc∗(�)  [� op,�]  ⌟  This means that the category of involution algebroids in � is monadic over the category of anchored  bundles in � using the monad-theory correspondence from Bourke and Garner (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  108  The final section looks at the category of � -groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Essentially, the free groupoid over the lin-  ear approximation of a graph will now give an infinitesimal object Weil1 → Gpd(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The nerve of  ∂ : Weilop  1  → Gpd(� )—the infinitesimal approximation—has a right adjoint via the realization of the  nerve functor from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is used to prove the culminating Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The individual pieces of categorical machinery used in this chapter are not new (the enriched per-  spective on tangent categories, enriched nerve constructions, enriched theories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, all of the  results dealing with the application of enriched nerve constructions and enriched theories to tangent  categories is original work of the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Tangent categories via enrichment  This section gives a quick introduction to Garner’s enriched perspective on tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  enriched approach to tangent categories first appeared in Garner (2018) and builds on the category  perspective on tangent categories introduced in Leung (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Garner was able to exhibit some of the  major results from synthetic differential geometry as pieces of enriched category theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' for example,  the Yoneda lemma implies the existence of a well-adapted model of synthetic differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of Weil spaces is the site of enrichment for tangent categories and is closely related to  Dubuc’s Weil topos from his original work on models of synthetic differential geometry Dubuc (1981);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  a deeper study of this topos may be found in Bertram (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that the category Weil1 is the free  tangent category over a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of Weil spaces is the cofree tangent category over  Set, which is the category of transverse-limit-preserving functors Weil1 → Set by Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Call this the category of Weil spaces, and write it � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Just as a simplicial set S : ∆ → Set is a gadget  recording homotopical data, a Weil space records infinitesimal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Weil space is a functor Weil1 → Set that preserves transverse limits (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5):  that is, the ⊗-closure of the set of limits  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  Tn+m  Tm  Tn  �  ⌟  ,  T2  T 2  �  T  0  µ  ⌟  ,  Tn  Tn  Tn  Tn  \uf8fc  \uf8f4  \uf8f4  \uf8fd  \uf8f4  \uf8f4  \uf8fe  A morphism of Weil spaces is a natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the category of Weil spaces as � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Every commutative monoid may be regarded as a Weil space canonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Observe that for every V ∈  Weil1 and commutative monoid M , one hasthe free V -module structure on M given by |V |⊗CMonM  (here |V | is the underlying commutative monoid of V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The commutative monoid |V | is exactly  �dimV, so that  |V |⊗CMon M ∼= ⊕dimV M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This agrees with the usual tangent structure on a category with biproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Following (i), any tangent category � that is concrete—that is, admitting a faithful functorU : � →  Set—will have a natural functor into Weil spaces (copresheaves on Weil1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every object A will have  an underlying Weil space V �→ USet(T V (A)), and whenever U preserves connected limits (such as  the forgetful functor from commutative monoids to sets), each of the underlying copresheaves will  be a Weil space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  109  (iii) Consider a symmetric monoidal category with an infinitesimal object, which by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 is  a transverse-colimit-preserving symmetric monoidal functor D : Weil1 → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then for every object  X , the nerve (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1) ND (X ) : �(D −,X ) : Weil1 → Set is a Weil space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) For any pair of objects A,B in a tangent category, �(A,T (−)B) : Weil1 → Set is a Weil space by the  continuity of �(B,−) : � → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Unlike the category of simplicial sets, the category of Weil spaces is not a topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of  Weil spaces does, however, inherit some nice properties from the topos of copresheaves on Weil1 by  applying results from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3, as it is locally presentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The basics of locally presentable categories  can be found in the Appendix 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Roughly speaking, a cocomplete category � has a subcategory of  finitely presentable objects �f p, those C so that  �(C ,−) : � → Set  preserves filtered colimits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' those colimit diagrams that commute with all finite limits in Set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � is  locally finitely presentable whenever every object is given by the coend  C ∼=  � X ∈�f p  �(X ,C ) · X  where S · X is the (possibly infinite) product X |S| with |S| the cardinality of the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This means that  � = Lex(�op  f p,Set), where Lex means the category of finite-limit-preserving functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The third pointofCorollary5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 below, that� islocallyfinitelypresentable asa cartesian monoidal  category, means that the category �f p is closed under products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This implies that � is locally pre-  sentable as a � -category (this ends up being an important technical condition whereby locally pre-  sentable � -categories make sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Whenever we discuss an arbitrary � -category, we will assume  that � is locally presentable as a monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 (Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of Weil spaces is a cartesian-monoidal reflective sub-  category of [Weil1,Set].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of Weil spaces is  (i) a cartesian closed category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) a representable tangent category, where the infinitesimal object is given by the restricted Yoneda  embedding � : Weilop  1  → � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Locally finitely presentable as a cartesian monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The cofree tangent structure on � is given by precomposition, that is:  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (V ) = M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (U ⊗ V ) = M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This tangent coincides with the representable tangent structure induced by the Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The proof is an application of the Yoneda lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Observe that  [D,M ](V ) ∼= [Weil1,Set](D × � (V ),M ) ∼= [Weil1,Set](� (W V ),M ) ∼= M (W V )  where D ×� (V ) = � (W V ) follows because the tensor product in Weil1 is cocartesian and the reflector  is cartesian monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  110  Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent category � is a representable tangent category, where D = � W (Definition  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We will write the Yoneda functor � : Weilop  1  → � as D(−), so it is closer to the usual notation  used in representable tangent categories or synthetic differential geometry (a single D may be used as  shorthand for D(W ), D (n) for D(Wn), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that  D(V ) = D (⊗kWn(i)) =  K  �  D(ni)  and that D(ℓ) = ⊗,D(c ) = (π1,π0), D(+) = δ, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  At this point, we are ready to move to the enriched perspective on tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The basics  of enriched category theory may be found in the Appendix 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, but one definition in particular is im-  portant to include here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a � -category for � a closed symmetric monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For J ∈ � , the  power by J of an object C ∈ � is an object C J so that the following is an isomorphism:  ∀D : � (J ,�(D,C )) ∼= �(D,C J )  whereas the copower is given by  ∀D : � (J ,�(C ,D)) ∼= �(J • C ,D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Let � ←� � be a full monoidal subcategory of � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A � -category � has coherently chosen powers by � if  there is a choice of � -powers (−)J so that  (C J )K = C J ⊗K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Likewise, coherently chosen copowers are a choice of � -copowers so that  K • (J • C ) = (K ⊗ J ) • C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The sub-2-categories of � -categories equipped with coherently chosen powers and copowers are � Cat�  and � Cat� , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Wood (1978) proved that the 2-category of actegories over a monoidal � is equivalent to the 2-  category of [�,Set]-enriched categories with powers by representable functors (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6), us-  ing the monoidal structure on [�,Set] induced by Day convolution1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Moreover, Garner (2018) showed  that a monoidal reflective subcategory � �→  ˆ  � exhibits the 2-category of � -categories as a reflective  sub-2-category of � -categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this proves that tangent categories are equivalent to a particular class  of enriched category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 (Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent category is exactly a � -category with powers by repre-  sentables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every A,B ∈ �0, the Weil space is defined as  �(A,B) :=U �→ �(A,T U B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1The Day convolution tensor product of presheaves X ,Y : �op → Set is given by X �⊗Y := Lan⊗(X ⊠Y ), where (X ×Y )(V ) =  X (V ) × Y (V ) (Day (1970)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  111  The functor �(A,−) is continuous and T −B is an infinitesimally linear functor, so this is a Weil space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The following diagram gives the composition map :  �(B,C ) × �(A,B)  �(A,C )  �(B,T U C ) × �(A,T V B)  �(A,T V U C )  �(T V B,T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U C ) × �(A,T V B)  T V ×id  m  Note that it is natural in U and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By the Yoneda lemma (as the internal hom in � is the internal hom of copresheaves on Weil1),  �(A,T V B) = (U �→ �(A,B)(V ⊗U )) = [D (V ),�(A,B)]  so this category has coherently chosen powers by representable functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now the original notions of (lax, strong, strict) tangent functors can be shown to correspond to  power-preservation properties of � -functors between � -categories with coherently chosen powers:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 (Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We have the following equivalences of 2-categories:  (i) the 2-category of � -categories with coherently chosen powers and TangCatLax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) the 2-category of � -categories with coherently chosen powers and power-preserving � functors  and TangCatStrong;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The 2-category of � -categories with coherently chosen powers and chosen-power-preserving �  functors and TangCatStrict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that for a lax tangent functor (F,α) : � → �, the map  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X : F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X  can be seen as the unique morphism induced by universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Conversely, for a strong tangent functor,  the natural isomorphism α is the isomorphism from a power F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X to the coherently chosen power  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In contrast, a strict tangent functor preserves the coherent choice of powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The ability to work with � -categories that do not have powers by representables allows for signif-  icant flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is useful to observe that there are � -categories which are not tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Every monoid (M ,m,e ) in � gives rise to a one-object � -category whose hom-object is M , com-  position is m, and unit is e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given a tangent category �, it is possible to take the full � -category over some set of objects �,  even though D may not be closed under iterated applications of the tangent functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The dual of a � -category is a � category, where  �op (A,B) = �(B,A) : Weil1 → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Dually, in the case that � is a tangent category, �op will have coherent copowers by representables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  112  (iv) If a cartesian category � hasan infinitesimal object D (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7), it hasa natural � -category  structure (with coherent copowers by representables)  �(A,B) : Weil1 → Set := �(A × D (−),B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using the dual tangent structure on �op from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8, the enrichment on � is exactly the  enrichment found by regarding � as the dual � -category of the tangent category �op .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  As a final remark, note that the Yoneda lemma applies to � -categories, so there is an embedding  � �→ [�op ,Set].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The powers and copowers by representables are computed pointwise in a presheaf category, so they  inherit the coherent choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus the following holds:  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10 (Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every tangent category embeds into a � -cocomplete representable  tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (In fact, showing that the Yoneda embedding applies to tangent categories is the main theorem of  Garner (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  Differential and anchored bundles as enriched structures  This section gives an enriched-categorical reinterpretation of the work in Chapter 2 regarding differen-  tial bundles and Chapters 3 and 4 regarding anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Differential bundles as enriched structures  As a first case study using the enriched perspective for tangent categories, consider differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Most of the work in Chapter 2 uses the intuition that differential bundles are some sort of tangent-  categorical algebraic theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this section will make that intuition concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that a lift (Definition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3) is a map λ : E → T E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Using the enriched perspective and treating T E as a power (recall Defini-  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6), this gives the following correspondence:  λ : 1 → �(E ,E D )  ˆλ : D → �(E ,E )  The commutativity condition for ˆλ, then, is translated as follows:  E  T E  D × D  �(E ,E ) × �(E ,E )  T E  T 2E  D  �(E ,E )  ⊙  mE ,E ,E  ˆλ×ˆλ  ˆλ  λ  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  ℓ  That is, ˆλ is a semigroup morphism D → �(E ,E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In any cartesian closed category with coproducts,  a semigroup may be freely lifted to a monoid using the "exception monad" (−) + 1 from functional  programming (see, for example, Seal (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Regard the following monoid as the one-object � -category Λ:  D × D + D + D + 1  (ιL ◦m|ιL|ιL |ιR )  −−−−−−−−→ D + 1  m : (D + 1) × (D + 1) → D + 1  113  Thus, a lift λ is exactly a functor ˆλ : Λ → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now check that morphisms are tangent natural transfor-  mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the semigroup D is commutative, so Λ = Λop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the choice of using Λop in the next  lemma is to be consistent with conventions used in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of lifts in a tangent category � is isomorphic to the category of � -functors  and � -natural transformations Λop → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Check that a � -natural transformation is exactly a morphism of lifts f : λ → λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Start with the  � -naturality square:  D + 1  �(A,A) × �(A,B)  �(A,B) × �(B,B)  �(A,B)  (λ,f ◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  (f ◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',l )  mAB B  mAAB  Now, rewriting D + 1 → �(A,B) as a semigroup map D → �(A,B), we have:  D  (λ,f ◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  −−−→ �(A,A) × �(A,B)  mAAB  −−−→ �(A,B)  1  (λ′,f )  −−→ �(A,T A) × �(A,B)  1×T  −−→ �(A,T A) × �(T A,T B)  mA,T A,T B  −−−−−→ �(A,T B)  1  T f ◦λ′  −−−→ �(A,T B)  Similarly, the other path is exactly λ′ ◦ f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, a � -natural transformation is exactly a morphism of  lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is a classical result in synthetic differential geometry that the object D has only one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the  case of � , this follows from the Yoneda lemma (regarding � as a Set-category):  � (1,D) = Weil1(�[x]/x 2,�) = {!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' : �[x]/x 2 → �}  Note that the natural idempotent e : id ⇒ id from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8, then, must be the point 0 : 1 → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that this idempotent is an absorbing element of the monoid D + 1, so for any f : X → D + 1, it  follows that m(f ,0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=') = m(0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', f ) = 0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='. A pre-differential bundle is exactly a lift with a chosen splitting  of the natural idempotent p ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every lift ¯λ : Λop → �, the natural idempotent e = p ◦ λ is exactly  1  0−→ D  ιL−→ D + 1 → �(E ,E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now that the natural idempotent is understood as a map in Λ, that idempotent splits to give the  theory of a pre-differential object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The � -category Λ+ is given by the set of objects {0,1} with hom-Weil-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Specifi-  cally,  The hom-spaces are Λ+(1,1) = D + 1, otherwise Λ+(i, j) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Composition (writing the original composition from Λ as m) is given by  m111 : (D + 1) × (D + 1)  m  −→ (D + 1)  m101 : 1× 1  ιR−→ (D + 1)  otherwise: mi j k =!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  114  Idempotent splittings are absolute (co)limits, and are preserved by all functors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' as this is a limit  completion, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of pre-differential bundles is exactly the category of �-functors Λ+ → � (that  is, �-valued presheaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is straightforward to exhibit the category of differential bundles as a reflective subcategory of  pre-differential bundles in [Λ+,�] (so long as � has equalizers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of differential bundles in a tangent category � with T -equalizers and  T -pullbacks is a reflective subcategory of [Λ+,�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of pre-differential bundles in � is isomorphic to [Λ+,�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8, the  category of differential bundles is the category of algebras for an idempotent monad on the category  of pre-differential bundles in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The reflector sends a pre-differential bundle to the T -equalizer:  A  T E  T E  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  This equalizer will always exist if � has equalizers, and pullbacks of the projection will exist if � has  pullbacks, so the pullback is a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This reflection gives a left-exact idempotent monad  on [Λ+,�] whose algebras are differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now, in the case that � is a locally presentable tangent category (such as � ), [Λ+,�] is locally pre-  sentable and so is the reflective subcategory of differential bundles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If � is locally presentable, then DBun(�) is a locally presentable category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Anchored bundles  There are two ways to think about anchored bundles:  (i) an anchored bundle is a differential bundle with an anchor A → T M , or  (ii) an anchored bundle is an involution algebroid without an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  These two perspectives can be unified by regarding A as a cylinder for the weighted limit T M , so that  the anchor is induced by the unique map A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0:  Λ+  �  A  T M  ̺  That is, the syntactic category for anchored bundles is constructed as a full � -category of Weil1 that  doesn’t include the map c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This may be found by taking the full subcategory of Weil1 whose objects  are constructed out of Wn,n ∈ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Weil algebra has width k ∈ � if it can be written  V = ⊗0≤i<kWn(i), n(i) ∈ �  The category of k-truncated Weil algebras, written Weilk  1 , is the full � -subcategory of Weil1 of Weil  algebras of width k or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The full subcategory whose objects are {�,W } will be given the special notation Weil∗  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  115  Note that for each V in Weilk  1 , the enrichment is given by  Weilk  1 (U ,V ) := (X �→ Weil1(U ,X ⊗ V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The maps in Weil1  1—that is, the full subcategory of Weil1 whose objects are  {Wn|n ∈ �}  —have the useful property that they may be written without the flip c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This makes Weil1  1 a natural  candidate for the syntactic category of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every morphism  Wn → V ∈ Weil1  may be written without c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, it is generated by the set of maps {p,+,0,ℓ} closed under tensor, com-  position, and maps induced by transverse limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every map Wn → V in Weil1 is the finite sum  v �→  �  x  (Ax v) • x  where x ∈ var(V ), and Ax ∈ �n so that An v is the ordinary dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Each term can be written  without c , and the whole term is then constructed by adding each component using the appropriate  U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V -symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 2  It is possible, then, to show that the category of anchored bundles in � is a full sub-tangent-category  of functors Weil1  1 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' note that W acts as a cone for (−)D , so this induces a map  ̺ : F (W ) → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F (�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every anchored differential bundle in � determines a functor Weil1  1 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' an an-  chored bundle morphism is exactly an enriched natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Start with an anchored bundle (q : E → M ,ξ,λ,̺);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' then for hom-objects with domain �,  Weil1  1(�,�) = 1 �→ idM ,  Weil1  1(�,T ) = 1 �→ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Forthe hom-objectswith domain T , the problem isslightlymore difficult, asWeil1  1(T,�)(T V ) = Weil1(T,T V )  and Weil1  1(T,T )(T ) = Weil1(T,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This part of the proof amounts to constructing maps  Weil1(T,T V ) → �(E ,T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ),  Weil1(T,T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ) → �(E ,T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E )/  The first mapping is straightforward: send θ to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ◦̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For the second map, observe that the following  diagrams commute:  T A  T 2M  A  T M  A  T M  A2  T2M  A  T M  M  M  M  M  A  T M  ̺  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  λ  ℓ  ξ  0  ̺  ̺  π  p  +q  ̺2  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  ̺  2This may also be regarded as a consequence of the graphical notation for maps in Weil1 in Table 1 on page 308 of Leung  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  116  The idea is to take an anchored bundle morphism f and rewrite it as a string of compositions that does  not include c , switching out every occurrence of  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ,θ ∈ {p,0,+,ℓ}  and replacing it with T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (θ ′), where θ ′ is the corresponding map in {q,ξ,+q,λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This induces a map  Weil1  1(W,W ) → �(E ,E ) ∈ �  and the ̺ is exactly the unique α : F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F induced by universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For morphisms, the inclusion (Λ+)op → Weil1 ensures that any tangent natural transformation will  be a linear morphism on the underlying differential bundle, and the tangent natural transformations  coherences ensure that a tangent natural transformation will preserve the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Conversely, an an-  chored bundle morphism will preserve each of the constructed morphisms E → T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,E → T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  (as it preserves each of {q,+q,ξ,λ}), giving a natural tangent transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus there is a faithful  embedding Anc(�) �→ [Weil1  1,�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The converse, identifyingthose functorsWeil1  1 → � thatdetermine anchored bundles, isimmediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of anchored bundles comprises precisely the � -functors and � -natural  transformations A : Weil∗  1 → � (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8) so that the precomposition Λ+ → Weil∗  1 → � determines  a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' That is to say, the category of anchored bundles in � is the following pullback in  � Cat:  Anc(�)  [Weil1  1,�]  DBun(�)  [Λ+,�]  ⌟  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  Enriched nerve constructions  Nerve constructionspresenta powerful generalization of the Yoneda functor that sends an object C ∈ �  to the representable presheaf �(−,C ) ∈ [�op ,� ] (for a general reference on nerve constructions and  their realizations, see Chapter 3 of Loregian (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Yoneda lemma states that this functor is an  embedding (that is, it is fully faithful), so no information is lost when embedding a category into its cat-  egory of presheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nerve constructions, then, move this towards an approximation of the original  category � by some subcategory � �→ �, or more generally by some functor K : � → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5,  the “infinitesimal” approximation of a Lie groupoid as a Lie algebroid will be exhibited as approxima-  tion by a � -functor ∂ : Weilop  1  → Gpd(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We work with a � that is locally presentable as a monoidal category, such as � ,Set, or the category  of commutative monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The nerve of a � -functor K : � → � is the functor  NK : � → [� op,� ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  C �→ �(K −,C )  Any presheaf A : � → � that is in the image of NK is a K -nerve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  117  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The “K -nerve” terminology seems to go back to Grothendieck/Segals’s original intuition  for the nerve construction of a category3 (Segal (1974)) and appears in, for example, Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2012)  and Bourke and Garner (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, the “approximation” of a category by a functor K : � → �  originally used in topology seems to be a more intuitive description of the functor NK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The nerve of the identity functor � = � is the usual Yoneda embedding � �→ [�op ,� ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The first example of a nerve construction in the mathematical literature is the simplicial approxi-  mation of a topological space by Kan (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall the original construction of the simplicial nerve  of a topological space X , where Xn = Top(∆n,X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is exactly the nerve of the functor ∆ → Top  that sends n to the n-simplex  �  x ∈ �n|  �  xi = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The following example figures into Segal’s original nerve construction for a category, and is revisited  in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define a reflexive graph to be a presheaf over the full subcategory ofCatwhose objects  are the two preorders  [0] = 0, [1] = 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This is equivalent to the free category with two parallel arrows and a common retract,  [1]  [0]  t  s  e  so that a graph in a category has an object-of-vertices V and an object-of-edges E ,along with source  and target maps s,t : E → V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' morphisms of graphs are maps (fE , fV ) that commute with the source  and target maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3, any full subcategory containing the representables will be dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define the  category of paths, Pth, to be full subcategory of graphs generated by  [0]  [0]  [1]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [1]  [n]  [t ]  [s]  [s]  [t ]  ⌟  ,n ∈ �  so that Gph([n],G ) picks out the set of paths of length n in a graph G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We call this subcategory  P : Pth → Gph, and see that NP sends a graph to its Pth-presheaf of composable paths:  En  E  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  E  V  V  t  s  t  s  ⌟  3Segal published the result, but seems to have credited the theorem to Grothendieck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  118  The pushout [n] is precisely the graph  0 −→ 1 −→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' −→ (n − 1) −→ n  where the reflexive map e at each vertex is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall the functor sending reflexive graphs to anchored bundles from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This may  be restated as a nerve construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a functor  ∂ : Weil∗  1 → Gph(� )  so that N∂ : Gpd(� ) → [Weil∗  1,� ] is the linear approximation of a reflexive graph as described in Exam-  ple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let s,t : C → M ,e : M → C be a reflexive graph, and recall that the anchored bundle C ∂ → C is  induced by the equalizer  C ∂  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e s  The category Gph is a representable tangent category, where D is the discrete graph D = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By the  Yoneda lemma, we have that  [Gph,� ](� 1,G ) = G1,[Gph,� ](� 0,G ) = G0,[Gph,� ](D × � (i),G ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Gi,  so the limit diagram defining the infinitesimal approximation of a graph becomes  C ∂  Gph(D × I ,C )  Gph(D × I ,C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (e ×I )∗  (D×e −)∗  Now observe thatGph(� )op with the dual tangentstructure from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 hasa reflexive graph  object  s,t : 1 → I ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' : I → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Construct its linear approximation as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7(iv) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' take the coequalizer of � -graphs):  D × I  D × I  ∂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  e ×I  D×e s  By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10, this determines a functor  ∂ : Weil1  1 → Gph(� )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By the continuity of the hom-functor, the nerve N∂ : Gph(� ) → [Weil∗  1,� ] lands in the category of  anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The continuity of the hom-functor ensures that this is indeed the linear approxi-  mation from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The simplicial localization in Kan (1958) has a left adjoint, the geometric realization, that constructs  a topological space using the data of a simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This realization may be constructed using a left  Kan extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  119  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let K : � → � be a � -functor for a cocomplete �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The realization of K is the left Kan  extension  A  �A  �  K  �  |−|:=LanK �  |C |K :=  � A  �(K A,C ) • K A  A nerve/realization context is a � -functor K : � → � from a small A to a cocomplete �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The adjunction between simplicial sets and topological spaces follows from general categorical  machinery as a nerve/realization context:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every nerve/realization context K : � → �, the realization of K is left adjoint to the  nerve of K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4  Nervous monads and algebroids  Transposing the characterization of algebroids from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9 to the enriched perspective intro-  duces the challenge of finding an appropriate framework to describe algebroids as enriched structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Kelly’s theory of enriched sketches (see Chapter 6 of Kelly (2005)), small � -categories equipped with  a chosen class of limits cones, seems to be a natural candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, when regarding a tangent  functor as a � -functor, the natural part  α : F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  is the unique morphism induced by the universality of T as a weighted limit, so it becomes unclear  how to translate the condition that α is a cartesian natural transformation (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A clue for how to proceed may be found in Kapranov (2007), which proved that Lie algebroids  are monadic over anchored bundles (when allowing for infinite-dimensional vector bundles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recent  work in Bourke and Garner (2019) and Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2012) has developed the appropriate notion of (en-  riched) theories that correspond to monads over general locally presentable categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A critical in-  sight is that for a filtered-colimit-preserving monad � on Set, the opposite category of the Lawvere  theory � is precisely the full subcategory of the Kleisli category whose objects are [n] =  �  n 1, and the  nerve of the inclusion  � op �→ Set�  is fully faithful;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, when the functor K is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Formally, a functor is dense whenever its nerve  behaves like the Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The theory of dense functors is developed in Chapter 5 of Kelly  (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We continue to assume that an arbitrary site of enrichment � is locally presentable as a � -  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A functor K : � → � is dense whenever the nerve of K is fully faithful, and a subcate-  gory inclusion that is dense will be called a dense subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Dense functors are poorly behaved under composition, but there is a useful cancellativity result  from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 of Kelly (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider a diagram of � -categories  X  Z  Y  K  F  J  α  120  where α is a natural isomorphism and K is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If this diagram exhibits (α, J ) as the left Kan extension  of K along F , then J is also dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If K is dense, F is fully faithful, and J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F = K , then F is a dense subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Generally speaking, we will often refer to dense subcategories rather than dense functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is  achieved by factoring the dense functor  K = �  K ′  −→ im(K ) �→ �  where im(K ) is the � -category whose objects are those of � and whose hom-objects are given by  im(K )(A,B) = �(K A,K B), so that the inclusion of im(K ) into � is fully faithful and therefore dense  by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The category of finite cardinals Σ is the full subcategory of Set whose objects are given by finite  coproducts of the terminal object, [n] =  �  n 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is a skeleton of the category of finite sets, and a  dense subcategory of Set (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Bourke and Garner (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Recall that the category of � -presheaves on � , [� op,� ] is the free colimit completion of � (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kelly (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If � is cocomplete, and K : � → � is dense, then the realization |− |K exhibits �  as a reflective subcategory of [� op,� ], and is this a locally presentable category4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Conversely, if �  is a reflective subcategory of [� op ,� ] that contains the representable functors, then � is a dense  subcategory of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7, the category of differential bundles in � is a reflective subcategory of [Λ+op,� ],  so � : Λ+ �→ [Λ+op,� ] is a dense subcategory of differential bundles in � following the argument  in the above example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Nervous theories (Bourke and Garner (2019)) are generalizations of Lawvere theories that extend to  arbitrary locally finitely presentable � -categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that a classical Lawvere theory is a bijective-  on-objects, product-preserving functor  t : Σ → �  where � is a cartesian category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nervous theories replace Σ with a dense subcategory of some locally  presentable � -category, and the product preservation condition with conditions on the nerve of the  theory map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let K : � → � be a dense � -subcategory of a locally finitely presentable �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We call  the replete image of NK in [� op,� ] the category of K -nerves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An � -theory is a bijective-on-objects  � -functor J : � → � , where each  � (J −,a) : � → �  is a K -nerve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of models for an � -theory is the pullback in � CAT:  ��  �  �  �  �  �  J ∗  NK  ⌟  (These are called concrete models in Bourke and Garner (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  4In fact, an equivalent definition of a locally presentable category is as a cocomplete category with a dense subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  121  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of models for a theory is monadic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The core of the argument is due to Weber’s  nerve theorem, found in Weber (2007), but an exposition on that result is beyond the scope of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) A functor t : Σ �→ � is a Σ-theory if and only if � is a Lawvere theory, where we use the fact that  Σ (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4) a skeletal subcategory of finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The nerve conditions in this case identify the  models of the Lawvere theory, as the nerve of Σ �→ Set sends a set to the strict product-preserving  functor [n] �→ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) As shown in Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2012) and Bourke and Garner (2019), the original nerve construction  from Segal (1974) may be restated as saying that small categories arise as models of a Pth-theory  (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The set Gph([n],G ) is the set of paths of through the graph G that have n non-  identity elements, as  G t ×s G ∼= G t ×id M id ×s G ∼= G t ×s◦e ◦sG t ◦e ◦t ×sG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Next, recall that the category ∆ may be regarded as follows:  Objects: A strict order [n] = 0 < 1 < ··· < n, for n ≥ 0, regarded as a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Maps: Functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There is a bijective-on-objects functor from Pth → ∆ that sends the graph [n] to the pre-order [n],  as every graph homomorphism between paths will be order-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now observe that a model  is precisely a simplicial set, where  X ([0]) = M ,X ([1]) = C ,X ([2]) = C t ×s C ,X ([3]) = C t ×s C t ×s C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore, the map  0  0  1  1  2  in ∆ becomes a composition map:  X ([2]) → X ([1])  C t ×s C → C  Associativity and unitality (for the section e : M → C ) follow by functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, a model of  Pth → ∆ is a small category, and a morphism is exactly a functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11, then, gives a monadicity result for � -anchored bundles over � -differential bun-  dles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � -anchored bundles are models of the theory (Λ+)op → (Weil∗  1)op (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  122  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the inclusion Λ+ → Weil1  1 is a differential bundle, so it is a nerve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then the diagram:  Anc(� )  [Weil∗  1,� ]  DBun(� )  [Λ+,� ]  ⌟  exhibits � -anchored bundles as models of a (Λ)op -theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  As a corollary, we see that the category of anchored bundles is locally presentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of anchored bundles is locally presentable, and (Weil∗  1)op �→ Anc(� ) is  dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In Bourke and Garner (2019), the authors identify exactly those monads on a locally presentable  � -category that correspond to the models of a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall the notation that the category of algebras  for a monad is �T and the category of free coalgebras is �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For a dense subcategory � �→ �, use �T  for the category of free algebras over objects in � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that the Lawvere theory  K : FinSet → �  for a filtered-colimit-preserving monad � on Set may be re-derived as the full subcategory of free alge-  bras over finite sets, KT : FinSet� �→ Set� �→ Set�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nervous monads abstract this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a locally presentable � -category, with K : � → � a dense sub-� -category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A � -monad � over � is K -nervous if  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the inclusion KT : �T �→ �T is dense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the following diagram is a pullback in � CAT:  �T  [� op  T  ,� ]  �  [� op,� ]  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11 (Bourke and Garner (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let K : � �→ � be a dense sub-� -category of a cocomplete  � -category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an equivalence of categories between � -theories and � -nervous monads on �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first step in showing that algebroids are monadic over anchored bundles is to construct a dense  subcategory of Anc(� ) that plays the role of the V -prolongations from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The enriched  framework makes this straightforward: prolongations are weighted limits, with which we may freely  complete Weil∗  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider the category Anc(� ) as a representable tangent category, and take the dual  tangent structure on Anc(� )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category � is defined as the full subcategory of Anc(� )op whose  objects are generated by the prolongations of the Yoneda embedding � : Weil∗  1 → Anc(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For any small � -category �, the free completion of � is the opposite � -category of the category of  copresheaves on �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 Thus, � is the free completion of Weil1  1 with prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Therefore, a choice  5This is the dual statement of the classical theorem that the category of presheaves is the free cocompletion of a small  category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  123  of prolongations on an anchored bundle A determines a unique functor � → � given by right Kan  extension (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kelly (1982)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The continuity of  Anc(� )(−,A) : Anc(� )op → �  ensures that Anc(� )(LV ,A) is AV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9, the category of involution algebroids in � is a full sub-� -category of [Weil1,�]  that has a forgetful functor down to the category of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A consequence of Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 is that a functor Weil1 → � is an involution algebroid if and only if the precomposition  Weil∗  1 �→ Weil1 → �  restricts to an anchored bundle, with each V sent to the V -prolongation of this anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In  other words, algebroids are models of the following enriched theorem (as in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define the � op-theory of algebroids as the functor  a : � → Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thisfunctorisbijective-on-objectsby definition, and satisfiesthe nerve condition because the V -prolongation  of the tangent bundle is T V , so each presheaf  Weil1(a−,V ) : � → �  is an � -nerve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The Weil nerve, then, translates to the following characterization of the category of involution alge-  broids in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of involution algebroids with chosen prolongations in a tangent category  � is precisely the pullback in � CAT:  I nv ∗(�)  [Weil1,�]  Anc ∗(�)  [� ,�]  ⌟  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2)  Consequently, in � involution algebroids are monadic over anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall the correspondence  ¯A : Weil1 → � ∈ � Cat  ( ˆA,α) : Weil1 → � ∈ TangCatl a x  If  Weil∗  1 → � → Weil1 → �  determines an anchored bundle (π : A → M ,ξ,λ,̺) whose V -coprolongation is ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V , so that ˆA is the  nerve of an involution algebroid by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus, we may characterize � -involution algebroids as algebras for a � -nervous monad on the  category of � -anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  124  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of involution algebroids in � is equivalent to the category of algebras  for the � -nervous monad on Anc(� ) generated by the theory  a : � → Weil1  from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13, using Theorem 19 from Bourke and Garner (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The construction of involution algebroids as the category of models for a � -theory is  remarkably similar to the original nerve construction for categories in Segal (1974), with the symmetric  nerve construction for groupoids replacing ∆ with the category of finite sets Σ (see Example 44 (iv) of  Bourke and Garner (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Each construction truncates the original category to two objects and builds  a new category with a bijective set of objects by freely adding limits to the two-object truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  The infinitesimal approximation of a groupoid  This section contains the main theorem of the chapter, namely that there is an adjunction  Gpd(� )  Inv(� )  ⊣  This adjunction follows by inducing a nerve/realization context on the category of groupoids in � :  ∂ : Weilop  1  → Gpd(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Asimplified version ofthisfunctorappeared in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4, the linearapproximation ofa reflexive  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In fact, ∂ will be exactly the free groupoid over the graph ∂ from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 (if we had  worked in the category of reflexive graphs equipped with an involution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that groupoids and  Weil spaces are both models of a sketch (recall Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The symmetry of the theories gives two  equivalent presentations of groupoids in � :  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent category of � -groupoids is equivalently  (i) the tangent category of internal groupoids in � , where the tangent structure is computed point-  wise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) the tangent category of transverse-limit-preserving functors Weil1 → Gpd(Set) with the cofree tan-  gent structure from Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Itisimportanttonote thatthe tangentstructure on � -groupoidsisrepresentable, asitisa cartesian  closed category with an infinitesimal object given by Weil1 �→ � �→ Gpd(� ) (a proof that Gpd(�) is  cartesian closed for a cartesian closed category � with sufficient limits a may be found in Section B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  of Johnstone (2002)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There are two classes of “trivial” � -groupoids that will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Every small groupoid G may be regarded as a groupoid in � as the constant functor Weil1 →  Gpd(Set) sending V to the groupoid G preserves transverse limits, we may then apply the symmetry  of theories to find a groupoid in the category of Weil spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  125  (ii) Every object M in a category has a cofree groupoid given by s,t : M = M , this is the discrete  groupoid whose only morphisms are the identity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, every Weil space M ∈ � has a corre-  sponding discrete groupoid (this will often be written M = M to denote the discrete groupoid over  M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using the cartesian closure of � and the full subcategories of trivial and discrete groupoids, we  have the following:  Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of � -groupoids is both  (i) a � -category, with powers by � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The power of a � -groupoid � = s,t : G → M by a Weil space E  is given by the � -groupoid  [E ,�] = {[E ,s],[E ,t ] : [E ,G ] → [E ,M ], [E ,e ] : [E ,M ] → [E ,G ]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) aGpd-enriched category with powersby small groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The powerof a � -groupoid � : Gpd → �  by a groupoid � is the � -groupoid  [� ,�](V ) = [� ,�(V )]Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The following two � -groupoids form the basic building block for the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Set D := � W in � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The discrete groupoid D = D represents the tangent functor internally, and the  copresheaf Gpd(D,−) : Gpd(� ) → � sends a groupoid s,t : G → M to T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The arrow groupoid is the free groupoid generated by the graph:  → •  Write the trivial � -groupoid on this groupoid as I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the power by I will send a groupoid  to its “arrow groupoid” � →, the groupoid whose objects are arrows in �, with a map u → v being  a commuting square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It follows that Gpd(I ,�) is the space of arrows of the groupoid, Gpd(I × I ,�)  the space of commuting squares, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The next proposition gives the necessary properties of I to construct the Lie derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For every groupoid � = s,t : G → M , there is an isomorphism  G t ×t G s ×t G ∼= Gpd(� )(I × I ,�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  this corresponds to the unique filler for the diagram u  v  w  ���!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='w −1◦v◦u  126  (ii) The arrow groupoid has a semigroup with a zero structure (like an infinitesimal object D), and the  multiplication is the coequalizer  1× I  I × I  I × I  I  I × 1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='×I  s×I  I ×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  I ×s  m  The dual result of part (ii) is somewhat more obvious to see, as the following fork is always an  equalizer in � for a groupoid G :  G  G □  G □  G e [s]×I  G I ×e [s]  G m  (where we write G as the object of arrows and Gpd(� )(I × I ,G ) as G □).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that G e [s]×I and G I ×e [s]  correspond to the idempotents on G □: q  u  v  w  u  u  q  q  G I ×e [s]  G e [s]×I  Also note that a commuting square is equalized by these two maps if and only if u = q = id , which  forces w = v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' these commuting squares are exactly the image of G m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The semigroup structure on I represents the map sending  Y  X  Y  X  X  X  u  u  u  Now look at the ∂ defined in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4, and take the same colimit in Gpd(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Groupoids are  monadic over reflexive graphs with an involution, so this is essentially the free groupoid over that graph  (as the free functor is a left adjoint and therefore preserves colimits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define the � -groupoid ∂ to be the � -coequalizer  D × I  D × I  ∂  e ×I  D×e [s]  where e = 0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',e [s] = i ◦ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that just as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4, ∂ is an anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The properties of the arrow groupoid make it possible to prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The object ∂ determines a cartesian � -functor  ∂ (−) : Weilop  1  → Gpd(� )  that is both an infinitesimal object in � -groupoids and an involution algebroid in Gpd(� )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  127  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the pushout powers of ξ : 1 → ∂ as ∂ (n), and for any Weil algebra V = Wn(1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ⊗ Wn(k),  write ∂ (V ) = ∂ (n1)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='×∂ (nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Composition for internal categories will be written in the diagramattic  order, writing composition as an infix semicolon “;”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' to keep it distinct from composition of morphisms  in � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The proof has two main steps:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every Weil algebra V , there is an isomorphism  ∂V ∼= ∂ (V )  where ∂V is the V -prolongation of the anchored bundle in Gpd(� )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The universal lift for the anchored bundle, given as a coequalizer  ∂ × ∂  ∂ × ∂  ∂  e ∂ ×∂  e ∂ ×∂  ⊡  is a semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  From (1) and (2), we may infer that the object ∂ is an infinitesimal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The semigroup map ⊡ is  commutative and has a zero by (2), and satisfies all of the couniversality axioms by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The proof of this step follows by induction on the width (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8) of the Weil algebra V ,  where the cases 0,1 both hold by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the case n = 2, we need only check this holds for  ∂ × ∂ ∼= ∂W W :  D × ∂  D  L2  ∂ × ∂  ∂  (id,0)  δ  ⌟  (δ,id)  (id,0)  For any G , a map X : ∂ × ∂ → G corresponds to a commuting square in T 2G of the form T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦u  v  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦v  u  so that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ v = id and e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ u = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It follows that u = (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 (i)),  and that precomposition with the uniquely induced map determines ( ¯u,v) : X → G L(2), where  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observe that any pair ( ¯u,v) : X → G L(2) determines a square T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u  v  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦v  128  where T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ v = id and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ ¯u = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 ◦ ¯u = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 ◦ ¯u, and that the bottom  horn is u = (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' now check  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ u = (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ v)  = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ v = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u  and  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ u = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v  = (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)  = (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v) = id  thus determining a map X → G ∂ ×∂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The two maps are inverse to each other, giving an isomor-  phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For the inductive case, look at the prolongations of anchored bundles and recall that  AU V ⊠M AZ = AU V ⊠� (U ,A) AU Z  Where AU V and AU Z are treated as spans over AU , so that  AU  id⊠πV  ←−−− AU V  an c U ⊠AV  −−−−−−→ T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='AV  AU  id⊠πZ  ←−−− AU Z  an c U ⊠AZ  −−−−−−→ T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='AZ  and observe that their span composition is  AU V Z  AU V  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='AU Z  AU  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='AU  T Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='AU  ̺U ⊠AV  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (̺U ⊠AZ )  ⌟  So it now suffices to prove that the diagram  ∂U × DV  ∂U × (DV × ∂Z )  ∂U × ∂V  ∂U × (∂V × ∂Z )  ⌟  is a pushout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' But by the inductive hypothesis,  DV  DV × ∂Z  ∂V  ∂V × ∂Z  ⌟  so the result follows by the cocontinuity of ∂U × (−), and Gpd(� ) is a cartesian closed category  (so X × (−) is cocontinuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  129  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that the fork  I × I  I × I  I  m  e [s]×I  I ×e [s]  is a coequalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Also note that e ∂ is the coequalizer of e ,e [s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The commutativity of colimits  then ensures that there is a multiplication map induced by the following diagram:  (D × I )2  (D × I )2  ∂ 2  (D × I )2  (D × I )2  ∂ 2  D × I  D × I  ∂  Thus the multplication is associative, is commutative, and has a zero given by 1  0×s  −→ D × I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Aan immediate corollary of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 is that the ∂ determines a nerve/ realization where the  nerve factorsthrough the categoryofinvolution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thisputsthe Lie functorintoa nerve/realization  context (recall Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Lie realization is the left Kan extension  |− |∂ = Lan∂ : Inv(� ) → Gpd(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This functor is well behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, ote that it preserved products:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The realization functor preserves products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Product preservation is a consequence of |−|∂ being the left Kan extension of a cartesian functor  along a cartesian functor (Day and Street (1995)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that this implies  � v ∂ (v) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Next, we see that the realization of an involution algebroid has the same base space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The base space of the groupoid ∂ is D v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' When constructing the colimit of groupoids, the colimit’s base space is the ordinary colimit for  the diagram of the base spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The reflector from simplicial objects to groupoids preserves products,  so it suffices to check that the base space of ∂ (n) is D(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The base space of I is 1+1, and the map e s  0 is given by δ−◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='. Since D is a discrete cubical object, its  base space is D × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus the coequalizer defining ∂ is  D + D  D + D  ∂0  0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  id+0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A map γ : D + D → M is a pair of maps γ0,γ1 : D → M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We can see that γ0 and γ1 agree at 0 (they are  both γ0(0)), and γ(0) is a constant tangent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It follows that γ1 ◦ (id |id ) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  130  Now recall the co-Yoneda lemma: for any � -presheaf A : �op → � ,  F (C ) =  � C ′∈�  �(C ,C ′) ⊗ F (C ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In particular, for an involution algebroid in � (or any � -presheaf on Weilop  1 ),  A(U ) =  � V ∈Weil1  Weilop  1 (U ,V ) × A(V ) =  � V ∈Weil1  Weil1(V,U ) × A(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For any involution algebroid A,  A(R) =  � V ∈Weil1  A(V ) × D V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Use the tangent structure on Weil1 to regard it as a � -enriched category:  D V = � (V ) = Weil1(V,−) = (U �→ Weil1(V,U ))  = (U �→ Weil1(V,U ⊗ R)) = Weil1(V,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The following computation gives the result:  A(R) =  � V  Weil1(V,R) × A(V ) =  � V  D V × A(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We can now see that the base space of an involution algebroid is isomorphic to the base space of  its realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' That is to say, the realization sends an involution algebroid over a Weil space M to a  groupoid over the Weil space M (up to isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be an involution algebroid in � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then |A|([0]) = A(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Use the Yoneda lemma, and the fact that � [0] = 1 is a small projective so that Gpd(1,−) is � -  cocontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11:  |A|([0]) = Gpd(1,|A|)  = Gpd  �  1,  � v∈Weil1  A(v) • ∂ v  �  =  � v  A(v) ×Gpd(1,∂ v)  =  � v  A(v) × D v = A(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus, as a final result, we have achieved an adjunction between the category of involution alge-  broids and groupoids in � that is product-preserving and stable over the base spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  131  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13 (The Lie Realization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a tangent adjunction between the category of involution  algebroids and groupoids in � , where each functor preserves products and the base spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Gpd(� )  Inv(� )  N∂  |−|∂  ⊣  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Just as the introduction to this thesis begins with the work of Charles Ehresmann, we  should take a moment to see how the Lie realization relates to his original research into sketch theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Rather than sketches, we use their closely related cousins, essentially algebraic theories, which in this  case are small, finitely complete � -categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We may regard Gpd(� ) and Inv(� ) as � -functor cate-  gories  Lex(�Gpd,� ),  Inv(�Gpd,� ),  respectively (where Lex means finite-limit-preserving � -functors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The functor ∂ : Weilop  1  → Gpd(� )  induces a left-exact � -functor  ˆ∂ : �Inv → �Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This means the functor from Lie groupoids to Lie algebroids is induced by a morphisms of essentially  algebraic theories, and may thus be presented as a morphism of sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  132  Chapter 6  Conclusions and future work  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Conclusions  We now take stock of what this thesis has accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Enrichedessentiallyalgebraicpresentationsofgeometricstructures  Thisthesishasbeen concerned  with categories of vector bundles and Lie algebroids in smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' While these are not models  of a limit sketch in the category of smooth manifolds, classical results such as the Serre–Swann theo-  rem or Vaintrob’s presentation of Lie algebroids indicate that these categories do have some algebraic  description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By combining the results in Chapters 2 and 3, we can see that vector bundles and Lie  algebroids are characterized by tangent categorical gadgets, differential bundles (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1) and  involution algebroids (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By combining these observations with the enriched perspec-  tive on tangent categories, the results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 exhibit vector bundles and Lie algebroids as models  of enriched sketches (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14), as can also be found in Chapter 6 of Kelly  (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that from this perspective, the construction of the Weil nerve of an involution algebroid  is mostly important as an intermediate step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  New tangent structures for Lie algebroids and groupoids  It is well known that the categories of  Lie algebroids and groupoids have tangent structures induced by post-composition with the tangent  functor, yielding the tangent algebroid and tangent groupoid, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The key result in Chapter  4 exhibiting the category of Lie algebroids as transverse-limit-preserving cartesian-tangent functors  Weil1 → SMan, then, gives the category of Lie algebroids a second tangent structure that corresponds  to Martinez’s prolongation Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Similarly, the construction of the infinitesimal nerve of a lo-  cal groupoid restricts to a new tangent structure on the category of Lie groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' in this case, the  base space is the Lie algebroid of the groupoid G , and the total space is the Lie algebroid of the arrow  groupoid G 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  While this may seem like a piece of formal category theory, it is closely related to symmetry re-  duction in classical mechanics using Lie theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This goes back to Poincaré’s celebrated note Poincaré  (1901), which introduced the Euler-Poincare formulation ofLagrangian mechanicson a manifold equipped  with a Lie group action, where the tangent bundle is replaced with a Lie algebra action (see Marle  (2013) for a modern exposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Poincaré’s formalism has been extended to Lie groupoids and Lie  algebroids in Weinstein (1996) and Martínez (2001), and the thesis Fusca (2018) investigates fluid me-  chanics through this lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This work may be interpreted as using the novel tangent structures on Lie  groupoids and Lie algebroids given in Chapters 4 and 5, and warrants further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  133  Functorial semantics of Lie theory  The work in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 puts the Lie derivative on a new formal  grounding:  (i) In the enriching category � , the Lie derivative functor now arises as a nerve/realization context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This guarantees the existence of a left adjoint—the realization—that constructs a Lie groupoid  from an algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The realization preserves products and is stable over the base space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) It is only a small extension of the work in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 to show that the Lie derivative of a groupoid  in a tangent category may generally be regarded as a weighted limit, where AV = {∂ (V ),G } for  each Weil prolongation of the involution algebroid of the groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  These appear to be new results, although in a nerve/realization approach they had previously appeared  in Lie theory in the form of Sullivan’s construction (Sullivan (1977)), which takes the differential graded  algebra of a Lie algebroid and constructs a simplicial set:  DGAop(Ω(∆n),A)  where Ω(∆n) is the de Rham cohomology of the n-simplex in cartesian space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  Future work  This section outlines various lines of research that were either cut from the thesis while writing due to  time and/or space constraints, or new lines of research that the thesis-writing process has motivated  but which have not yet been pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Enriched sketches and Mackenzie theory  Following Voronov (2012), Mackenzie theory refers to the  body of research developed by Kirill Mackenzie and his collaborators into Lie theory, particularly struc-  tures like double Lie algebroids (Mackenzie (1992)), VB-Lie algebroids (Bursztyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2016)), double  Lie groupoids and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Intuitively, a double Lie algebroid is a Lie algebroid in the category of Lie alge-  broids, while a VB-Lie algebroid is a vector bundle in the category of Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These structures  have an intuitive relationship with tensor products of sketches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, for limit sketches A,B there is  a chain of isomorphisms of categories:  Mod(A,Mod(B,Set)) ∼= Mod(A ⊗ B,Set) ∼= Mod(B,Mod(A,Set)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We have already demonstrated that the Lie algebroids and vector bundles are � -sketchable, so the  natural next step is to revisit Mackenzie theory via this lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Certain results such as the symmetry of  partial Lie derivatives from Mackenzie (1992) (given a double Lie groupoid, the order in which the Lie  functor is applied doesn’t matter) should be immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A tangent categorical formalization of mechanics  Several papers in synthetic differential geometry  have translated aspectsofLagrangian and Hamiltonian mechanicsintothe syntheticsetting(Bunge and Heggie  (1984);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nishimura (1997)), and to a degree this has made it difficult to build enthusiasm for a tangent  categorical presentation of mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These approaches generally emphasize the ability to construct  function spaces, or the use of a topos-theoretic internal language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The novel tangent structures for  Lie algebroids and groupoids presented here, however, provide a new line of inquiry: to develop a uni-  fied framework for Lagrangian or Hamiltonian mechanics in a tangent category that agrees with the  algebroid and groupoid approaches to mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  134  Appendix: Background on locally  presentable � -categories  Synthetic differential geometry uses a topos-theoretic framework, such as can be found in the final  chapters of Lavendhomme (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The internal language of a topos is a key tool, and models of syn-  thetic differential geometry are constructed using sheaf constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tangent categories have an  analagous toolbox, drawing from enriched locally presentable categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We begin with locally pre-  sentable categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' basic material on these may be found in Adámek and Rosický (1994), and on en-  riched categories in Kelly (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following class of colimits is foundational in the theory of locally  presentable categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A category is filtered whenever any finite diagram in � has a cocone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A colimit whose  diagram category is filtered is called a filtered colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that a category � is filtered if and only if  every colimit diagram � → Set commutes with every finite limit in Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Any finite category � with a terminal object is filtered: for any diagram D : � → � , the natural  transformation !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' : id ⇒ K1 is a cocone via whiskering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  �  �  �  D  K1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Any category with finite colimits is filtered, because each diagram has a colimit (and therefore a  cocone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An object in a cocomplete � is finitely presentable whenever �(C ,−) preserves filtered  colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A cocomplete category � is locally finitely presentable whenever it has a small subcategory  �f p of finitely presentable objects so that every object in � is given by the coend  C ∼=  � X ∈�f p  �(X ,C ) · X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the category Set, the subcategory of finitely presentable objects is precisely the category  of finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every set is a filtered colimit of finite sets, so that Set is locally finitely presentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Intuitively, this means that every object in � is generated by gluing together finitely presentable  objects in a canonical way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' hence the name locally finitely presentable categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The subcategory of finitely presentable objects in a cocomplete category is itself finitely cocom-  plete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This follows from the fact that �(−,X ) : �op → Set is a continuous functor (it sends colimits in �  135  to limits in Set) and also that filtered colimits commute with finite limits in Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, for any filtered  colimit of representables,  �(colD,limF ) ∼= limX limY �(D(X ),F (Y )) ∼= limY limX �(D(X ),F (Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There are several ways to characterize locally finitely presentable categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' these may be found in  Adámek and Rosický (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For a category � the following are equivalent:  (i) � is a locally finitely presentable category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For a small cocomplete category � , � = Lex(� ,Set) (where Lex means finite-limit-preserving) and  � = �op  f p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) � is the category of models for some limit sketch (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' a small category � equipped with a class of  cones � , where a model is a functor into Set sending cones to limit diagrams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) � is a reflective subcategory of a presheaf topos for some small category � (this holds tautologically  for presheaf topoi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) � is a full subcategory of a locally presentable category that is closed under limits (this relies on a  large cardinal axiom known as Vopenka’s principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  One of the remarkable aspects of the theory of locally finitely presentable categories is that its re-  sults translate over to enrichment in a monoidal category � without any significant changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A � -graph � is given by a (large) set of objects �0, and a  map  � : �0 × �0 → � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Given a collection of maps  mABC : �(B,C ) × �(A,B) → �(A,C ),  jA : I → �(A,A)  we say � is a � -category whenever the following associativity and unitality diagrams commute:  �(B,D) ⊗ �(A,B)  (�(C ,D) ⊗ �(B,C )) ⊗ �(A,B)  �(A,D )  �(C ,D) ⊗ (�(B,C ) ⊗ �(A,B))  �(C ,D ) ⊗ �(A,C )  m  m⊗id  m  α  id⊗m  I ⊗ �(B,C )  �(B,B) ⊗ �(B,C )  �(B,C )  �(B,C )  �(B,C ) ⊗ I  �(B,C ) ⊗ �(C ,C )  λ  j ⊗id  m  ρ  id⊗j  m  136  Examples of enrichment abound throughout category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Any category with biproducts is enriched in the category of commutative monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) A symmetric monoidal closed category is self-enriched using the internal hom [−,−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) A 2-category is a category enriched in Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The unit � category is the 1-object � category whose hom-object is I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) When � is symmetric monoidal, we can construct the opposite � -category of � , where �op (A,B) =  �(B,A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The new composition is defined by  mop : �op (B,C ) ⊗ �op(A,B)  σ  −→ �(B,A) ⊗ �(C ,B)  m  −→ �(C ,A) = �op (A,C )  There is a 2-category of � -categories for any � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A � -functor F : � → � is an assignment on objects F0 : �0 → �0 and a collection of  maps FA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B : �(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B) → �(F0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0B) so that the following diagrams are satisfied:  �(B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C ) ⊗ �(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B)  �(F0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C ) ⊗ �(F0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0B)  I  �(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A)  �(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C )  �(F0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C )  �(F0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0A)  m C  F ⊗F  m D  F  F  j C  A  j D  F0A  A � -natural transformation F ⇒ G is a collection of maps γA : I → �(F0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G0A) so that the following  diagram commutes:  �(F0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C )  �(F0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C ) ⊗ I  �(F0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C ) ⊗ �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0B)  �(B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C )  �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C )  �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G0C )  I ⊗ �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G0C )  �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0B) ⊗ �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G0C )  G  F  ρ  id⊗γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B  m  λ  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C ⊗id  m  � Cat is the 2-category of � -categories, � -functors and � -natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Enriched categories have a richer notion of (co)limit than ordinary categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A limit for a functor  is defined as a universal cone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, a right Kan extension along !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' :  1  �  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  KA  F  β  so that the universal property  ∀C : �(C ,limF ) ∼= [�,Set](KC ,F −)  137  is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Dually, a colimit is a universal cocone, and so a left Kan extension:  �  �  1  F  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  KA  β  where the couniversal property  ∀C : [�op ,Set](F −,KC ) ∼= �(colimF,C )  is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This does not translate over to � -categories, for a few reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An obvious one is that �  need not be cartesian monoidal, so there is no reason a � -category should have a functor � → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  notion of a weighted colimit is more appropriate:  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a � -category, and call W : � → � the weight and D : � → � the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  limit of D weighted by W , or the weighted limit {W,�}, is an object satisfying the following universal  property:  ∀C : � (W,[�,� ](F −,KC )) ∼= �(F,W ,C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Dually, given a diagram P : �op → �, the colimit of P weighted by W , W ∗P, is an object in � satisfying  the universal property  ∀C : � (W,[�,� ](KC ,F −)) ∼= �(C ,F ∗ W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) A conical (co)limit is weighted by the constant functor into the unit I , KI : � → w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This coincides  with the definition of ordinary (co)limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The power of C by an object V ∈ � is an object C V satisfying the universal property that  �(B,C V ) ∼= � (V,�(B,C )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Dually, the copower by V is given by  �(B • V,C ) ∼= � (V,�(B,C )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) In ordinary categories, the power of an object C by V is given by the (possibly infinitary) V -indexed  coproduct:  C V ∼=  �  v∈V  C  whereas the copower is the possibly infinitary V -index product of C :  V • C ∼=  �  v∈V  C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) In a symmetric monoidal closed category � , the power by � is given by [V,−] and the copower by  V ⊗ _.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Following Kelly’s work, we observe that every finite weighted limit is composed of powers and con-  ical limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every weighted (co)limit is the composition of conical (co)limits and (co)powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  138  Corollary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Weighted limits are exactly conical/ordinary limits in ordinary categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The previous definitions relating to locally presentable categories transfer over mostly unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) A monoidal closed � is presentable as a closed category if �f p is a monoidal subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given � as in (i), a weight W : � → � is filtered whenever W ⋆ (−) : �  � → � preserves finite limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) An object in a cocomplete � category is finitelypresentable when �(C ,−) preservesfiltered weighted  colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) A cocomplete � -category � is locally finitely presentable whenever every object C is the coend  C ∼=  � X ∈�f p  �(X ,C ) · X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14 (Freyd and Kelly (1972);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kelly (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' All of the previous results about locally pre-  sentable categories lift up to locally presentable � -categories, provided that � is presentable as a closed  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  139  Bibliography  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Adámek and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Rosický.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Locally presentable and accessible categories, volume 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cambridge: Cam-  bridge University Press, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 0-521-42261-2/pbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Adámek, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Borceux, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lack, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Rosický.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A classification of accessible categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Algebra, 175(1):7–30, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0022-4049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1016/S0022-4049(02)00126-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='com/science/article/pii/S0022404902001263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Spe-  cial Volume celebrating the 70th birthday of Professor Max Kelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Adámek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Gumm, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Trnková.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Presentation of set functors: a coalgebraic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Logic  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 20(5):991–1015, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0955-792X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1093/logcom/exn090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Barr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Acyclic models, volume 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' American Mathematical Society (AMS), Providence, RI, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN  0-8218-2877-0/hbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Bauer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Johnson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Osborne, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Riehl, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tebbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Directional derivatives and higher-order chain  rules for abelian functor calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  Mathematics:  Theory & Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Translated from the second Portuguese edi-  tion by Francis Flaherty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' C ∞-schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Catégories topologiques et catégories différentiables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Librairie universitaire, 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' ISSN 0960-1295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1017/S0960129502003729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Regnier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The differential lambda-calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theoretical Computer Science, 309(1–3):  1–41, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' MacLane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' General theory of natural equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  Magdalini K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Mackenzie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' (unpublished  preprint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  ISSN 0022-4049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' A groupoid approach to geometric mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Ulmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lokal präsentierbare kategorien, volume 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Springer-Verlag, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Garner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Linear logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 50(1):1–101, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Mackenzie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Johnson and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Johnstone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 1, volume 43 of Oxford Logic  Guides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Functors involving c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' ISBN 978-0-521-68738-6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Slovák.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Berlin: Springer-Verlag,  1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' The convenient setting of global analysis, volume 53 of Mathematical Surveys  and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' American Mathematical Society, Providence, RI, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 0-8218-0780-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lavendhomme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Basic concepts of synthetic differential geometry, volume 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Dordrecht: Kluwer  Academic Publishers, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 0-7923-3941-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lawvere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categorical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Topos theoretic methods in geometry, volume 30 of Various  Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', pages 1–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Aarhus: Aarhus University, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  143  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lawvere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Qualitative distinctions between some toposes of generalized graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Categories in  computer science and logic (Boulder, CO, 1987), volume 92 of Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', pages 261–299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', Providence, RI, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1090/conm/092/1003203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Leung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Classifying tangent structures using Weil algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 32(9):286–337, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theorie der transformationsgruppen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=': abschnitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Die endlichen continuirlichen gruppen der  geraden linie und der ebene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Die endlichen continuirlichen gruppen des gewöhnlichen raumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Die projectiven gruppen des gewöhnlichen raumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Untersuchungen über verschiedene arten von  gruppen des n-fach ausgedehnten raumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Untersuchungen über die grundlagen der geometrie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Allgemeine betrachtungen über endliche continuirliche gruppen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 1893, volume 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' BG Teubner, 1893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Fosco Loregian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (Co)end Calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' London Mathematical Society Lecture Note Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cambridge Uni-  versity Press, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1017/9781108778657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lucyshyn-Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' On the geometric notion of connection and its expression in tangent cate-  gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 33(28):832–866, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' MacAdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Vector bundles and differential bundles in the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Categ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', pages 1–26, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Mackenzie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Double Lie algebroids and second-order geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 94(2):180–239,  1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0001-8708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' General theory of Lie groupoids and Lie algebroids, volume 213 of London Mathe-  matical Society Lecture Note Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' ISBN 978-0-521-  49928-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Ehresmann doubles and Drinfel’d doubles for Lie algebroids and Lie bialgebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 658:193–245, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0075-4102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Mackenzie and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Integration of Lie bialgebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Topology, 39(3):445–467,  2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Proving the Jacobi identity the hard way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Geometric methods in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' XXX  workshop, Poland, June 26 – July 2, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Selected papers based on the presentations at the workshop,  pages 357–366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Categories for the working mathematician.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 4th corrected printing, volume 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' New York:  Springer-Verlag, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', Madrid, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Characteristic classes, volume 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Princeton, NJ: Princeton University  Press, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1515/9781400881826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  ISBN 0-387-97489-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' ISBN 0-387-  95543-7/hbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Synthetic Hamiltonian mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1007/BF02435785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Osoekawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' General Jacobi identity revisited again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' ISSN 0020-7748.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1007/s10773-007-9397-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Groupoids and the integration of Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Pradines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Calcul différenetiel dans la catégorie  des groupoïdes infinitésimaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Paris Sér.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Abstract tangent functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' A survey of graphical languages for monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  D Sullivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Mathématiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' 1618-1913/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An introduction to manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Universitext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' ISBN  978-1-4419-7399-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Voronov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Q-manifolds and Mackenzie theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  Centre National de la Recherche Scientifique, Paris, 1953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' ISBN 0-8218-  0261-5/hbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Regular pairings of functors and weak (co)monads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Algebra Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' ISSN 1726-3255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Springer, Berlin, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='CT]  31 Dec 2022  UNIVERSITY OF CALGARY  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  A THESIS  SUBMITTED TO THE FACULTY OF GRADUATE STUDIES  IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE  DEGREE OF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  Abstract  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  ii  Table of Contents  Abstract .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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