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+arXiv:2301.05542v1  [math.CT]  13 Jan 2023
+Tangent categories as a bridge between
+differential geometry and algebraic geometry
+G.S.H. Cruttwell∗ and Jean-Simon Pacaud Lemay†
+January 16, 2023
+Abstract
+Discussions of tangent vectors, tangent spaces, and differentials are important in both differential
+geometry and algebraic geometry. In this paper, we use the abstract notion of a tangent category to
+make some of these commonalities precise. In particular, we focus on the idea of a differential bundle in
+a tangent category, which gives a new way to compare smooth vector bundles and modules. The results
+of this paper also give a new characterization of the opposite category of modules over a commutative
+ring and the opposite category of quasicoherent sheaves.
+Contents
+1
+Introduction
+2
+2
+Tangent Categories
+4
+2.1
+Basics of Tangent Categories
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.2
+Commutative rings as a Tangent Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+2.3
+Affine schemes as a Tangent Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+3
+Differential Bundles
+18
+3.1
+Differential Bundles and Differential Objects
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+3.2
+Differential Bundles as Pre-Differential Bundles . . . . . . . . . . . . . . . . . . . . . . . . . .
+22
+3.3
+Morphisms and Categories of Differential Bundles . . . . . . . . . . . . . . . . . . . . . . . . .
+24
+4
+Differential Bundles for Commutative Rings
+26
+4.1
+From Differential Bundles to Modules
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+26
+4.2
+From Modules to Differential Bundles
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+26
+4.3
+Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+29
+5
+Differential Bundles for (Affine) Schemes
+34
+5.1
+From Differential Bundles to Modules
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+34
+5.2
+From Modules to Differential Bundles
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+34
+5.3
+Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+37
+5.4
+Differential bundles in schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+42
+6
+Future work
+44
+∗Partially supported by an NSERC Discovery grant.
+†For this research, author was financially supported by NSERC Postdoctoral Fellowship - Award #: 456414649 and a JSPS
+Postdoctoral Fellowship, Award #: P21746
+1
+
+1
+Introduction
+What exactly is the relationship between differential geometry and algebraic geometry? While there are
+many differences between these two subjects, one common thread is the use of “differential” methods.
+Indeed, discussions of tangent vectors, tangent spaces, and differentials are important in both subjects. A
+natural question to ask then is: can we precisely relate and contrast how differential geometry and algebraic
+geometry use these ideas? This paper gives one way to approach this question, via the theory of tangent
+categories, and in particular through investigating differential bundles in tangent categories.
+Tangent categories were first introduced by Rosick´y in [25], and later generalized and further developed by
+Cockett and Cruttwell in [5]. A tangent category (Definition 2.1) is a category equipped with an endofunctor
+T which for every object A associates an object T(A) that “behaves like a tangent bundle” for A. More
+precisely, this behaviour is captured through various natural transformations related to the endofunctor T
+which encode basic properties such as linearity of the derivative and symmetry of mixed partial derivatives.
+The canonical example of a tangent category is the category of smooth manifolds, where the endofunctor
+is the tangent bundle functor (Example 2.6). But there are many other interesting examples of tangent
+categories. In fact, almost any category which has some form of “differentiation” for its morphisms can be
+given the structure of a tangent category. Examples of tangent categories include:
+• Most generalizations of smooth manifolds form tangent categories.
+The category of “convenient”
+manifolds [19], the category of C∞ rings [20], and any model of synthetic differential geometry (SDG)
+[18], all give tangent categories.
+• Any Cartesian differential category [3], which formalizes differential calculus over Euclidean spaces,
+gives a tangent category. In particular, there are many examples of Cartesian differential categories
+from computer science, such as models of the differential lambda-calculus [12].
+• The category of commutative rings and the category of commutative algebras are tangent categories,
+with a particularly simple tangent structure induced by dual numbers. This example will be discussed
+in more detail below (Section 2.2).
+• “Tangent infinity” categories model ideas in Goodwillie functor calculus [1].
+The theory of tangent categories is now well-established with a rich literature. There are also many things
+one can do in an arbitrary tangent category. In particular, one can discuss:
+• Vector fields (Definition 2.10), and prove important ideas such as the Jacobi identity [6].
+• The analogue of vector bundles, known as differential bundles [8], which is one of the main structures
+of focus in this paper (Section 3).
+• Connections on differential bundles and corresponding results such as the Bianchi identities and the
+existence of geodesics [7].
+• Solutions to differential equations and dynamical systems [9].
+• Differential forms and de Rham cohomology [11].
+It is worth noting that translating these ideas from the category of smooth manifolds to an arbitrary tangent
+category is not a trivial process. The definition of a tangent category involves various natural transformations
+which appear in the category of smooth manifolds, but are not generally seen as central to differential
+geometry. In a tangent category, these natural transformations are the core part of the structure, and so
+to translate a desired notion to an arbitrary tangent category, one must translate the definition to make
+appropriate use of those natural transformations. The fact that one can do so with many of the central
+notions of differential geometry provides evidence that the abstract notion of a tangent category is indeed
+a good categorical generalization of differential geometry. A central question of tangent category theory
+2
+
+is to understand what these notions look like in the various examples of tangent categories. In categories
+which generalize the category of smooth manifolds (such as convenient manifolds, or synthetic differential
+geometry), these generally reconstruct existing definitions in these subjects. But in other areas, what these
+notions give is less obvious.
+One particular focus of this paper is examples of tangent categories in algebra and algebraic geometry,
+and investigating what tangent structure definitions look like in these particular examples. In particular,
+categories of (affine) schemes (potentially over some fixed ring or scheme). All such categories are also tangent
+categories, with the endofunctor given by the Spec of the symmetric algebra of the Kahler differentials of a
+scheme, which is precisely what Grothendieck himself called the “tangent bundle” of a scheme [15, Definition
+16.5.12.I]. While this example was mentioned in [5], as a corollary to a more general result, the tangent
+structure was not explored explicitly. One of the contributions of this paper is an explicit description of the
+natural transformations for this tangent structure (Section 2.3). This is a necessary component to further
+understand how the theory of tangent categories applies to algebraic geometry. Given this, then, the next
+important question is: what do the concepts which can be applied to any tangent category give you when
+applied to the algebraic geometry examples?
+Do they recreate existing notions?
+Do they give us new
+perspectives on existing ideas?
+The main focus of this paper is on differential bundles (Definition 3.1) in the examples of tangent cate-
+gories in algebra and algebraic geometry. Differential bundles are a central structure in tangent categories,
+as they generalize smooth vector bundles in the category of smooth manifolds (Example 3.3). However,
+intriguingly, they are defined quite differently than vector bundles. The definition of a differential bundle
+contains no mention of either vector spaces, a base field, or local triviality. Instead, their central structure is
+the existence of a vertical lift, which is a map from the total space to its tangent bundle, which satisfies a key
+universal property. That such a structure, when looked at in the category of smooth manifolds, gives exactly
+smooth vector bundles [22], is already interesting enough, as structures like the vector spaces in each fibre,
+and the local triviality, all come “for free” from the universality of the vertical lift. But what are differential
+bundles in the tangent categories of (affine) schemes? It is not immediately obvious what they should be.
+The main objective of this paper is to answer this question, and in doing also providing new and interesting
+results which hopefully opens up the possibility for many future investigations in this area. In summary, the
+main results of this paper are that:
+• Proposition 4.5 and Theorem 4.7: In the tangent category of commutative rings, differential bundles
+over a commutative ring R correspond to modules over R, and the category of differential bundles over
+R is equivalent to the category of modules over R.
+• Proposition 5.5 and Theorem 5.7: In the tangent category of affine schemes (or equivalently the opposite
+category of commutative rings), differential bundles over a commutative ring R correspond to modules
+over R, and the category of differential bundles over R is equivalent to the opposite category of modules
+over R.
+• Theorem 5.17: In the tangent category of schemes, differential bundles over a scheme A correspond to
+quasicoherent sheaves of modules over A, and the category of differential bundles over A is equivalent
+to the opposite of the category of quasicoherent sheaves of modules over A.
+These results are fascinating for several reasons. For one, they show how diverse differential bundles can
+be. In the canonical tangent category example of smooth manifolds, differential bundles are exactly smooth
+vector bundles, which includes the strict condition of local triviality. However, for these algebra or algebraic
+geometry examples of tangent categories, differential bundles still give categories of central importance
+(modules) but in which the objects have no sort of local triviality condition. Independently, these results
+are also interesting as they give a new characterization of these categories. In particular, differential bundles
+provide a novel characterization of the opposite of the category of (quasicoherent sheaves of) modules over
+a commutative. To the best of the authors’ knowledge, there is no known previous characterization of the
+opposite of the category of modules for an arbitrary commutative ring (though there are some results in
+3
+
+special cases, like characterizations of the opposite category of Abelian groups).
+These results are thus
+interesting in and of themselves.
+Even more promising than the results themselves is what future results and ideas they can lead to. As
+described above, in any tangent category one can define and prove results about connections on such bundles;
+again, when applied to the tangent category of smooth manifolds, this recreates the usual notion. But now
+via tangent categories, we get a notion of connection on modules - what do these look like? What examples
+of them are there? Do they recreate existing notions of connections in algebraic geometry? We hope to
+explore these questions in future work (Section 6), and continue to use these ideas to bridge the gap between
+differential geometry and algebraic geometry.
+Outline:
+In section 2, we review the definition of tangent categories and explore some of their basic
+examples and theory. We also explicitly describe the tangent structure of the category of (affine) schemes,
+which as noted above, has not previously been given.
+In section 3, we recall the theory of differential
+bundles in tangent categories, and review MacAdam’s characterization of differential bundles in the tangent
+category of smooth manifolds [22].
+In Section 4, we give our first major result: a characterization of
+differential bundles in the tangent category of commutative rings. Section 5 contains our most important
+results: characterizations of differential bundles in the tangent categories of affine schemes and schemes. As
+mentioned above, as far as we know, these results provide new characterizations of the opposites of categories
+of (quasicoherent sheaves of) modules. Lastly, in Section 6, we describe future work that we hope to pursue
+that builds on the ideas presented in this paper.
+Conventions:
+We assume the reader is familiar with the basic notions of category theory such as categories,
+opposite categories, functors, natural transformations, and (co)limits like (co)products, pullbacks, pushouts,
+terminal/initial objects, etc. In an arbitrary category, we denote identity maps as 1A : A −→ A, and we use
+the classical notation for composition, g ◦ f, as opposed to diagrammatic order which was used in other
+papers on tangent categories (such as in [5, 8] for example). For pullbacks and products (which recall are
+specific kinds of pullbacks), we use πj for the projections and ⟨−, −⟩ for the pairing operation which is
+induced by the universal property.
+2
+Tangent Categories
+In this section, we review the basics of tangent categories and provide full detailed descriptions of the main
+tangent categories of interest for this paper. Tangent categories were first defined by Rosick´y [25], then later
+generalized by Cockett and Cruttwell [5]. We begin by providing the full definition of a tangent category, both
+the Cockett and Cruttwell version without negatives (Definition 2.1), and the Rosick´y version with negatives
+(Definition 2.2). Afterwards, we provide a detailed description of the tangent categories of commutative
+rings (Section 2.2) and (affine) schemes (Section 2.3), the latter of which has not been previously done in
+full. We also discuss some basic, but important, concepts in these tangent categories, like tangent spaces
+(Definition 2.8) and vector fields (Definition 2.10).
+2.1
+Basics of Tangent Categories
+The following definition of a tangent category is the one provided in [8, Definition 2.1], which when compared
+to the original definition provided in [5, Definion 2.3] is the same except for the universality of the vertical
+lift which is presented as a pullback instead of an equalizer. That said, these two axiomatizations are indeed
+equivalent [8, Lemma 2.10].
+Definition 2.1 [5, Definion 2.3] A tangent structure on a category X is a sextuple T := (T, p, +, 0, ℓ, c)
+consisting of:
+(i) An endofunctor T : X −→ X, called the tangent bundle functor;
+4
+
+(ii) A natural transformation pA : T(A) −→ A, called the projection, such that for each n ∈ N, the pullback
+of n copies of pA exists, which we denote as Tn(A) with n projections πj : Tn(A) −→ T(A), for all
+1 ≤ j ≤ n, so pA ◦ πj = pA ◦ πi for all 1 ≤ i, j ≤ n, and for all m ∈ N, Tm preserves these pullbacks;
+(iii) A natural transformation1 +A : T2(A) −→ T(A), called the sum;
+(iv) A natural transformation 0A : A −→ T(A), called the zero;
+(v) A natural transformation ℓA : T(A) −→ T2(A), called the vertical lift;
+(vi) A natural transformation cA : T2(A) −→ T2(A), called the canonical flip;
+and such that:
+[T.1] (pA, +A, 0A) is an additive bundle over A [5, Definition 2.1], that is, the following diagrams commute:
+T2(A)
+πj
+�
++A
+� T(A)
+pA
+�
+A
+0A
+�
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+T(A)
+pA
+�
+T(A)
+pA
+� A
+A
+T3(A)
+⟨+A◦⟨π1,π2⟩,π3⟩ �
+⟨π1,+A◦⟨π2,π3⟩⟩
+�
+T2(A)
++A
+�
+T(A)
+⟨0A◦pA,1T(A)⟩
+�
+⟨1T(A),0A◦pA⟩
+�
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+T2(A)
++A
+�
+T2(A)
++A
+� T(A)
+T2(A)
++A
+� T(A)
+T2(A)
++A
+�◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+⟨π2,π1⟩
+� T2(A)
++A
+�
+T(A)
+(1)
+[T.2] The vertical lift ℓA preserves the additive bundle structure, that is, the following diagrams commute:
+T(A)
+ℓA
+�
+pA
+�
+T2(A)
+T(pA)
+�
+A
+0A
+� T(A)
+T2(A)
+⟨ℓA◦π1,ℓA◦π2⟩
+�
++A
+�
+TT2(A)
+T(+A)
+�
+A
+0A
+�
+0A
+�
+T(A)
+ℓA
+�
+T(A)
+ℓA
+� T2(A)
+T(A)
+T(0A)
+� T2(A)
+(2)
+1Note that by the universal property of the pullback, it follows that we can define functors Tn : X −→ X.
+5
+
+[T.3] The canonical flip cA preserves the additive bundle structure, that is, the following diagrams commute:
+T2(A)
+cA
+�
+T(pA)
+�
+T2(A)
+pT(A)
+�
+T(A)
+T2(A)
+TT2(A)
+⟨cA◦T(π1),cA◦T(π2)⟩
+�
+T(+A)
+�
+T2T(A)
++T(A)
+�
+T(A)
+T(0A)
+�
+T(A)
+0T(A)
+�
+T2(A)
+cA
+� T2(A)
+T2(A)
+cA
+� T2(A)
+(3)
+[T.4] The following diagrams commute:
+T2(A)
+cA
+�
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+T2(A)
+cA
+�
+T3(A)
+cT(A)
+�
+T(cA)
+� T3(A)
+cT(A)
+� T3(A)
+T(cA)
+�
+T2(A)
+T3(A)
+T(cA)
+� T3(A)
+cT(A)
+� T3(A)
+(4)
+[T.5] The following diagrams commute:
+T(A)
+ℓA
+�
+ℓA
+�
+T2(A)
+ℓT(A)
+�
+T(A)
+ℓA
+�
+ℓA
+�■
+■
+■
+■
+■
+■
+■
+■
+■
+T2(A)
+cA
+�
+T2(A)
+ℓT(A) �
+cA
+�
+T3(A)
+T(cA) � T3(A)
+cT(A)
+�
+T2(A)
+T(ℓA)
+� T3(A)
+T2(A)
+T2(A)
+T(ℓA)
+� T3(A)
+(5)
+[T.6] Universality of the vertical lift ℓA, that is, the following square is a pullback:
+T2(A)
+pA◦πj
+�
+νA
+� T2(A)
+T(pA)
+�
+A
+0A
+� T(A)
+(6)
+where νA : T2(A) −→ T2(A) is defined as follows:
+νA := T2(A)
+⟨ℓ◦π1,0T(A)◦π2⟩ � TT2(A)
+T(+A)
+� T(A)
+(7)
+and such that the above pullback square is preserved by all Tn.
+A tangent category is a pair (X, T) consisting of a category X equipped with a tangent structure T on X.
+Tangent categories formalize the properties of the tangent bundle on smooth manifolds from classical
+differential geometry, as we will review in Examples 2.5 and 2.6 below. An object A should be interpreted
+as a base space, and T(A) as its abstract tangent bundle. The projection pA is the analogue of the natural
+projection from the tangent bundle to its base space, making T(A) an abstract bundle over A. The sum +A
+and the zero 0A make T(A) into a generalized version of a smooth additive bundle, and so each fiber is a
+6
+
+commutative monoid. To make this more precise, this notion is captured by the concept of an additive bundle
+[5, Section 2.1], which can be defined in any arbitrary category. Briefly, additive bundles are commutative
+monoid objects in slice categories. So in the case of a tangent category, pA is a commutative monoid in the
+slice category over A with binary operation +A and unit 0A. The pullbacks of n copies of pA is required to
+sum multiple times, while preservation by Tm implies that Tm(pA) is also an additive bundle with Tm(+A)
+and Tm(0A). The top two diagrams in [T.1] simply say that +A and 0A are maps in the slice category,
+while the remaining three diagrams are the axioms of a commutative monoid: associativity of the sum, that
+zero is a unit, and commutativity of the sum.
+To explain the vertical lift, recall that in differential geometry, the double tangent bundle (that is, the
+tangent bundle of the tangent bundle) admits a canonical sub-bundle called the vertical bundle which is
+isomorphic to the tangent bundle. Thus, the vertical lift ℓA is an analogue of the embedding of the tangent
+bundle into the double tangent bundle via the vertical bundle. The canonical flip cA is an analogue of the
+natural canonical flip, which is a smooth involution on the double tangent bundle. The diagrams in [T.2]
+and [T.3] say respectively that ℓA and cA are additive bundle morphisms [5, Definition 2.2], that is, monoid
+morphisms in the slice category over A. The diagrams in [T.4] express that the canonical flip cA is a sort of
+symmetry map: the left diagram says that cA is a self-inverse isomorphism, while the right diagram is the
+Yang-Baxter associativity identity. The diagrams in [T.5] are compatibility relations between the vertical lift
+and canonical flip. The universality of the vertical lift in [T.6] is essential for generalizing desired important
+properties of the tangent bundle from differential geometry, see [5, Section 2.5] for more details on this
+axiom. Lastly, for maps, T(f) is interpreted as the differential of f, and so the functoriality of T represents
+the chain rule. The naturality of p says that T(f) is a bundle map between the tangent bundles, and the
+naturality of + and 0 implies that T(f) preserves the additive structure, while the naturality of ℓ represents
+that the differential is linear, and the naturality of c represents the symmetry of the partial differentials.
+We now add negatives to the story and obtain Rosick´y’s original definition of a tangent category [25,
+Section 2], which is essentially the same as the above definition but with an added natural transformation
+which makes each fiber of the tangent bundle into an Abelian group. In [5, 22], such a setting was simply
+called a tangent category with negatives. Here, we introduce new terminology and call such a setting a
+Rosick´y tangent category.
+Definition 2.2 [5, Section 3.3] A tangent structure with negatives on a category X is a septuple
+T := (T, p, +, 0, ℓ, c, −) consisting of:
+(i) A tangent structure (T, p, +, 0, ℓ, c) on X
+(ii) A natural transformation −A : T(A) −→ T(A), called the negative;
+such that:
+[T.N] (pA, +A, 0A, −A) is an Abelian group bundle over A [25, Section 1], that is, the following diagrams
+commute:
+T(A)
+−A
+�
+pA
+�◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+T(A)
+pA
+�
+T(A)
+pA
+�P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+⟨1T(A),−A⟩
+�
+⟨−A,1T(A)⟩
+�
+T2(A)
++A
+�
+A
+A
+0A
+�P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+T2(A)
++A
+� T(A)
+(8)
+A Rosick´y tangent category is a pair (X, T) consisting of a category X and tangent structure with negatives
+T on X.
+7
+
+In a Rosick´y tangent category, the negative nA makes each fiber an Abelian group. The left diagram of
+[T.N] says that the negative −A is a map in the slice category over A, while the right diagram is the extra
+axiom about inverses required for Abelian groups. It is also worth mentioning that in a Rosick´y tangent
+category, the universality of the vertical lift can be replaced with the following which expresses the vertical
+lift as an equalizer [5, Lemma 3.13]:
+[T.6’] The following is an equalizer diagram:
+T(A)
+ℓA
+� T2(A)
+pT(A)
+�
+T(pA)
+�
+pT(A)
+� T(A)
+pA
+� A
+0A
+� T(A)
+(9)
+and such that the above equalizer is preserved by all Tn.
+Furthermore, in a Rosick´y tangent category, the vertical lift ℓA and the canonical flip cA also preserve
+the group structure, that is, they are Abelian group bundle morphisms.
+Indeed, recall that in classical
+group theory that morphisms which preserve the group’s addition also preserve inverses. The same is true
+for Abelian group bundles in the sense that additive bundle morphisms between Abelian group bundles
+automatically also preserve inverses. Explicitly:
+Lemma 2.3 In a Rosick´y tangent category (X, T), the following diagrams commute:
+T(A)
+ℓA
+�
+−A
+�
+T2(A)
+T(−A)
+�
+T2(A)
+cA
+�
+T(−A)
+�
+T2(A)
+−T(A)
+�
+T(A)
+ℓA
+� T2(A)
+T2(A)
+cA
+� T2(A)
+(10)
+We now discuss Cartesian tangent categories, which are tangent categories that also have finite products
+which are compatible with the tangent structure. The extra coherences for a Cartesian tangent category
+ensure that the tangent bundle of a product is naturally isomorphic to the product of the tangent bundles
+and that the tangent bundle of the terminal object is the terminal object.
+Definition 2.4 [5, Definition 2.8] A Cartesian (Rosick´y) tangent category is a (Rosick´y) tangent
+category (X, T) such that X has finite products, with binary product × and terminal object ∗, and that the
+canonical natural transformation ⟨T(π1), T(π2)⟩ : T(A × B) −→ T(A) × T(B) is a natural isomorphism, and
+the unique map T(∗) −→ ∗ is an isomorphism, so T(A × B) ∼= T(A) × T(B) and T(∗) ∼= ∗.
+The main example of a (Cartesian) tangent category is the category of smooth manifolds, where the
+tangent structure is induced by the tangent bundle of a smooth manifold. This example provides a direct
+link between tangent categories and differential geometry. Here we review in full the tangent structure on
+the subcategory of Euclidean spaces, as it is simpler to describe in detail. For lists of other examples of
+tangent categories see [8, Example 2.2] and [14, Example 2].
+Example 2.5 The category of Euclidean spaces and smooth functions is a Cartesian Rosick´y tangent cate-
+gory where the tangent structure is induced by the total derivative of smooth functions. Let SMOOTH be the
+category whose objects are Euclidean spaces Rn and whose maps are smooth functions. SMOOTH has finite
+products where the binary product is given by the standard Cartesian product, Rm×Rn = Rm+n, and where
+the terminal object is the singleton, R0 = {0}. To define the tangent structure, recall that for a smooth
+8
+
+function F : Rm −→ Rn, which is actually an n-tuple F = ⟨f1, . . . , fn⟩ of smooth functions fi : Rm −→ R, that
+the total derivative of F is the smooth function D[F] : Rm × Rm −→ Rn defined as the sum of the partial
+derivatives of the fi:
+D[F](⃗x, ⃗y) =
+� m
+�
+j=1
+∂f1
+∂uj
+(⃗x)yj, . . . ,
+m
+�
+j=1
+∂fn
+∂uj
+(⃗x)yj
+�
+The total derivative D[F] can also be expressed in terms of the Jacobian of F. We define a Rosick´y tangent
+structure T on SMOOTH as follows:
+(i) The endofunctor T : SMOOTH −→ SMOOTH is defined on a Euclidean space as T(Rn) = Rn × Rn and
+on a smooth function F : Rm −→ Rn as the smooth function T(F) : Rm × Rm −→ Rn × Rn defined as:
+T(F)(⃗x, ⃗y) =
+�
+F(⃗x), D[F](⃗x, ⃗y)
+�
+(ii) The projection pRn : Rn × Rn −→ Rn is defined as the projection of the first component:
+pRn(⃗x, ⃗y) = ⃗x
+(iii) The pullback of m copies of pRn is given by taking the product of m + 1 copies of Rn:
+Tm(Rn) = Rn × . . . × Rn
+�
+��
+�
+m+1 times
+and where the projection πj : Tm(Rn) −→ Rn × Rn projects out the first and j-th components:
+πj(⃗x, ⃗y1, . . . , ⃗
+ym) = (⃗x, ⃗yj)
+(iv) The sum +Rn : Rn × Rn × Rn −→ Rn × Rn adds the second and third components:
++Rn(⃗x, ⃗y,⃗z) = (⃗x, ⃗y + ⃗z)
+(v) The zero 0Rn : Rn −→ Rn × Rn inserts the zero vector into the second component:
+0Rn(⃗x) = (⃗x,⃗0)
+(vi) The vertical lift ℓRn : Rn × Rn −→ Rn × Rn × Rn × Rn inserts zero vectors into the middle components:
+ℓRn(⃗x, ⃗y) = (⃗x,⃗0,⃗0, ⃗y)
+(vii) The canonical flip cRn : Rn × Rn × Rn × Rn −→ Rn × Rn × Rn × Rn flips the middle two components:
+cRn(⃗x, ⃗y,⃗z, ⃗w) = (⃗x,⃗z, ⃗y, ⃗w)
+(viii) The negative −Rn : Rn × Rn −→ Rn × Rn makes the second component negative:
+−Rn(⃗x, ⃗y) = (⃗x, −⃗y)
+So T = (T, p, s, z, l, c, n) is a tangent structure with negatives on SMOOTH.
+Lastly, we also have that
+Rn × Rn × Rm × Rm ∼= Rn × Rm × Rn × Rm and R0 × R0 ∼= R0. So (SMOOTH, T) is a Cartesian Rosick´y
+tangent category. In fact, SMOOTH is a Cartesian differential category [3], and every Cartesian differential
+category is a Cartesian tangent category by generalizing the above construction [5, Proposition 4.7].
+9
+
+Example 2.6 The category of smooth manifolds is a Cartesian Rosick´y tangent category where the tangent
+structure is given by the classical tangent bundle (here we follow [27, Defn. 5.9] and allow our manifolds
+to have different dimensions in different connected components). Let SMAN be the category whose objects
+are (finite-dimensional real) smooth manifolds M and whose maps are smooth functions between them. For
+a smooth manifold M, for each point x ∈ M let Tx(M) be the tangent space at x. Then recall that the
+tangent bundle of M is the smooth manifold T(M) which is the (disjoint) union of each tangent space:
+T(M) :=
+�
+x∈M
+Tx(M)
+This induces a functor T : SMAN −→ SMAN which is part of a tangent structure with negatives T on SMAN,
+which in local coordinates is defined in the same way as in Example 2.6. So (SMAN, T) is a Cartesian Rosick´y
+tangent category, for which (SMOOTH, T) is a sub-Cartesian Rosick´y tangent category.
+There are many ways to make new tangent categories from existing ones, but one of the most fundamental
+(assuming the existence of certain well-behaved limits) is by slicing.
+We will use this construction, in
+particular, to construct tangent categories of algebras from tangent categories of rings.
+Proposition 2.7 [5, Proposition 2.5] Suppose that (X, T) is a tangent category, and A is an object of X.
+Then the slice category X/A can be given the structure of a tangent category, where the tangent bundle of an
+object f : X −→ A, TA(f), is given by the pullback
+TA(f)
+�
+�
+T X
+T (f)
+�
+A
+0A
+� T A
+(assuming such pullbacks exist and are preserved by each T n).
+We end this section by reviewing two simple concepts from differential geometry that can be generalized
+to any (Cartesian) tangent category: vector fields and tangent spaces. In the examples above, vector fields
+and tangent spaces correspond precisely to their namesakes from classical differential geometry. Below, we
+will also discuss these tangent spaces and vector fields for the tangent categories of commutative rings and
+(affine) schemes.
+Recall that in a category with finite products, a point of an object A is a map from the terminal object
+to A. In a Cartesian tangent category, the tangent space at a point is given by the pullback (if it exists) of
+said point and tangent structure projection.
+Definition 2.8 [5, Definition 4.13] In a Cartesian tangent category (X, T), if A is an object of X and
+a : ∗ −→ A is a point of A, then the tangent space of A at a is an object Ta(A) equipped with a map
+πa : Ta(A) −→ T(A) such that the following diagram is a pullback:
+Ta(A)
+πa
+�
+�
+T(A)
+pA
+�
+∗
+a
+� A
+and is preserved by all Tn for all n ∈ N.
+Example 2.9 In SMOOTH, a point of Rn in the categorical sense corresponds precisely to elements of
+⃗x ∈ Rn. So in (SMOOTH, T), the tangent space of Rn at ⃗x ∈ Rn is T⃗x(Rn) = Rn and π⃗x : Rn −→ Rn × Rn
+is defined as the injection π⃗x(⃗y) = (⃗x, ⃗y). Similarly, in SMAN, a point of a smooth manifold M in the
+categorical sense is precisely a point x ∈ X. So in (SMAN, T), the tangent space of M at x ∈ M is the
+classical tangent space Tx(M) = M.
+10
+
+Tangent spaces are commutative monoid objects, where monoid structure is induced by the tangent
+bundle [5, Theorem 4.15], and in a Cartesian Rosick´y tangent category, tangent spaces are also Abelian
+group objects. The category of tangent spaces is a Cartesian differential category [5, Corollary 4.16]. Also
+observe that, trivially, for the terminal object ∗ the only point is the identity 1∗ : ∗ −→ ∗ and that T1∗(∗) = ∗.
+We now turn our attention to vector fields. In a tangent category, a vector field is simply a section of the
+tangent bundle’s projection.
+Definition 2.10 [5, Definition 3.1] In a tangent category (X, T), a vector field on an object A of X is a
+map v : A −→ T(A) which is a section of pA, that is, the following diagram commutes:
+A
+v
+�
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+T(A)
+pA
+�
+A
+Example 2.11 In (SMOOTH, T), a vector field on Rn is given by a smooth function v : Rn −→ Rn ×Rn such
+that v(⃗x) = (⃗x, f(⃗x)) for some smooth functions f : Rn −→ Rn. Therefore, vector fields in (SMOOTH, T)
+correspond precisely to endomorphisms in SMOOTH. In (SMAN, T), vector fields in the tangent category
+sense correspond precisely to vector fields in the usual sense.
+In any tangent category, the zero 0A : A −→ T(A) is a vector field, and the map νA : T2(A) −→ T2(A)
+from [T.6] induces a vector field for the tangent bundle LA : T(A) −→ T2(A) which generalizes the canonical
+vector field on the tangent bundle, also called the Liouville vector field [5, Section 3.1]. One can also define
+the sum of vector fields using the sum + [5, Proposition 3.2], as well as a new category whose objects are
+vector fields and whose maps commute with vector fields in the obvious way [9, Definition 2.8], and said
+category of vector fields is also a tangent category [9, Proposition 2.10]. In a Rosick´y tangent category, it
+is possible to define the Lie Bracket of vector fields [5, Definition 3.14], which in particular also satisfies the
+Jacobi identity [6]. Vector fields can also be used to describe differential equations, dynamical systems, and
+their solutions in a tangent category [9].
+There are numerous other interesting properties and concepts that one can discuss in a tangent category
+such as the fact the tangent bundle functor T admits a canonical monad structure [5, Proposition 3.4] or
+the notion of a representable tangent category [5, Section 5], which provides a link to synthetic differential
+geometry (SDG). Furthermore, there are many other concepts from differential geometry that one can
+generalize to a tangent category, such as connections [7], and de Rham cohomology [11].
+2.2
+Commutative rings as a Tangent Category
+In this section, we provide a full description of the tangent category of commutative rings, whose tangent
+bundle is given by the ring of dual numbers. This was one of the main examples in Rosick´y’s original paper
+[25, Example 2].
+By a commutative ring, we mean a commutative, unital, and associative ring.
+For a
+commutative ring R and a, b ∈ R, we denote the addition by a + b, the zero by 0 ∈ R, the negation by −a,
+the multiplication by ab, and the unit by 1 ∈ R. Let CRING be the category whose objects are commutative
+rings and whose maps are ring morphisms.
+For a commutative ring R, its ring of dual numbers is the commutative ring R[ε] defined as follows:
+R[ε] = {a + bε| ∀a, b ∈ R, ε2 = 0}
+where a and bε will be used respectively as shorthand for a + 0ε and 0 + bε. Then R[ε] is a commutative
+ring with multiplication induced by ε2 = 0, that is, the addition, multiplication, and negative are defined
+respectively as follows:
+(a + bε) + (c + dε) = (a + c) + (b + d)ε
+(a + bε)(c + dε) = ac + (ad + bc)ε
+−(a + bε) = −a − bε
+and where the zero is 0 and the unit is 1. Using the ring of dual numbers, we define a tangent structure with
+negatives
+T
+= (
+T
+, p, +, 0, ℓ, c, −) on CRING.
+11
+
+(i) The endofunctor
+T
+: CRING −→ CRING maps a commutative ring R to its ring of dual numbers
+T
+(R) = R[ε] and a ring morphism f : R −→ S is sent to the ring morphism
+T
+(f) : R[ε] −→ S[ε] defined
+as follows:
+T
+(f)(a + bε) = f(a) + f(b)ε
+(ii) The projection pR : R[ε] −→ R sends ε to zero, and so is defined as projecting out the first component:
+pR(a + bε) = a
+To describe the pullbacks of the projection, first recall that CRING is a complete category, and therefore
+all pullbacks exist in CRING. In particular, if R and R′ are commutative rings, then for any ring morphism
+f : R′ −→ R, the general construction of a pullback of n copies of f in CRING is given by:
+R′
+n = {(x1, . . . , xn)| xj ∈ E s.t. f(xi) = f(xj) for all 1 ≤ i, j ≤ n}
+and whose ring structure is given coordinate-wise. However for the projection of the ring of dual numbers,
+one can instead describe these pullbacks in terms of multivariable dual numbers. So for a commutative ring
+R, define R[ε1, . . . , εn] as follows:
+R[ε1, . . . , εn] = {a + b1ε1 + . . . + bnεn| ∀a, bi ∈ R and εiεj = 0}
+Then R[ε1, . . . , εn] is a commutative ring whose structure is defined in the obvious way, so in particular the
+multiplication is induced by εiεj = 0. We leave it as an exercise for the reader to check for themselves that
+R[ε1, . . . , εn] is indeed isomorphic to the pullback of n copies of pR. So we can continue to describe the
+tangent structure as follows:
+(iii) The pullback of n copies of pR is given by
+T
+n(R) = R[ε1, . . . , εn] and where πj : R[ε1, . . . , εn] −→ R[ε]
+sends εj to ε and the other nilpotents to zero, that is, πj projects out the first component and j-th
+nilpotent component:
+πj(a + b1ε1 + . . . + bnεn) = a + bjε
+(iv) The sum +R : R[ε1, ε2] −→ R[ε] maps both ε1 and ε2 to ε, which results in adding the nilpotent parts
+together:
++R(a + bε1 + cε2) = a + (b + c)ε
+(v) The zero 0R : R −→ R[ε] is the injection of R into its ring of dual numbers:
+0R(a) = a
+(vi) The negative −R : R[ε] −→ R[ε] maps ε to −ε, which results in making the nilpotent part negative:
+−R(a + bε) = a − bε
+It may be worth briefly discussing what additive bundles and Abelian group bundles are in CRING.
+In
+fact, Abelian group bundles in CRING were characterized by Beck in [2, Example 8], where it was explained
+that Abelian group bundles over a commutative ring are equivalent to modules over said commutative ring.
+Furthermore, it turns out that in CRING, additive bundles are always Abelian group bundles, so we also get
+an equivalence between additive bundles and modules.
+To describe the vertical lift and the canonical flip, let us first describe
+T
+2(R), the ring of dual numbers
+of the ring of dual numbers in terms of two nilpotent elements ε and ε′:
+T
+2(R) = R[ε][ε′] = {a + bε + cε′ + dεε′| ∀a, b, c, d ∈ R and ε2 = ε′2 = 0}
+where the multiplication is induced by ε2 = ε′2 = 0. So we define:
+12
+
+(vii) The vertical lift ℓR : R[ε] −→ R[ε][ε′] maps ε to ε′, and so maps the nilpotent component to the outer
+nilpotent component:
+ℓR(a + bε) = a + bεε′
+(viii) The canonical flip cR : R[ε][ε′] −→ R[ε][ε′] swaps ε and ε′, and so interchanges the middle nilpotent
+components:
+cR(a + bε + cε′ + dεε′) = a + cε + bε′ + dεε′
+So
+T
+= (
+T
+, p, +, 0, ℓ, c, −) is a tangent structure with negatives on CRING. Also, CRING has finite products
+where the binary product is given by the Cartesian product of rings R×S, where recall that the ring structure
+is given pointwise, and where the terminal object is the zero ring 0. We also have that (R×S)[ε] ∼= R[ε]×S[ε]
+and 0[ε] ∼= 0. So we have that:
+Lemma 2.12 (CRING,
+T
+) is a Cartesian Rosick´y tangent category.
+Remark 2.13 This tangent category construction nicely generalizes to other natural settings:
+• Instead of commutative rings, we could have considered commutative semirings (also called rigs, for
+rings without negatives), which are of particular interest throughout all of computer science. So the
+category of commutative semirings will be a Cartesian tangent category via dual numbers, but not a
+Rosick´y tangent category since we dropped negatives.
+• For any commutative (semi)ring R, the category of commutative R-algebras will also be a Cartesian
+tangent category; this follows from Proposition 2.7.
+• The Eilenberg-Moore category of a codifferential category (or dually the opposite category of the
+coEilenberg-Moore category of a differential category) is a Cartesian tangent category [10, Theorem
+22], whose tangent structure is indeed a generalization of the above dual numbers tangent structure. In
+fact, these tangent categories of commutative (semi)rings/algebras are precisely the Eilenberg-Moore
+categories of the appropriate polynomial models of codifferential categories.
+We conclude this section by discussing tangent spaces and vector fields.
+We first explain how in
+(CRING,
+T
+), there are no non-trivial tangent spaces. Indeed, since the zero ring 0 is the terminal object, a
+point of a commutative ring R would be a ring morphism of type f : 0 −→ R. However, since ring morphisms
+are required to preserve the unit and zero, and these are the same in the zero ring, we would have that
+1 = f(1) = f(0) = 0, which implies 1 = 0 in R. But the zero ring is the only ring for which the zero and
+unit are equal. Therefore, it follows that the only ring morphism with domain 0 is the identity 10 : 0 −→ 0,
+and the tangent space at this point is 0.
+Lemma 2.14 In (CRING,
+T
+), the only commutative ring with a tangent space at a point is the zero ring 0
+at the identity 10 : 0 −→ 0, and
+T
+10(0) = 0.
+Next we discuss vector fields in (CRING,
+T
+) and explain how they correspond precisely to derivations.
+Recall that for a commutative ring R, a derivation on R is a linear map D : R −→ R which satisfies the product
+rule: D(ab) = aD(b) + D(a)b. A commutative differential ring is a pair (R, D) consisting of a commutative
+ring R and derivation D on R. On the other hand, for a commutative ring R, a vector field on R is a ring
+morphism v : R −→ R[ε] such that pR ◦v = 1R, which implies that v(a) = a+Dv(a)ε for some Dv : R −→ R. It
+is a well-known result that ring morphisms v : R −→ R[ε] such that pR ◦ v = 1R, i.e. vector fields, correspond
+precisely to derivations on R. Indeed, if v is a vector field, then since it preserves the multiplication, it follows
+that Dv satisfies the product rule. Thus Dv is a derivation and (R, Dv) is a commutative differential ring.
+Conversely, given a derivation D : R −→ R on R, define the vector field vD : R −→ R[ε] as vD(a) = a + D(a)ε.
+Since these constructions are inverses of each other, we obtain the desired equivalence.
+Lemma 2.15 For a commutative ring R, vector fields on R in (CRING,
+T
+) are in bijective correspondence
+with derivations on R.
+13
+
+Therefore, it follows that the category of vector fields of (CRING,
+T
+) is equivalent to the category of
+commutative differential rings. Thus by [9, Proposition 2.10], the category of commutative differential rings
+is also a Cartesian Rosick´y tangent category, whose tangent structure is induced by dual numbers as above.
+Lastly, the tangent category Lie bracket corresponds precisely to the standard Lie bracket of derivations,
+[D1, D2] = D1 ◦ D2 − D2 ◦ D1.
+2.3
+Affine schemes as a Tangent Category
+In this section, we discuss the tangent categories of (affine) schemes, where the tangent structure is induced
+by K¨ahler differentials. In this paper, by the category of affine schemes we mean the opposite category of
+commutative rings CRINGop. As such, we will be working directly with CRINGop, so we will write in terms
+of commutative rings R instead of affine schemes Spec(R). So in this section, we provide a full description of
+the tangent structure on CRINGop. While CRINGop has been mentioned as an example of a tangent category
+in other papers [5, 14], a full explicit description of its tangent structure has not previously been given in
+the literature. We provide such a description here as it will both be useful for the story of this paper, and
+for future work on applications of tangent category theory in algebraic geometry.
+To give a tangent structure with negatives on CRINGop, we must give a “co-tangent structure with nega-
+tives” on CRING. Explicitly, this means giving a functor T : CRING −→ CRING and natural transformations,
+and so in particular ring morphisms, of type: pR : R −→ T(R), +R : T(R) −→ T2(R), 0R : T(R) −→ R,
+ℓR : T2(R) −→ T(R), cR : T2(R) −→ T2(R), and −R : T(R) −→ T(R).
+For a commutative ring R, its tangent bundle T(R) is the free symmetric R-algebra over its modules of
+K¨ahler differentials Ω(R):
+T(R) := SymR
+�
+Ω(R)
+�
+=
+∞
+�
+n=0
+Ω(R)⊗s
+R
+n = R ⊕ Ω(R) ⊕
+�
+Ω(R) ⊗s
+R Ω(R)
+�
+⊕ . . .
+where ⊗s
+R is the symmetrized tensor product over R. In [15, Definition 16.5.12.I], Grothendieck calls T(R)
+the “fibr´e tangente” (French for tangent bundle) of R, while in [17, Section 2.6], Jubin calls T(R) the tangent
+algebra of R. For the story of this paper, it will be useful to have a more explicit description of T(R). So
+equivalently, T(R) is the free R-algebra generated by the set {d(a)| a ∈ R} modulo the equations:
+d(1) = 0
+d(a + b) = d(a) + d(b)
+d(ab) = ad(b) + bd(a)
+which are the same equations that are modded out to construct the module of K¨ahler differentials of R. Thus,
+an arbitrary element of T(R) is a finite sum of monomials of the form ad(b1) . . . d(bn). So the ring structure of
+T(R) is essentially the same as polynomials. Furthermore, T(R) also has a universal property similar to that
+of the module of K¨ahler differentials, but instead for algebras. For a commutative R-algebra A, a derivation
+evaluated in A is a linear map D : R −→ A which satisfies the product rule D(ab) = a · D(b) + b · D(a), where
+· is the R-module action on A. Now T(R) is a commutative R-algebra A via the R-module action given by
+multiplication, a · w = aw for a ∈ R and w ∈ T(R). Then the map d : R −→ T(R), which maps a to d(a), is
+a derivation and is universal in the sense that for any commutative R-algebra A equipped with a derivation
+D : R −→ A, there exists a unique R-algebra morphism D♭ : T(R) −→ A such that D♭(d(a)) = D(a).
+Example 2.16 In may be useful to work out some basic examples of tangent bundles:
+(i) For the ring of integers Z, its tangent bundle is itself: T(Z) = Z
+(ii) For the polynomial ring in n-variables Z[x1, . . . , xn], its tangent bundle is the 2n-variable polynomial
+ring: T(Z[x1, . . . , xn]) = Z[x1, . . . , xn, d(x1), . . . , d(xn)], with no added assumptions on the variables
+d(xi);
+(iii) For coordinate rings of varieties, that is, the polynomial rings quotiented by some finitely generated ideal
+Z[x1, . . . , xn]/⟨p(⃗x), . . . , q(⃗x)⟩, its tangent bundle is the polynomial ring’s tangent bundle quotiented
+14
+
+by the ideal generated by the same polynomials and their total derivatives:
+T
+�
+Z[x1, . . . , xn]/⟨p(⃗x), . . . , q(⃗x)⟩
+�
+=Z[x1, . . . , xn, d(x1), . . . , d(xn)]/⟨p(⃗x), . . . , q(⃗x), d(p)(⃗x), . . . , d(q)(⃗x)⟩
+For example, for Z[x, y]/⟨x2 − xy2⟩, its the tangent bundle is:
+T
+�
+Z[x, y]/⟨x2 − xy2⟩
+�
+= Z[x, y, d(x), d(y)]/⟨x2 − xy2, 2xd(x) − y2d(x) − 2xyd(y)⟩
+To define the necessary ring morphisms for the tangent structure, note that T(R) is generated by a and
+d(a), for all a ∈ R. Therefore, to define ring morphisms with domain T(R), it suffices to define them on
+generators a and d(a). Using this to our advantage, we can define a tangent structure on CRINGop.
+(i) The endofunctor T : CRING −→ CRING maps a commutative ring R to its tangent bundle T(R) as
+defined above, and a ring morphism f : R −→ S is sent to the ring morphism T(f) : T(R) −→ T(S)
+defined as on generators as follows:
+T(f)(a) = f(a)
+T(f)(d(a)) = d(f(a))
+(ii) The projection pR : R −→ T(R) is defined as the injection of R into T(R):
+pR(a) = a
+We also need the pullback of n copies of pR in CRINGop, which means that we need the pushout of n
+copies of pR in CRING. Recall that CRING is cocomplete, and therefore admits all pushouts. To describe
+the desired pushout, note that for commutative rings R and R′, any ring morphism f : R −→ R′ induces an
+R-algebra structure on R′ via the R-module action a · x = f(a)x for all a ∈ R and x ∈ R′. Therefore, the
+pushout of n-copies of f : R −→ R′ is given by taking the tensor product over R of n copies of R′ viewed as
+an R-moudle: R′
+n := R′ ⊗R . . . ⊗R R′
+�
+��
+�
+n times
+, where ⊗R is the tensor product over R of R-modules. The induced
+R-algebra structure on T(R) via pR is precisely given by multiplication, as described above.
+(iii) The pushouts of n copies of pR is given by Tn(R) := T(R) ⊗R . . . ⊗R T(R)
+�
+��
+�
+n times
+and where the pushout
+injections πj : T(R) −→ Tn(R) injects T(R) into the j-th component:
+πj(w) = 1 ⊗R . . . ⊗R 1 ⊗R w ⊗R 1 ⊗R . . . ⊗R 1
+To describe the sum, zero, and negative, let us first explain what additive bundles and Abelian group
+bundles are in CRINGop. An additive bundle over R in CRINGop corresponds precisely to a commutative R-
+bialgebra over the tensor product ⊗R. The sum and zero of the additive bundle are the comultiplicaiton and
+counit respectively of the R-coalgebra structure. The fact that they are ring morphisms and they commute
+with the additive bundle’s projection will further imply that we obtain a commutative R-bialgebra. An
+Abelian group bundle over R in CRINGop corresponds precisely to a commutative R-Hopf algebra over the
+tensor product ⊗R. The negative of the Abelian group bundle gives the antipode for the R-Hopf algebra
+structure. So to give the sum, zero, and negative for our tangent structure, we must give a R-Hopf algebra
+structure on the tangent bundle T(R). Luckily, free symmetric R-algebras have a canonical commutative
+R-Hopf algebra.
+(iv) The sum +R : T(R) −→ T(R) ⊗R T(R) is given by the comultiplication of the canonical R-coalgebra
+structure of free symmetric R-algebras, that is, defined on generators as follows:
++R(a) = a ⊗R 1 = 1 ⊗R a
++R(d(a)) = d(a) ⊗R 1 + 1 ⊗R d(a)
+15
+
+(v) The zero 0R : T(R) −→ R is the counit of the canonical R-coalgebra structure of free symmetric
+R-algebras, that is, defined on generators as follows:
+0R(a) = a
+0R(d(a)) = 0
+(vi) The negative −R : T(R) −→ T(R) is the antipode of the canonical R-Hopf algebra structure of free
+symmetric R-algebras, that is, defined on generators as follows:
+−R(a) = a
+−R(d(a)) = −d(a)
+To describe the vertical lift and the canonical flip, let us first describe T2(R) as the free commutative
+R-algebra over the set {d(a)| a ∈ R} ∪ {d′(a)| a ∈ R} ∪ {d′d(a)| a ∈ R}, modulo the relations:
+d(1) = 0
+d(a + b) = d(a) + d(b)
+d(ab) = ad(b) + bd(a)
+d′(1) = 0
+d′(a + b) = d′(a) + d′(b)
+d′(ab) = ad′(b) + bd′(a)
+d′d(1) = 0
+d′d(a + b) = d′d(a) + d′d(b)
+d′d(ab) = d(b)d′(a) + ad′d(b) + d(a)d′(b) + bd′d(a)
+These relations say that d and d′ are derivations, and that d′d is the composite of derivations. Therefore, to
+define a ring morphism with domain T2(R), it suffices to define it on the four types of generators a, d(a),
+d′(a), and d′d(a) for a ∈ R.
+(vii) The vertical lift ℓR : T2(R) −→ T(R) is defined on generators as follows:
+ℓR(a) = a
+ℓR(d(a)) = 0
+ℓR(d′(a)) = 0
+ℓR(d′d(a)) = d(a)
+(viii) The canonical flip cR : T2(R) −→ T2(R) is defined on generators as follows:
+cR(a) = a
+cR(d(a)) = d′(a)
+cR(d′(a)) = d(a)
+cR(d′d(a)) = d′d(a)
+So T = (T, p, +, 0, ℓ, c, −) is a tangent structure with negatives on CRINGop. Also, CRING has finite coprod-
+ucts where the binary coproduct is given by the tensor product of rings R ⊗ S and where the initial object is
+the ring of integers Z. Thus CRINGop has finite products. Since one has that Ω(R⊗S) ∼= R⊗Ω(S)⊕S⊗Ω(R)
+and Ω(Z) ∼= 0, it follows that that T(R ⊗ S) ∼= T(R) ⊗ T(S) and T(Z) ∼= Z. So we have that:
+Lemma 2.17 (CRINGop, T) is a Cartesian Rosick´y tangent category.
+Remark 2.18 It is worth mentioning that the tangent structures for commutative rings and the tangent
+structure for affine schemes are related to one another via the adjoint tangent structure theorem. Per [5,
+Proposition 5.17], if the tangent bundle of a tangent category has a left adjoint, then this induces a tangent
+structure on the opposite category where the left adjoint is the tangent bundle. This is precisely what is
+happening between the tangent categories (CRING,
+T
+) and (CRINGop, T). Indeed, T : CRING −→ CRING is a
+left adjoint to
+T
+: CRING −→ CRING, so we have a natural bijective correspondence between ring morphisms
+of type R −→ R′[ε] and T(R) −→ R′.
+Explicitly, given a ring morphism f : R −→ R′[ε], which is of the
+form f(a) = f1(a) + f2(a)ε, define the ring morphism f ♯ : T(R) −→ R′ on generators as f ♯(a) = f1(a)
+and f ♯(d(a)) = f2(a), and conversely, given a ring morphism g : T(R) −→ R′, define the ring morphism
+g♭ : R −→ R′[ε] as: g♭(a) = g(a) + g(d(a))ε.
+Furthermore, (CRINGop, T) is not only a Cartesian Rosick´y
+tangent category but also a representable tangent category [5, Section 5.2]. Briefly, a representable tangent
+category is a Cartesian category whose tangent bundle functor T is a representable functor T ∼= (−)D for
+some object D, that is, T is a right adjoint for the functor ×D. The object D is called the infinitesimal object
+[5, Definition 5.6], and note that the opposite category of a representable category is a tangent category with
+tangent bundle functor
+× D (where × becomes a coproduct in the opposite category). (CRINGop, T) is a
+representable tangent category where the infinitesimal object is the ring dual of numbers for the integers,
+Z[ε]. So we have that T(R) ∼= RZ[ε] in CRINGop, and
+T
+(R) ∼= R ⊗ Z[ε] in CRING.
+16
+
+Using Proposition 2.7, we then get that for each commutative ring R, the slice category CRINGop/R is
+also a tangent category. But as is well-known, this slice category is equal to the (opposite of) the category
+of commutative R-algebras. Thus we have:
+Corollary 2.19 For any commutative ring R, the opposite of the category of algebras over R, (CALGR)op
+is a Cartesian Rosick´y tangent category, with tangent functor given as in Proposition 2.7.
+In particular, this tangent structure on objects is given by (the symmetric algebra of) the “relative” Kahler
+differentials: this is the same construction as seen earlier in this section, except with d(r) = 0 for all r ∈ R.
+Remark 2.20 There are also other ways to generalize the tangent category structure of CRINGop:
+• The opposite category of commutative semirings and the opposite category of commutative algebras
+over a commutative (semi)ring will be representable tangent categories via K¨ahler differentials in a
+similar fashion.
+• The coEilenberg-Moore category of a differential category (or dually the opposite category of the
+Eilenberg-Moore category of a codifferential category) is a (representable) tangent category [10, The-
+orem 26], and these tangent categories of opposite categories of commutative (semi)rings/algebras
+are precisely the coEilenberg-Moore categories of the appropriate polynomial models of differential
+categories.
+The category of schemes SCH is also a Cartesian Rosick��y tangent category in a similar fashion to the
+category of affine schemes. Indeed, recall that a scheme is by definition the gluing of affine schemes. So the
+tangent bundle of a scheme is defined as the gluing of the tangent bundles of each affine piece of the said
+scheme. Full details can be found in [14, Example 2.(iii)].
+Proposition 2.21 (SCH, T) is a Cartesian Rosick´y tangent category.
+As with affine schemes, we can apply Proposition 2.7 to tangent structure on each category of relative
+schemes:
+Corollary 2.22 For each scheme A, the slice category SCH/A has the structure of a Cartesian Rosick´y
+tangent category.
+We now discuss tangent spaces in (CRINGop, T). The terminal object in CRINGop is the initial object
+in CRING, which is the integers Z. So for a commutative ring R, a point of R in CRINGop corresponds to
+a ring morphism r : R −→ Z, which are better known as augmentations. Thus a tangent space of R at a
+point r in (CRINGop, T) corresponds to the pushout of r and pR in CRING. This essentially amounts to
+applying r on the R parts of the tangent bundle T(R). So let im(r) = {r(a)| ∀a ∈ R} be the image of r,
+which is a sub-ring of Z. Then the tangent space Tr(R) can explicitly be described as the free commutative
+im(r)-algebra generated by the set {d(a)| a ∈ R} modulo the equations:
+d(1) = 0
+d(a + b) = d(a) + d(b)
+d(ab) = r(a)d(b) + r(b)d(a)
+So an arbitrary element of Tr(R) is a finite sum of monomials of the form r(a)d(b1) . . . d(bn).
+Example 2.23 Here are some examples of tangent spaces:
+(i) The only point for Z is the identity, and T1Z(Z) = Z;
+(ii) For the polynomial ring in n-variables Z[x1, . . . , xn], points correspond precisely to evaluating polyno-
+mials at a point ⃗a ∈ Zn. However, for any point, the tangent space at that point is the polynomial ring
+in n-variables, T⃗a(Z[x1, . . . , xn]) = Z[d(x1), . . . , d(xn)];
+17
+
+(iii) For a polynomial ring quotiented by a finitely generated ideal Z[x1, . . . , xn]/⟨p(⃗x), . . . , q(⃗x)⟩, points
+correspond to points ⃗a ∈ Zn which are solutions to p(⃗a) = 0, ..., q(⃗a) = 0. The resulting tangent
+bundle is the polynomial ring in n-variables quotiented by the ideal generated by the evaluation of
+the polynomials d(p)(⃗x), . . . , d(q)(⃗x) in the xi variables at ⃗a. For example, for Z[x, y]/⟨xy⟩, its tangent
+bundle is
+Z[x, y, dx, dy]/⟨xdy + ydx⟩
+and thus its tangent space at the point (1, 1) is
+Z[dx, dy]/⟨dy + dx⟩
+which is isomorphic to the polynomial ring in one variable. However, at the point (0, 0), evaluating the
+relation xdy + ydx gives 0, and so in this case the tangent space is simply
+Z[dx, dy]
+the polynomial ring in two variables.
+This example is important as it shows that in this tangent
+category, the tangent spaces at different points can have different dimensions, even if the original space
+is connected. This is not true in the tangent category of smooth manifolds.
+Next, we discuss vector fields in (CRINGop, T) and explain how, like in the commutative ring case, they
+correspond precisely to derivations. This is expected since for a tangent category whose tangent bundle
+admits a left adjoint, vector fields for the left adjoint correspond precisely to vector fields for the right
+adjoint. So for a commutative ring R, a vector field on R in (CRINGop, T) is a ring morphism v : T(R) −→ R
+such that v ◦ pR = 1R, which implies that v(a) = a. Then define Dv : R −→ R as Dv(a) = v(d(a)). It
+follows that Dv is a derivation and (R, Dv) is a commutative differential ring. Conversely, given a derivation
+D : R −→ R on R, define the vector field vD : T(R) −→ R as the ring morphism defined on generators as
+vD(a) = a and vD(d(a)) = D(a). Since these constructions are inverses of each other, we obtain the desired
+equivalence.
+Lemma 2.24 For a commutative ring R, vector fields on R in (CRINGop, T) are in bijective correspondence
+with derivations on R.
+Therefore, it follows that the category of vector fields of (CRINGop, T) is equivalent to the opposite
+category of commutative differential rings. Thus by [9, Proposition 2.10], the opposite category of commu-
+tative differential rings is a Cartesian Rosick´y tangent category whose tangent structure is given by the free
+symmetric algebra over the module of K¨ahler differentials as above.
+3
+Differential Bundles
+In this section, we review differential bundles, as introduced by Cockett and Cruttwell in [8]. Differential
+bundles generalize the notion of smooth vector bundles to an arbitrary tangent category. We provide the full
+definition (Definition 3.1), review how smooth vector bundles do indeed correspond to differential bundles
+(Example 3.3, as shown by MacAdam in [22]), and also consider differential object (Definition 3.5), which
+are differential bundles over the terminal object. We then discuss morphisms between differential bundles
+and categories of differential bundles (Section 3.3). We will also review MacAdam’s notion of pre-differential
+bundles, as introduced in [22], which then allows for an equivalent alternative characterization of differential
+bundles that requires fewer structure data (Section 3.2). MacAdam’s characterization of differential bundles
+as pre-differential bundles is very important for the story of this paper, as we will use this approach to
+characterize differential bundles for commutative rings and (affine) schemes.
+18
+
+3.1
+Differential Bundles and Differential Objects
+One way of understanding the definition of a differential bundle over an object A in a tangent category is
+that it is a generalization of the structure involving the projection, sum, zero, and vertical lift on the tangent
+bundle T(A) in the definition of a tangent category (Definition 2.1).
+Definition 3.1 [8, Definion 2.3] In a tangent category (X, T), a differential bundle is a quadruple
+E = (q : E −→ A, σ : E2 −→ E, z : A −→ E, λ : E −→ T(E)) consisting of:
+(i) Objects A and E of X;
+(ii) A map q : E −→ A of X, called the projection, such that for each n ∈ N, the pullback of n copies of q
+exists, which we denote as En with n projection maps πj : En −→ E, for all 1 ≤ j ≤ n, so q ◦ πj = q ◦ πi
+for all 1 ≤ i, j ≤ n, and for all m ∈ N, Tm preserves these pullbacks;
+(iii) A map σ : E2 −→ E of X, called the sum;
+(iv) A map z : A −→ E of X, called the zero;
+(v) A map λ : E −→ T(E) of X, called the lift;
+and such that:
+[DB.1] (q, σ, z) is an additive bundle over A [5, Definition 2.1], that is, the following diagrams commute:
+E2
+πj
+�
+σ
+� E
+q
+�
+A
+z
+�
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+E
+q
+�
+E
+q
+� A
+A
+E3
+⟨σ◦⟨π1,π2⟩,π3⟩�
+⟨π1,σ◦⟨π2,π3⟩⟩
+�
+E2
+σ
+�
+E
+⟨z◦q,1E⟩
+�
+⟨1E,z◦q⟩
+�
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+E2
+σ
+�
+E2
+σ
+�◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+◆
+⟨π2,π1⟩
+� E2
+σ
+�
+E2
+σ
+� E
+E2
+σ
+� E
+E
+(11)
+[DB.2] The lift λ preserves the additive structure, that is, the following diagrams commute:
+E
+λ
+�
+q
+�
+T(E)
+T(q)
+�
+A
+0A
+� T(A)
+E2
+⟨λ◦π1,λ◦π2⟩
+�
+σ
+�
+T(E2)
+T(σ)
+�
+A
+0A
+�
+0A
+�
+T(A)
+T(z)
+�
+E
+λ
+� T(E)
+T(A)
+λ
+� T(E)
+(12)
+19
+
+[DB.3] The lift λ preserves the other possible additive structure, that is, the following diagrams commute:
+E
+λ
+�
+q
+�
+T(E)
+pE
+�
+A
+z
+� E
+E2
+⟨λ◦π1,λ◦π2⟩
+�
+σ
+�
+T2(E)
++E
+�
+A
+z
+�
+z
+�
+E
+0E
+�
+E
+λ
+� T(E)
+E
+λ
+� T(E)
+(13)
+[DB.4] The following diagram commutes:
+E
+λ
+�
+λ
+�
+T(E)
+T(λ)
+�
+T(E)
+ℓE
+� T2(E)
+(14)
+[DB.5] The following square is a pullback:
+E2
+q◦πj
+�
+µ
+� T(E)
+T(q)
+�
+A
+0A
+� T(A)
+(15)
+where µ : E2 −→ T(E) is defined as follows:
+µ := E2
+⟨λ◦π1,0E◦π2⟩ � T(E2)
+T(σ)
+� T(E)
+(16)
+and such that the above pullback square is preserved by all Tn.
+If E = (q : E −→ A, σ : E2 −→ E, z : A −→ E, λ : E −→ T(E)) is a differential bundle in (X, T), we also say
+that E is a differential bundle over A. When there is no confusion, differential bundles will be written as
+E = (q : E −→ A, σ, z, λ), and when the objects are specified simply as E = (q, σ, z, λ).
+Differential bundles generalize smooth vector bundles over smooth manifolds in the context of a tangent
+category, as we will review in Example 3.3 below. If E = (q : E −→ A, σ, z, λ) is a differential bundle, the
+object A is interpreted as a base space and the object E as the total space. The projection q is the analogue
+of the bundle projection from the total space to the base space, making E an “abstract bundle over A”.
+The sum σ and the zero z make each fiber into a commutative monoid, more precisely, make the projection
+q into additive bundle [5, Section 2.1], which recall is a commutative monoid in the slice category over A,
+which is what the diagrams of [DB.1] state.
+The lift and its universal property [DB.5] is related to local triviality for smooth vector bundles. Indeed,
+given a smooth vector bundle, each fibre Ea (for a ∈ A) is a vector space, and hence the tangent space at
+any point of said fibre is isomorphic to the fibre itself. As a result, it follows that the tangent bundle of
+the total space E, T E, admits a sub-bundle which is isomorphic to E itself. The lift λ (sometimes called
+the small vertical lift [24, Section 1]) is an analogue of the resulting embedding of the total space into its
+20
+
+tangent bundle. The fibres of the tangent bundle of the total space admit two monoid structures: one being
+the canonical one of a tangent bundle, and the other induced by the monoid structure of the fibres of the
+smooth vector bundle. Then [DB.2] and [DB.3] say the lift λ preserves both of these monoid structures, or
+more precisely, that λ is an additive bundle morphism [5, Definition 2.2], which recall is a monoid morphism
+in the slice category. Lastly, [DB.4] is the compatibility between the lift of a smooth vector bundle and the
+vertical lift of the tangent bundle.
+In any tangent category, for ever object A, its tangent bundle T(A) is a differential bundle over A, that
+is, (pA : T(A) −→ A, +A, 0A, ℓA) is a differential bundle [8, Example 2.4]. Also, if (q : E −→ A, σ, z, λ) is a
+differential bundle, then the tangent bundle of E is a differential bundle over the tangent bundle of A, that
+is,
+�
+T(q) : T(E) −→ T(A), T(σ), T(z), cE ◦ T(λ)
+�
+is a differential bundle [8, Lemma 2.5]. For more properties
+of differential bundles, see [8, Section 2.4].
+We now define differential bundles with negatives, which are differential bundles with an added structure
+map which makes each fiber into an Abelian group.
+Definition 3.2 [22, Lemma 5] In a tangent category (X, T), a differential bundle with negatives is a
+quintuple E = (q : E −→ A, σ : E2 −→ E, z : A −→ E, λ : E −→ T(E), ι : E −→ E) consisting of:
+(i) A differential bundle (q : E −→ A, σ, z, λ) in (X, T);
+(ii) A map ι : E −→ E of X, called the negative;
+such that:
+[D.N] (q, σ, z, ι) is an Abelian group bundle over A [25, Section 1], that is, the following diagrams commute:
+E
+ι
+�
+q
+�❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+E
+q
+�
+E
+q
+�❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+⟨1E,ι⟩
+�
+⟨ι,1E⟩
+�
+E2
+σ
+�
+A
+A
+z
+�❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+E2
+σ
+� E
+(17)
+If E = (q : E −→ A, σ : E2 −→ E, z : A −→ E, λ : E −→ T(E), ι : E −→ E) is a differential bundle with
+negatives in (X, T), we also say that E is a differential bundle over A.
+Note that a differential bundle can have negatives in any arbitrary tangent category and that the negative
+is necessarily unique. As was shown in [22], we will review below that in a Cartesian Rosick´y tangent category,
+every differential bundle comes equipped with a (necessarily unique) negative (Proposition 3.9). Therefore in
+a Cartesian Rosick´y tangent category, differential bundles are the same as differential bundles with negatives.
+We now review how smooth vector bundles correspond precisely to differential bundles (with negatives)
+in the tangent category of smooth manifolds, as was shown by MacAdam in [22, Theorem 1]. The result is
+somewhat surprising, as the definition of differential bundles contains no mention of local triviality.
+Example 3.3 For a smooth vector bundle q : E −→ M, as in the definition of manifolds, we allow the vector
+bundle to have different dimensions in different connected components. Given such a smooth vector bundle,
+in local co-ordinates we can represent an element of E as a pair (m, v) and an element of T E as a quadruple
+(m, v, w, a), and we can define a lift λ : E −→ T E by λ(m, v) = (m, 0, 0, v). Proposition 4.1.2 of [22] shows
+that this indeed gives a differential bundle in the category of smooth manifolds. To go the other direction,
+MacAdam proves a general result: in a Rosick´y tangent category, every differential bundle is a retract of a
+pullback of a tangent bundle [22, Corollary 3.1.4] Then every differential bundle is a smooth vector bundle,
+since the tangent bundle is a smooth vector bundle, and smooth vector bundles are closed under pullbacks
+and retracts.
+21
+
+The following result about differential bundles in slice tangent categories is easy to check:
+Proposition 3.4 If (X, T) is a tangent category with an object A which satisfies the requirements of Propo-
+sition 2.7, then in the slice tangent category X/A, a differential bundle over f : X −→ A is the same as a
+differential bundle over X in (X, T).
+We conclude this section by briefly discussing differential objects, which are the differential bundles over
+the terminal object. Differential objects were first defined in [5, Definition 4.8], before the introduction
+of differential bundles. Differential objects are quite important since they provide the link from tangent
+categories to Cartesian differential categories [3]. Indeed, the subcategory of differential objects (and all
+maps between them) is a Cartesian differential category [5, Theorem 4.11]. Conversely, every Cartesian
+differential category is a tangent category in which every object is a differential object [8, Lemma 3.13].
+From this, it follows that we obtain an adjunction between the category of Cartesian tangent categories and
+the category of Cartesian differential categories [5, Theorem 4.12]. Later, it was shown in [8, Proposition
+3.4] that differential objects were precisely the same thing as differential bundles over the terminal object.
+Since the focus of this paper is on differential bundles, we take this approach to defining differential objects.
+Definition 3.5 [8, Proposition 3.4] In a Cartesian tangent category (X, T), a differential object is a
+differential bundle over the terminal object ∗.
+Alternatively, a differential object can also be described as an object A equipped with maps ˆp : T(A) −→ A,
++ : A×A −→ A, and 0 : ∗ −→ A such that (A, +, 0) is a commutative monoid, T(A) ∼= A×A via pA and ˆp, and
+the diagrams from [5, Definition 4.8] commute. In a Cartesian Rosick´y tangent category, every differential
+object is automatically an Abelian group.
+Example 3.6 In (SMAN, T), the differential objects are precisely the Euclidean spaces since in particular
+T(Rn) = Rn × Rn. Therefore (SMOOTH, T) is equivalent to the resulting Cartesian differential category of
+differential objects of (SMAN, T).
+3.2
+Differential Bundles as Pre-Differential Bundles
+In this section, we review MacAdam’s pre-differential bundles as introduced in [21]. These allow for an
+alternative characterization of differential bundles, which in particular requires less data and axioms. Indeed,
+MacAdam cleverly observed that in the definition of a differential bundle, the sum (and negative), and any
+axioms involving it, can be replaced by a pullback square, called Rosick´y’s universality diagram. From this
+special pullback, the sum (and negative) for the differential bundle can be constructed from the sum (and
+negative) of the tangent bundle. MacAdam then introduced pre-differential bundles, which are defined using
+only the projection, zero, and lift, and showed that differential bundles are precisely pre-differential bundles
+such that the n-fold pullbacks of the projection exist and the Rosick´y’s universality diagram holds. This
+pre-differential bundle approach to differential bundles is quite useful since it requires less data and fewer
+axioms to check when one wants to construct a differential bundle. This will be particularly useful when we
+will characterize differential bundles for commutative rings and (affine) schemes.
+The definition of a pre-differential bundle is what remains from the definition of a differential bundle
+after removing the sum (and negative) and any required pullback.
+Definition 3.7 [21, Definition 10] In a tangent category (X, T), a pre-differential bundle is a triple
+(q : E −→ A, z : A −→ E, λ : E −→ T(E)) consisting of objects A and E of X, and maps q : E −→ A, z : A −→ E,
+and λ : E −→ T(E) of X such that the following diagrams commute:
+A
+z
+�
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+E
+q
+�
+E
+λ
+�
+q
+�
+T(E)
+pE
+�
+A
+z
+�
+z
+�
+E
+0E
+�
+E
+λ
+�
+λ
+�
+T(E)
+T(λ)
+�
+A
+A
+z
+� E
+E
+λ
+� T(E)
+T(E)
+ℓE
+� T2(E)
+(18)
+22
+
+If (q : E −→ A, z : A −→ E, λ : E −→ T(E)) is a pre-differential bundle, we say that it is a pre-differential
+bundle over A. When there is no confusion, pre-differential bundles will be denoted as (q : E −→ A, z, λ),
+and when the objects are specified simply as (q, z, λ).
+By definition, the projection, zero, and lift of a differential bundle gives a pre-differential bundle, since
+the diagrams in Definition 3.7 all appear in Definition 3.1. On the other hand, a pre-differential bundle
+is a differential bundle precisely when the pullback of n copies of the projection exists and certain squares
+are pullbacks [22, Proposition 6]. Since the main tangent categories of interest in this paper are Cartesian
+Rosick´y tangent, we review when a pre-differential bundle is a differential bundle in this setting, where
+only one square is required to be a pullback [22, Corollary 3]. This pullback is called Rosick´y’s universality
+diagram, and using the pullback universal property, we can construct the sum and negative for the differential
+bundle [22, Lemma 5].
+Proposition 3.8 [22, Corollary 3] Let (X, T) be a Cartesian Rosick´y tangent and let (q : E −→ A, z, λ) be a
+pre-differential bundle in (X, T) such that:
+(i) For each n ∈ N, the pullback of n copies of q exists, which we denote as En with n projection maps
+πj : En −→ E, for all 1 ≤ j ≤ n, so q ◦ πj = q ◦ πi for all 1 ≤ i, j ≤ n, and for all m ∈ N, Tm preserves
+these pullbacks;
+(ii) The following commuting square is a pullback, called the Rosick´y’s universality diagram:
+E
+λ
+�
+q
+�
+T(E)
+⟨T(q),pE⟩
+�
+A
+⟨0A,z⟩
+� T(A) × E
+(19)
+and for all m ∈ N, Tm preserves this pullback.
+Then define the maps σ : E2 −→ E and ι : E −→ E respectively as follows using the universal property of the
+above pullback:
+E2
+σ
+�❑
+❑
+❑
+❑
+❑
+❑
+πj
+�
+⟨λ◦π1,λ◦π2⟩
+� T2(E)
++E
+�
+E
+ι
+�❏
+❏
+❏
+❏
+❏
+❏
+q
+�
+λ
+� T(E)
+−E
+�
+E
+λ
+�
+q
+�
+T(E)
+⟨T(q),pE⟩
+�
+E
+λ
+�
+q
+�
+T(E)
+⟨T(q),pE⟩
+�
+E
+q
+� A
+⟨0A,z⟩
+� T(A) × E
+A
+⟨0A,z⟩
+� T(A) × E
+(20)
+Then E = (q, σ, z, λ, ι) is a differential bundle with negatives over A.
+Conversely, if E = (q : E −→ A, σ, z, λ) is a differential bundle in a Cartesian Rosick´y tangent category,
+then (q, z, λ) is a pre-differential bundle which satisfies (i) and (ii) in Proposition 3.8. Furthermore, the
+induced sum as constructed in Proposition 3.8 is precisely the sum σ one started with, and so (q, σ, z, λ, ι) is
+a differential bundle with negatives. Similarly, if (q : E −→ A, σ, z, λ, ι) is a differential bundle with negatives,
+then the induced negative as constructed in Proposition 3.8 is precisely the negative ι one started with.
+Therefore, in a Cartesian Rosick´y tangent category, every differential bundle is in fact a differential bundle
+with negatives. In conclusion, we have the following equivalence:
+Proposition 3.9 [21, Proposition 6 & Corollary 3] In a Cartesian Rosick´y tangent category (X, T), the
+following are in bijective correspondence:
+23
+
+(i) Differential bundles;
+(ii) Differential bundles with negatives;
+(iii) Pre-differential bundles that satisfy (i) and (ii) in Proposition 3.8.
+3.3
+Morphisms and Categories of Differential Bundles
+In this section, we discuss morphisms between differential bundles.
+There are two possible kinds: one
+where the base objects can vary and one where the base object is fixed. The former is used as the maps
+in the category of all differential bundles of a tangent category, while the latter is used in the category of
+differential bundles over a specified object. In either case, a differential bundle morphism is asked to preserve
+the projections and the lifts of the differential bundles.
+Definition 3.10 [8, Definion 2.3] Let (X, T) be a (Rosick´y) tangent category.
+(i) Let E = (q : E −→ A, σ, z, λ) and E′ = (q′ : E′ −→ A′, σ′, z′, λ′) be differential bundles in (X, T). A
+differential bundle morphism (f, g) : E −→ E′ is a pair of maps f : E −→ E′ and g : A −→ A′ such
+that the following diagram commutes:
+E
+f
+�
+q
+�
+E′
+q′
+�
+E
+λ
+�
+f
+� E′
+λ′
+�
+A
+g
+� A′
+T(E)
+T(f)
+� T(E′)
+(21)
+Let DBun
+�
+(X, T)
+�
+be the category whose objects are differential bundles in (X, T), maps are differential
+bundle morphisms between them, identity maps are pairs of identity maps (1E, 1A) : E −→ E, and
+composition is defined point-wise, that is, (f, g) ◦ (h, k) = (f ◦ h, g ◦ k).
+(ii) Let A be an object in X and E = (q : E −→ A, σ, z, λ) and E′ = (q′ : E′ −→ A, σ′, z′, λ′) be differential
+bundles over A in (X, T). A differential bundle morphism f : E −→ E′ over A is a map f : E −→ E′
+such that (f, 1A) : E −→ E′ is a differential bundle morphism.
+Explicitly, the following diagrams
+commute:
+E
+f
+�
+q
+�❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+❖
+E′
+q′
+�
+E
+λ
+�
+f
+� E′
+λ′
+�
+A
+T(E)
+T(f)
+� T(E′)
+(22)
+Let DBunT[A] be the category whose objects are differential bundles over A in (X, T) and whose maps
+are differential bundle morphisms over A between them, and where identity maps and composition are
+the same as in X.
+Differential bundle morphisms automatically preserve the sum and zero, and negatives if they exist:
+Lemma 3.11 [8, Proposition 2.16] Let (X, T) be a tangent category, and let E = (q : E −→ A, σ, z, λ) and
+E′ = (q′ : E′ −→ A′, σ′, z′, λ′) be differential bundles in (X, T), and let (f, g) : E −→ E′ be a differential
+bundle morphism between them. Then (f, g) is an additive bundle morphism [5, Definition 2.2], that is, the
+following diagrams commute:
+E2
+σ
+�
+⟨f◦π1,f◦π2⟩
+� E′
+2
+σ′
+�
+A
+z
+�
+g
+� A′
+z′
+�
+E
+f
+� E′
+E
+f
+� E′
+(23)
+24
+
+Similarly, let (q : E −→ A, σ, z, λ, ι) and (q′ : E′ −→ A′, σ′, z′, λ′, ι′) be differential bundles with negatives in
+(X, T), and let (f, g) : (q, σ, z, λ) −→ (q′, σ′, z′, λ′) be a differential bundle morphism between the underlying
+differential bundles. Then f preserves the negative, that is, the following diagram commutes:
+E
+ι
+�
+f
+� E′
+ι′
+�
+E
+f
+� E′
+(24)
+Other properties of differential bundle morphisms can be found in [8, Section 2.5].
+Note that since differential bundle morphisms preserve negatives, the notion of a morphism between
+differential bundles with negatives is the same as a differential bundle morphism. Therefore for a Rosick´y
+tangent category, it follows from Proposition 3.9 that its category of differential bundles is the same as its
+category of differential bundles with negatives. As such, abusing notation slightly, for a Rosick´y tangent cat-
+egory (X, T), we will consider DBun
+�
+(X, T)
+�
+and DBunT[A] to be the categories whose objects are differential
+bundles with negatives and whose maps are differential bundle morphisms.
+For a Cartesian (Rosick´y) tangent category, its category of differential objects is the category of differential
+bundles over the terminal object. Note that this is not the same as the Cartesian differential category of
+differential objects [5, Theorem 4.11], since in that category the morphisms are not required to preserve the
+lift, sum, zero, or negative.
+Definition 3.12 Let (X, T) be a Cartesian (Rosick´y) tangent category. Define DIFF
+�
+(X, T)
+�
+to be the cate-
+gory of differential objects and differential bundle morphisms over ∗ between them, DIFF
+�
+(X, T)
+�
+= DBunT[∗].
+We conclude this section by discussing differential bundle isomorphisms. If (X, T) is a (Rosick´y) tangent
+category, then a differential bundle isomorphism is an isomorphism in the category DBun
+�
+(X, T)
+�
+. Explicitly,
+this is a differential bundle morphism (f, g) such that there exists a differential bundle morphism of opposite
+type (f −1, g−1) such that (f, g) ◦ (f −1, g−1) = (1, 1) and (f −1, g−1) ◦ (f, g) = (1, 1). By definition of the
+composition in DBun
+�
+(X, T)
+�
+, this is precisely the same as requiring that f and g are isomorphisms in X.
+Similarly, for an object A, a differential bundle isomorphism over A is an isomorphism in the category
+DBunT[A], which is a differential bundle morphism f which is an isomorphism in X whose inverse f −1 is also
+a differential bundle morphism. We will now prove the converse, that if the underlying maps of a differential
+bundle morphism are isomorphisms in the base category, then their inverses are also a differential bundle
+morphism. This will allow us to reduce the number of things to check when characterizing differential bundles
+in various tangent categories.
+Lemma 3.13 Let (X, T) be a tangent category.
+(i) Let E = (q : E −→ A, σ, z, λ) and E′ = (q′ : E′ −→ A′, σ′, z′, λ′) be differential bundles in (X, T), and let
+(f, g) : E −→ E′ be a differential bundle morphism between them. If f : E −→ E′ and g : A −→ A′ are
+isomorphisms in X, then (f −1, g−1) : E′ −→ E is a differential bundle morphism. Therefore, (f, g) is a
+differential bundle isomorphism with inverse (f −1, g−1).
+(ii) Let A an object in X, and let E = (q : E −→ A, σ, z, λ) and E′ = (q′ : E′ −→ A, σ′, z′, λ′) be differential
+bundles over A in (X, T), and let f : E −→ E′ be a differential bundle morphism over A between them.
+If f : E −→ E′ is an isomorphism in X, then f −1 : E′ −→ E is a differential bundle morphism over A.
+Therefore f is a differential bundle isomorphism with inverse f −1.
+Proof: For (i), we compute:
+g−1 ◦ q′ = g−1 ◦ q′ ◦ f ◦ f −1 = g−1 ◦ g ◦ q ◦ f −1 = q ◦ f −1
+25
+
+The fact that (f −1, g−1) is then an isomorphism in the category of differential bundles follows from [8,
+Lemma 2.18.ii]. For (ii), if f is a differential bundle morphism over A, then (f, 1A) is a differential bundle
+morphism. The identity is always an isomorphism, so if f is also an isomorphism, it follows from (i) that
+(f −1, 1A) is a differential bundle morphism, which implies that f −1 is a differential bundle morphism over
+A as desired.
+✷
+4
+Differential Bundles for Commutative Rings
+In this section, we characterize differential bundles (with negatives) in the tangent category of commutative
+rings and prove that they correspond precisely to modules (Proposition 4.5). To go from a module to a
+differential bundle, we use a semi-direct product to build a sort of ring of dual numbers from said module
+(Lemma 4.2). To go from a differential bundle to a module, we take the kernel of the projection (Lemma 4.1).
+We then obtain that the category of differential bundles is equivalent to the category of modules (Theorem
+4.7 and Theorem 4.8). We will also explain how the only differential object is the zero ring (Corollary 4.6).
+For a commutative ring R, for a (left) R-module M, unless otherwise specified, we denote the action by
+a · m, where a ∈ R and m ∈ M.
+4.1
+From Differential Bundles to Modules
+We begin by unpacking what a differential bundle with negatives would consist of in (CRING,
+T
+). First
+recall that (CRING,
+T
+) is a Cartesian Rosick´y tangent category, so by Proposition 3.9, differential bundles
+are the same thing as differential bundles with negatives. Also, as discussed in Section 2.2, CRING admits
+all pullbacks, so for any ring morphism q : E −→ R between commutative rings, the general construction of
+a pullback of n copies of q in CRING is given by:
+En = {(x1, . . . , xn)| xj ∈ E s.t. q(xi) = q(xj) for all 1 ≤ i, j ≤ n}
+and whose ring structure is given coordinate-wise. In particular, E2 = {(x, y)| x, y ∈ E s.t. q(x) = q(y)}. So
+for a commutative ring R, a differential with negatives over R in (CRING,
+T
+) would consist of a commutative
+ring E and five ring morphisms: q : E −→ R, σ : E2 −→ E, z : R −→ E, λ : E −→ E[ε], and ι : E −→ E. These
+also need to satisfy the equalities and properties of Definition 3.1, many of which we will expand further in
+the proof of Lemma 4.4 below. To obtain an R-module, we take the kernel of the projection q.
+Lemma 4.1 Let R be a commutative ring and E = (q : E −→ R, σ, z, λ, ι) be a differential bundle with
+negatives over R in (CRING,
+T
+). Then the kernel of the projection ker(q) = {x| q(x) = 0} is an R-module
+with action a · x = z(a)x.
+Proof: Since q : E −→ R is a ring morphism, this induces an R-module structure on E with action
+a · e = z(a)e. Then viewing R as an R-module with action given by multiplication a · b = ab, this makes the
+projection q : E −→ R an R-linear morphism. Therefore since the kernel of an R-linear morphism is always
+an R-module, we indeed have that ker(q), with the same action as E, is an R-module.
+✷
+4.2
+From Modules to Differential Bundles
+We now construct a differential bundle from a module. For a commutative ring R and an R-module M,
+define M[ε] as follows:
+M[ε] = {a + mε| a ∈ R, m ∈ M and ε2 = 0}
+where a and mε will be used respectively as shorthand for a + 0ε and 0 + mε. Then M[ε] is a commutative
+ring with multiplication induced by ε2 = 0, that is, the addition, multiplication, and negative are defined
+respectively as follows:
+(a + mε) + (b + nε) = (a + b) + (m + n)ε
+(a + mε)(b + nε) = ab + (a · m + b · n)ε
+−(a + mε)=−a − mε
+26
+
+and where the zero is 0 and the unit is 1. Note that when M = R and the action is given by multiplication
+a · b = ab, then this construction gives us the ring of dual numbers over R, or in other words, the tangent
+bundle
+T
+(R) = R[ε]. We now define a differential bundle over R structure on M[ε].
+(i) The projection qM : M[ε] −→ R is defined as projecting out the R component:
+qM(a + mε) = a
+As noted above, there is a general construction of pullbacks in CRING.
+However for the projection
+qM : M[ε] −→ R, we can instead describe these pullbacks in terms of multivariable dual numbers, like for the
+pullbacks of the tangent bundle. So define M[ε1, . . . , εn] as follows:
+M[ε1, . . . , εn] = {a + m1ε1 + . . . + mnεn| ∀a ∈ R, mj ∈ M and εiεj = 0}
+Then M[ε1, . . . , εn] is a commutative ring whose structure is defined in the obvious way, so in particular
+the multiplication is induced by εiεj = 0. We leave it as an exercise for the reader to check for themselves
+that M[ε1, . . . , εn] is the pullback of n copies of pR. We can then describe the rest of the differential bundle
+structure as follows:
+(ii) The pullback of n copies of pR is given by M[ε]n = M[ε1, . . . , εn] and where the pullback projection
+πj : M[ε1, . . . , εn] −→ M[ε] sends εj to ε and the other nilpotents to zero, that is, πj projects out the
+R component and j-th M component:
+πj(a + m1ε1 + . . . + mnεn) = a + mjε
+(iii) The sum σ : M[ε1, ε2] −→ M[ε] maps both ε1 and ε2 to ε, which results in adding the M components
+together:
+σ(a + mε1 + nε2) = a + (m + n)ε
+(iv) The zero z : R −→ M[ε] is the injection of R into the R component:
+0R(a) = a
+(v) The negative ι : M[ε] −→ M[ε] maps ε to −ε, which results in making the M component negative:
+ι(a + mε) = a − mε
+To describe the lift, let us describe
+T
+�
+M[ε]
+�
+, the ring of dual numbers of M[ε] in terms of two nilpotent
+elements ε and ε′:
+T
+�
+M[ε]
+�
+= M[ε][ε′] = {a + mε + bε′ + nεε′| ∀a, b ∈ R, m, n ∈ M and ε2 = ε′2 = 0}
+where the multiplication is induced by ε2 = ε′2 = 0. So we define:
+(vii) The lift λ : M[ε] −→ M[ε][ε′] maps ε to ε′, and so maps the R component of M[ε] to the first R
+component of M[ε][ε′], and the M component of M[ε] to the second M component of M[ε][ε′]:
+λ(a + mε) = a + mεε′
+We leave it as an exercise for the reader to check that these are all well-defined ring morphisms.
+Lemma 4.2 For every commutative ring R and R-moulde M,
+M
+R(M) := (qM, σM, zM, λM, ιM) is a differ-
+ential bundle with negatives over R in (CRING,
+T
+).
+27
+
+Proof: To show that we have a differential bundle, we will instead show that we have a pre-differential
+bundle which satisfies (i) and (ii) in Proposition 3.8. To show that (qM, zM, λM) is a pre-differential bundle,
+we must show that the four equalities from Definition 3.7 hold, but these all follow from straightforward
+computation, which we leave to the reader.
+Next, we must show that this pre-differential bundle also satisfies the extra assumptions required to make
+it a differential bundle. Firstly, it is straightforward to observe that M[ε1, . . . , εn] is indeed the pullback
+of n copies of the projection q. Also, since
+T
+is a right adjoint, it preserves all limits, and therefore all
+T
+n
+preserves these pullbacks. So (qM, zM, λM) satisfies assumption (i) of Proposition 3.8. Next, we must show
+that the following square is a pullback:
+M[ε]
+λM
+�
+qM
+�
+M[ε][ε′]
+⟨
+T
+(qM),pM[ε]⟩
+�
+R
+⟨0R,zM⟩
+� R[ε] × M[ε]
+(25)
+So suppose S is a commutative ring, and we have ring morphisms f : S −→ M[ε][ε′] and g : S −→ R such
+that ⟨
+T
+(qM), pM[ε]⟩ ◦ f = ⟨0R, zM⟩ ◦ g, that is, for every x ∈ S the following equality holds:
+�
+T
+(qM)(f(x)), pM[ε](f(x))
+�
+=
+�
+g(x), g(x)
+�
+Now f(x) ∈ M[ε][ε′] is of the form: f(x) = f1(x) + f2(x)ε + f3(x)ε′ + f4(x)εε′ for some f1(x), f3(x) ∈ R and
+f2(x), f4(x) ∈ M. Then the above equality tells us that:
+(g(x), g(x)) =
+�
+T
+(qM)(f(x)), pM[ε](f(x))
+�
+=
+�
+T
+(qM)
+�
+f1(x) + f2(x)ε + f3(x)ε′ + f4(x)εε′�
+, pM[ε]
+�
+f1(x) + f2(x)ε + f3(x)ε′ + f4(x)εε′��
+=
+�
+qM(f1(x) + f2(x)ε) + qM(f3(x) + f4(x)ε)ε, pM[ε]
+�
+f1(x) + f2(x)ε + f3(x)ε′ + f4(x)εε′��
+=
+�
+f1(x) + f3(x)ε, f1(x) + f2(x)ε
+�
+So this implies that g(x) = f1(x) + f2(x)ε and g(x) = f1(x) + f3(x)ε. However, in both equalities, the
+left-hand side has no nilpotent component. Therefore, we have that g(x) = f1(x), f2(x) = 0, and f3(x) = 0.
+So f(x) = g(x) + f4(x)εε′. Then define ⟨f, g⟩ : S −→ M[ε] to be f but without ε′, that is, as follows:
+⟨f, g⟩(x) = g(x) + f4(x)ε
+(26)
+That ⟨f, g⟩ is a ring morphism essentially follows from the fact that f is a ring morphism. Next we compute
+that ⟨f, g⟩ also satisfies the following:
+λM(⟨f, g⟩(x)) = λM(g(x) + f4(x)ε) = g(x) + f4(x)εε′ = f(x)
+qM(⟨f, g⟩(x)) = qM(g(x) + f4(x)ε) = g(x)
+So λM ◦ ⟨f, g⟩ = f and qM ◦ ⟨f, g⟩ = g as desired. Lastly, it remains to show that ⟨f, g⟩ is the unique such
+ring morphism. So suppose we have a ring morphism h : S −→ M[ε] such that λM ◦ h = f and qM ◦ h = g.
+Now h(x) ∈ M[ε] is of the form h(x) = h1(x) + h2(x)ε for some h1(x) ∈ R and h2(x) ∈ M. By assumption,
+we have that:
+g(x) + f4(x)εε′ = f(x) = λM(h(x)) = λM(h1(x) + h2(x)ε) = h1(x) + h2(x)εε′
+So g(x) + f4(x)εε′ = h1(x) + h2(x)εε′, which implies that h1(x) = g(x) and h2(x) = f4(x). Therefore,
+h(x) = g(x) + f4(x)ε = ⟨f, g⟩(x), and so ⟨f, g⟩ is unique. So we conclude that the above square is a pullback
+28
+
+diagram. Furthermore, since
+T
+is a right adjoint, we also have that
+T
+n preserves these pullbacks. Thus
+(qM, zM, λM) satisfies assumption (ii) of Proposition 3.8. Therefore, (qM, zM, λM) will induce a differential
+bundle with negatives.
+It remains to construct the sum and the negative as in Proposition 3.8, and show that these are the same
+as the proposed σ and ι above. The sum σ will be given by:
+σ =
+�
++M[ε] ◦ ⟨λM ◦ π1, λM ◦ π2⟩, qM ◦ πj
+�
+We leave it to the reader to check for themselves that the following equalities hold:
++M[ε]
+�
+⟨λM ◦ π1, λM ◦ π2⟩(a + mε1 + nε2)
+�
+= a + (m + n)εε′
+Therefore by construction, we have that σ(a + mε1 + nε2) = a + (m + n)ε as desired. The negative ι will be
+given by:
+ι =
+�
+−M[ε] ◦ λM, qM
+�
+We then compute that:
+−M[ε](λM(a + mε)) = a − mεε′
+So by construction, we have that ι(a + mε) = a − mε. So we conclude that
+M
+R(M) = (qM, σM, zM, λM, ιM)
+is a differential bundle with negatives over R.
+✷
+4.3
+Equivalence
+We will now show that the constructions of Lemma 4.1 and Lemma 4.2 are inverses of each other.
+Beginning from the module side of things, let R be a commutative ring and M be an R-module. Consider
+ker(qM), the kernel of the projection of the induced differential bundle
+M
+R(M). However, qM(a + mε) = 0
+implies that a = 0. So the kernel of the projection consists solely of the M component, that is, ker(qM) =
+{mε| ∀m ∈ M}, which is clearly isomorphic to M. Explicitly, αM : M −→ ker(qM) is defined as αM(m) = mε,
+and α−1
+M : ker(qM) −→ M is defined as α−1
+M (mε) = m.
+Lemma 4.3 For every commutative ring R and R-module M, αM : M −→ ker(qM) is an R-linear isomor-
+phism with inverse α−1
+M : ker(qM) −→ M.
+Proof: Clearly for every R-module M, αM and α−1
+M are inverses of each other, that is, α−1
+M (αM(m)) = m
+and αM(α−1
+M (mε)) = mε. However, we must explain why αM and α−1
+M are also R-linear morphisms. Clearly,
+they are both linear, so we must show that they preserve the action. We start by showing that αM does,
+where recall that the action on ker(qM) is defined as a · (mε) = zM(a)mε:
+αM(a · m) = (a · m)ε = (a + 0ε)mε = zM(a)mε = a · mε = a · αM(m)
+So αM is an R-module morphism. Since αM and α−1
+M are inverses as functions, it then follows that α−1
+M will
+also be an R-module morphism. So we conclude that αM and α−1
+M are inverse R-linear isomorphisms.
+✷
+Let’s now start from the differential bundle side of the story.
+Let E = (q : E −→ R, σ, z, λ, ι) be a
+differential bundle with negatives over a commutative ring R in (CRING,
+T
+). To define differential bundle
+isomorphisms between E and
+M
+R
+�
+ker(q)
+�
+, we will first need to define a ring isomorphism between E and
+ker(q)[ε].
+To do so, we must first take a closer look at the lift λ : E −→ E[ε].
+Since the lift is a ring
+morphism whose codomain is a ring of dual numbers, it is well-known that it must be of the following form:
+λ(x) = pE(λ(x)) + Dλ(x)ε, where Dλ : E −→ E is a derivation. Now by the first diagram of [DB.3], we have
+that pE ◦ λ = z ◦ q. This implies that the lift is in fact of the form:
+λ(x) = z(q(x)) + Dλ(x)ε
+29
+
+and the product rule for the derivation Dλ is given by Dλ(xy) = z(q(x))Dλ(y) + z(q(y))Dλ(x). Then define
+the function βE : E −→ ker(q)[ε] as follows:
+βE(x) = q(x) + Dλ(x)ε
+(27)
+To define its inverse β−1
+E
+: ker(q)[ε] −→ E, we will need to make use of Rosick´y’s universality diagram,
+that is, the pullback square from Proposition 3.8. First, define the ring morphism ζE : ker(q)[ε] −→ E[ε]
+as ζE(a + xε) = z(a) + xε. By universality of the pullback, define β−1
+E
+: ker(q)[ε] −→ E as the unique ring
+morphism which makes the following diagram commute:
+ker(q)[ε]
+qker(q)
+�
+ζE
+�
+β−1
+E
+�P
+P
+P
+P
+P
+P
+P
+E
+q
+�
+λ
+� E[ε]
+⟨
+T
+(q),pE⟩
+�
+R
+⟨0R,z⟩
+� R[ε] × E
+(28)
+so β−1
+E
+= ⟨qker(q), ζE⟩. We will show below that β−1
+E
+is a differential bundle morphism, from which it follows
+from the compatibility with the lift that β−1
+E (a + xε) = z(a) + x.
+Lemma 4.4 For commutative ring R and a differential bundle with negatives E = (q : E −→ R, σ, z, λ, ι)
+over R in (CRING,
+T
+), βE : E −→
+M
+R
+�
+ker(q)
+�
+is a differential bundle isomorphism over R with inverse
+β−1
+E
+:
+M
+R
+�
+ker(q)
+�
+−→ E.
+Proof: We first explain why βE and β−1
+E
+are well-defined ring morphisms. Starting with βE, we must
+first explain why Dλ(x) is in the kernel of the projection q. By the first diagram of [DB.2], we have that
+T
+(q)◦ λ = 0R ◦ q. Therefore, for all x ∈ E, we have that q(z(q(x)))+ q(Dλ(x))ε = q(x). Since the right-hand
+side has no nilpotent component, this implies that q(Dλ(x)) = 0. So for all x ∈ E, Dλ(x) ∈ ker(q), and so
+βE is well-defined. We leave it to the reader to check for themselves that βE is indeed a ring morphism.
+Next we explain why β−1
+E
+is a well-defined. To do so, we must show that the outer diagram of (28)
+commutes. First, note that by the second diagram of [DB.1], q ◦ z = 1R, so q(z(a)) = a for all a ∈ R. Then
+for all a ∈ R and x ∈ ker(q) we compute:
+⟨
+T
+(q), pE⟩
+�
+ζE(a + xε)
+�
+= ⟨
+T
+(q), pE⟩
+�
+z(a) + xε
+�
+=
+�
+T
+(q)(z(a) + xε), pE(z(a) + xε)
+�
+=
+�
+q(z(a)) + q(x)ε), pE(z(a) + xε)
+�
+=
+�
+a, z(a)
+�
+=
+�
+0R(a), z(a)
+�
+= ⟨0R, z⟩(a) = ⟨0R, z⟩
+�
+qker(q)(a + xε
+�
+So ⟨
+T
+(q), pE⟩ ◦ ζE = ⟨0R, z⟩ ◦ qker(q). Therefore, by the universal property of the pullback square, there exists
+a unique ring morphism β−1
+E
+: ker(q)[ε] −→ E such that λ ◦ β−1
+E
+= ζE and q ◦ β−1
+E
+= qker(q). In particular,
+these imply that for every a ∈ R and x ∈ ker(q) the following equalities hold:
+q
+�
+β−1
+E (a + xε)
+�
+= a
+Dλ(β−1
+E (a + xε)) = x
+Next we show that βE and β−1
+E
+are inverses of each other. To show that βE ◦ β−1
+E
+= 1ker(q)[ε], we use the
+above identities:
+βE(β−1
+E (a + xε)) = q(β−1
+E (a + xε)) + Dλ(β−1
+E (a + xε))ε = a + xε
+To show that β−1
+E
+◦ βE = 1E, we will first show that q ◦ β−1
+E
+◦ βE = q and λ ◦ β−1
+E
+◦ βE = λ:
+q(β−1
+E (βE(x))) = q(βE(x)) = q((q(x) + Dλ(x)ε)) = q(x)
+30
+
+λ(β−1
+E (βE(x))) = ζE(βE(x)) = ζE(q(x) + Dλ(x)ε) = z(q(x)) + Dλ(x)ε = λ(x)
+Therefore, by the universal property of the pullback, it follows that β−1
+E
+◦ βE = 1E. So βE and β−1
+E
+are
+inverse ring isomorphisms.
+Lastly, we must show that βE and β−1
+E
+are also differential bundle morphisms over R. To do so, we will
+need to know a bit more about Dλ. The third diagram of [DB.3] is 0E ◦ z = λ ◦ z, which implies that for
+all a ∈ R, z(a) = z(a) + Dλ(z(a))ε. Since the left-hand side has no nilpotent component, it follows that
+Dλ(z(a)) = 0 for all a ∈ R. On the other hand, the diagram of [DB.4] says that
+T
+(λ)◦λ = ℓE ◦λ. Then using
+that q(Dλ(x)) = 0, q(z(a)) = a, and Dλ(z(a)) = 0, [DB.4] explicitly states that q(z(x)) + Dλ(Dλ(x))εε′ =
+q(z(x)) + Dλ(x)εε′. This implies that Dλ(Dλ(x)) = Dλ(x) for all x ∈ E. With these identities, we can now
+show that βE is a differential bundle morphism over R. So we show that the diagrams of (22) hold:
+(i) qker(q) ◦ βE = q:
+qker(q)(βE(x)) = qker(q)(q(x) + Dλ(x)ε) = q(x)
+(ii)
+T
+(βE) ◦ λ = λker(q) ◦ βE:
+T
+(βE)(λ(x)) =
+T
+(βE)(z(q(x)) + Dλ(x)ε) = βE(z(q(x)) + βE(Dλ(x)ε)ε′
+= q(z(q(x)))+Dλ(z(q(x)))ε+z(q(Dλ(x)))ε′+Dλ(Dλ(x))εε′ = q(x)+0ε+0ε′+Dλ(x)εε′ = q(x)+Dλ(x)εε′
+= λker(q)(q(x) + Dλ(x)ε) = λker(q)(βE(x))
+So βE is a differential bundle morphism. Since βE is a ring isomorphism, by Lemma 3.13 it then follows that
+β−1
+E
+is also a differential bundle morphism. In particular, note that this implies that λ◦β−1
+E
+=
+T
+(β−1
+E )◦λker(q).
+But by definition, we have that λ◦β−1
+E
+= ζE, and so we also have that
+T
+(β−1
+E )◦λker(q) = ζE. This implies that
+β−1
+E (a) + β−1
+E (xε)ε = z(a) + xε, and so β−1
+E (a) = z(a) and β−1
+E (xε) = x. Therefore, β−1
+E (a + xε) = z(a) + x.
+So we conclude that βE and β−1
+E
+are differential bundle isomorphisms over R.
+✷
+Therefore, the construction from a module to a differential bundle is the inverse of the construction from
+a differential bundle to a module. So we conclude that:
+Proposition 4.5 For a commutative ring R, there is a bijective correspondence between R-modules and
+differential bundles (with negatives) over R in (CRING,
+T
+).
+In CRING, recall that the terminal object is the zero ring 0. So differential objects in (CRING,
+T
+) corre-
+spond precisely to 0-modules. However, the only 0-module is 0. Therefore, there are no non-trivial differential
+objects in (CRING,
+T
+).
+Corollary 4.6 The only differential object in (CRING,
+T
+) is the zero ring 0.
+We now extend Proposition 4.5 to an equivalence of categories. For a commutative ring R, let MODR be
+the category of R-modules and R-linear morphisms between them. We define an equivalence of categories
+between MODR from DBUN
+T
+[R] as follows:
+(i) Define the functor
+M
+R : MODR −→ DBUN
+T
+[R] which sends an R-module M to the differential bundle
+M
+R(M), and sends an R-linear morphism f : M −→ M ′ to the differential bundle morphism over R
+M
+R(f) :
+M
+R(M) −→
+M
+R(M ′) where
+M
+R(f) : M[ε] −→ M ′[ε] is defined as:
+M
+R(f)(a + mε) = a + f(m)ε
+(ii) Define the functor
+M
+◦
+R : DBUN
+T
+[R] −→ MODR which sends a differential bundle with negatives over R,
+E = (q : E −→ R, σ, z, λ, ι) to the R-module
+M
+◦
+R(E) = ker(q), and sends a differential bundle morphism
+f : E = (q : E −→ R, σ, z, λ, ι) −→ E′ = (q′ : E′ −→ R, σ′, z′, λ′, ι′) over R to the R-linear morphism
+M
+◦
+R(f) : ker(q) −→ ker(q′) defined as:
+M
+◦
+R(f)(x) = f(x)
+31
+
+(iii) Define the natural isomorphism α : 1MODR ⇒
+M
+◦
+R ◦
+M
+R with inverse α−1 :
+M
+◦
+R ◦
+M
+R ⇒ 1MODR as αM
+α−1
+M defined in Lemma 4.3.
+(iv) Define the natural isomorphism β : 1DBUN
+T
+[R] ⇒
+M
+R ◦
+M
+◦
+R with inverse β−1 :
+M
+◦
+R ◦
+M
+R ⇒ 1DBUN
+T
+[R]
+as βE β−1
+E
+defined in Lemma 4.4.
+Theorem 4.7 For a commutative ring R, we have an equivalence of categories: MODR ≃ DBUN
+T
+[R].
+Proof: We must first explain why
+M
+R and
+M
+◦
+R are well-defined on morphisms.
+So given an R-linear
+morphism f : M −→ M ′, we must show that
+M
+R(f) is a differential bundle morphism over R. We leave it to
+the reader to check for themselves that
+M
+R(f) is a ring morphism. So it remains to show that the diagrams
+of (22) also hold:
+(i) qM′ ◦
+M
+R(f) = qM:
+qM′ �
+M
+R(f)(a + mε)
+�
+= qM′(a + f(m)ε) = a = qM(a + mε)
+(ii)
+T
+�
+M
+R(f)
+�
+◦ λM = λM′ ◦
+M
+R(f):
+T
+�
+M
+R(f)
+� �
+λM(a + mε)
+�
+=
+T
+�
+M
+R(f)
+� �
+a + mεε′�
+=
+M
+R(f)(a) +
+M
+R(f)(mε)ε′
+= a + f(m)εε′ = λM′ �
+a + f(m)ε
+�
+= λM′ �
+M
+R(f)(a + mε)
+�
+So we conclude that
+M
+R(f) is a differential bundle morphism over R. On the other hand, given a differential
+bundle morphism f : E −→ E′ over R, we must first explain why if x ∈ ker(q) then
+M
+◦
+R(f)(x) ∈ ker(q′).
+Note that since f is a differential bundle morphism over R, by definition this means that for all x ∈ E,
+q′(f(x)) = q(x). So it follows that if x ∈ ker(q), we have that:
+q′ �
+M
+◦
+R(f)(x)
+�
+= q′(f(x)) = q(x) = 0
+and so
+M
+◦
+R(f)(x) ∈ ker(q′). Thus
+M
+◦
+R(f) : ker(q) −→ ker(q′) is well-defined. To show that
+M
+◦
+R(f) is R-linear,
+clearly since f is linear,
+M
+◦
+R(f) will be linear, therefore it remains to show
+M
+◦
+R(f) preserves the action. Since
+f is a differential bundle morphisms over R, by Lemma 3.11, we have that f preserves the zero, that is,
+f(z(a)) = z′(a) for all a ∈ R. So we compute:
+a ·
+M
+◦
+R(f)(x) = a · f(x) = z′(a)f(x) = f(z(a)x) = f(a · x) =
+M
+◦
+R(f)(a · x)
+So we conclude that
+M
+◦
+R(f) is an R-linear morphism. So
+M
+R and
+M
+◦
+R are well-defined, and it is straightforward
+to see that they also preserve composition and identities, so
+M
+R and
+M
+◦
+R are indeed functors.
+Next we explain why α, α−1, β, and β−1 are natural transformations. In fact, it suffices to explain why
+α and β−1 are natural, and it will then follow that α−1 and β are as well since we have already shown they
+are isomorphisms on each object. So for an R-linear morphism f : M −→ M ′, we compute:
+M
+◦
+R
+�
+M
+R(f)
+� �
+αM(m)
+�
+=
+M
+◦
+R
+�
+M
+R(f)
+�
+(mε) =
+M
+R(f)(mε) = f(m)ε = αM′(f(m))
+So α is indeed a natural transformation, and so α−1 will also be a natural transformation. Therefore, α and
+α−1 are inverse natural isomorphism. On the other hand, for a differential bundle morphism f : E −→ E′
+over R, we compute:
+β−1
+E′
+�
+M
+R
+�
+M
+◦
+R(f)
+�
+(a + xε)
+�
+= β−1
+E′
+�
+a +
+M
+◦
+R(f)(x)ε
+�
+= β−1
+E′
+�
+a + f(x)ε
+�
+= z′(a) + f(x)ε = f(z(a)) + f(x) = f(z(a) + x) = f
+�
+β−1
+E (a + xε)
+�
+32
+
+So β−1 is indeed a natural transformation, and so β will also be a natural transformation. Therefore, β
+and β−1 are inverse natural isomorphism. So we conclude that we have an equivalence of categories, and so
+MODR ≃ DBUN
+T
+[R].
+✷
+We also obtain an equivalence of categories between the category of all differential bundles and the
+category of modules. Let MOD be the category whose objects are pairs (R, M) consisting of a commutative
+ring R and an R-module M, and where a map (g, f) : (R, M) −→ (R′, M ′) is a pair consisting of a ring
+morphism g : R −→ R′ and an R-linear map f : M −→ M ′, where M ′ is an R-module via the action
+a•m = g(a)·m, so explicitly, f(a·m) = g(a)·f(m). Composition is defined as (g′, f ′)◦(g, f) = (g′ ◦g, f ′ ◦f)
+and identities are (1R, 1M). We define an equivalence of categories between MOD and DBUN
+�
+(CRING,
+T
+)
+�
+as follows:
+(i) Define the functor
+M
+: MOD −→ DBUN
+�
+(CRING,
+T
+)
+�
+which sends an object (R, M) to the differential
+bundle
+M
+(R, M) =
+M
+R(M), and sends a map (g, f) : (R, M) −→ (R′, M ′) to the differential bundle
+morphism
+M
+(g, f) :
+M
+R(M) −→
+M
+R′(M ′) defined as:
+M
+(g, f)(a + mε) = g(a) + f(m)ε
+(ii) Define the functor
+M
+◦ : DBUN
+�
+(CRING,
+T
+)
+�
+−→ MOD which sends a differential bundle with negatives
+E = (q : E −→ R, σ, z, λ, ι) to the pair
+M
+◦(E) = (R, ker(q)), and sends a differential bundle morphism
+(f, g) : E = (q : E −→ R, σ, z, λ, ι) −→ E′ = (q′ : E′ −→ R′, σ′, z′, λ′, ι′) to the pair
+M
+◦(f, g) = (g,
+M
+◦
+R(f)).
+(iii) Define the natural isomorphism α : 1MOD ⇒
+M
+◦ ◦
+M
+as α(R,M) = (1R, αM), with inverse natural
+isomorphism α−1 :
+M
+◦ ◦
+M
+⇒ 1MOD defined as α−1
+(R,M) = (1R, α−1
+M ).
+(iv) Define the natural isomorphism β : 1DBUN
+�
+(CRING,
+T
+)
+� ⇒
+M
+◦
+M
+◦ as βE = (1, βE), with inverse natural
+isomorphism β
+−1 :
+M
+◦
+R ◦
+M
+R ⇒ 1DBUN
+�
+(CRING,
+T
+)
+� as β
+−1
+E
+= (1, β−1
+E ).
+Theorem 4.8 We have an equivalence of categories: MOD ≃ DBUN
+�
+(CRING,
+T
+)
+�
+.
+Proof: That
+M
+and
+M
+◦ are well-defined on morphisms is similar to the proofs that
+M
+R and
+M
+◦
+R were well-
+defined on morphisms in the proof of Theorem 4.7. So
+M
+and
+M
+◦ are indeed functors. Next, since αM and
+α−1
+M are R-linear morphisms, it follows that α(R,M) = (1R, αM) and α−1
+(R,M) = (1R, α−1
+M ) are indeed maps in
+MOD, so α and α−1 are well-defined. On the other hand, since βE and β−1
+E
+are differential bundle morphisms
+over the base commutative ring, it follows by definition that βE = (1, βE) and β
+−1
+E
+= (1, β−1
+E ) are differential
+bundle morphisms, so β and β
+−1 are well-defined. Lastly, that α, α−1, β, and β
+−1 are natural isomorphisms
+follows directly from the fact that α, α−1, β, and β−1 are natural isomorphisms. So we conclude that we
+have an equivalence of categories: MOD ≃ DBUN
+�
+(CRING,
+T
+)
+�
+.
+✷
+Remark 4.9 The equivalence between modules and differential bundles is also true in more general settings.
+Indeed, both for the tangent category of commutative semirings and the tangent category of commutative
+algebras over a (semi)ring, differential bundles correspond precisely to modules via the above constructions.
+However, in a setting where one does not have negatives, we would have also needed to prove [DB.5], since
+this is also required to make a pre-differential bundle a differential bundle in a Cartesian tangent category
+without negatives [22, Proposition 6]. Even more generally, in a codifferential category, every module of
+an algebra of the monad will induce a differential bundle in the Eilenberg-Moore category via [4, Theorem
+5.1] and a generalization of Lemma 4.2.
+If a codifferential category has kernels, then every differential
+bundle induces a module by generalizing Lemma 4.1, and so in the presence of kernels, differential bundles
+in the Eilenberg-Moore category also correspond precisely to modules. However, since not all codifferential
+categories have all kernels, there may be differential bundles which are not induced by modules.
+33
+
+5
+Differential Bundles for (Affine) Schemes
+In this section, we characterize differential bundles (with negatives) in the tangent category of affine schemes
+and prove that they also correspond to modules (Proposition 5.5). However, the constructions are quite
+different in this case. To go from a module to a differential bundle, we take the free symmetric algebra over
+said module (Lemma 5.2). To go from a differential bundle to a module, we take the image of the derivation
+induced by the lift (Lemma 5.1). Moreover, in contrast to the previous section, in this case, we obtain
+that the category of differential bundles is equivalent to the opposite category of modules (Theorem 5.7 and
+Theorem 5.9). To the best of our understanding, there is no general reason why the fact that differential
+bundles in commutative rings are equivalent to the category of modules would also imply that differential
+bundles in commutative rings opposite are equivalent to the opposite category of modules. We conclude the
+section by generalizing these results to the category of schemes, where differential bundles are equivalent to
+the opposite category of quasicoherent sheaves of modules.
+5.1
+From Differential Bundles to Modules
+Let us begin by unravelling what a differential bundle with negatives would be in (CRINGop, T). First recall
+that (CRINGop, T) is a Cartesian Rosick´y tangent category, so by Proposition 3.9, differential bundles are
+the same thing as differential bundles with negatives. Also, as discussed in Section 2.3, CRINGop admits all
+pullbacks, since CRING admits all pushouts. For any ring morphism q : R −→ E between commutative rings,
+recall that E becomes a commutative R-algebra, so, in particular, an R-module, with action a · x = q(a)x.
+Then the pushout of n copies of q in CRING is given by taking the tensor product over R of n copies
+of E: En = E ⊗R . . . ⊗R E
+�
+��
+�
+n times
+.
+Then a differential bundle with negatives over a commutative ring R in
+(CRINGop, T) viewed in CRING would consist of a commutative ring E and five ring morphisms: q : R −→ E,
+σ : E −→ E ⊗R E, z : E −→ R, λ : T(E) −→ E, and ι : E −→ E. These must also satisfy the dual equalities
+and properties of Definition 3.1. In particular, note that [DB.1] and [DB.N] imply that E is a commutative
+R-Hopf algebra, where the sum σ is the comultiplication, the zero z is the counit, and the negative ι is the
+antipode.
+To obtain an R-module from a differential bundle, we take the image of the map Dλ : E −→ E defined as
+Dλ(x) = λ(d(x)), which is, in fact, a derivation whose product rule is Dλ(ab) = λ(a)Dλ(b) + λ(b)Dλ(a).
+Lemma 5.1 Let R be a commutative ring, and let E = (q : E −→ R, σ, z, λ, ι) be a differential bundle (with
+negatives) over R in (CRINGop, T). Then the image of the derivation im(Dλ) = {Dλ(x) = λ(d(x))| ∀x ∈ E}
+is an R-module with action a · Dλ(x) = Dλ(q(a)x).
+Proof: Recall that for any R-linear map f : M −→ N, the image im(f) = {f(m)| ∀m ∈ M} is an R-module
+with action a · f(m) = f(a · m). Therefore, to prove that im(Dλ) is an R-module, it suffices to show that
+Dλ is an R-linear map. Clearly Dλ is linear, so it remains to show that Dλ also preserves the action. First
+note that by the dual of the first diagram of [DB.2], that λ ◦ T(q) = q ◦ 0R. In particular this implies
+that λ(q(a)) = q(a) and λ(d(q(a))) = 0 for all a ∈ R. Note that the second equality can be rewritten as
+Dλ(q(a)) = 0 for all a ∈ R. So we compute:
+Dλ(a · x) = Dλ(q(a)x) = λ(q(a))Dλ(x) + λ(x)Dλ(q(a)) = q(a)Dλ(x) + 0 = a · Dλ(x)
+So Dλ is R-linear and we conclude that im(Dλ) is an R-module.
+✷
+5.2
+From Modules to Differential Bundles
+We now construct a differential bundle from a module. For a commutative ring R and an R-module M, let
+SymR(M) be the free symmetric R-algebra over M, that is:
+SymR (M) =
+∞
+�
+n=0
+M ⊗s
+R
+n = R ⊕ M ⊕ (M ⊗s
+A M) ⊕ . . .
+34
+
+where ⊗s
+R is the symmetrized tensor product over R. Note that as a commutative ring, SymR(M) is generated
+by all a ∈ R and m ∈ M. Therefore, to define ring morphisms with domain SymR(M), it suffices to define
+them on generators a and m. Using this to our advantage, we define a differential bundle with negatives over
+R structure on SymR(M) viewed in CRING (so the differential bundle structure maps will all be backwards)
+as follows:
+(i) The projection qM : R −→ SymR(M) is defined as the injection of R into SymR(M):
+qM(a) = a
+(ii) The pushouts (which recall are pullbacks in CRINGop) are given by taking the tensor product over
+R of n copies of SymR(M), so SymR(M)n := SymR(M) ⊗R . . . ⊗R SymR(M)
+�
+��
+�
+n times
+, where the jth injection
+πj : SymR(M) −→ SymR(M)n injects SymR(M) into the j-th component:
+πj(w) = 1 ⊗R . . . ⊗R 1 ⊗R w ⊗R 1 ⊗R . . . ⊗R 1
+(iii) The sum σM : SymR(M) −→ SymR(M) ⊗R SymR(M) is the canonical comultiplication of the free
+symmetric R-algebras, that is, defined on generators as follows:
+σM(a) = a ⊗R 1 = 1 ⊗R a
+σM(m) = m ⊗R 1 + 1 ⊗R m
+(iv) The zero 0R : SymR(M) −→ R is the canonical counit of the free symmetric R-algebras, that is, defined
+on generators as follows:
+zM(a) = a
+zM(m) = 0
+(v) The negative ιM : SymR(M) −→ SymR(M) is the canonical antipode of the free symmetric R-algebras,
+that is, defined on generators as follows:
+ιM(a) = a
+ιM(m) = −m
+To describe the lift, note that T(SymR(M)) as a commutative ring is generated by a, m, d(a), and d(m) for
+all a ∈ R and m ∈ M (and modulo the appropriate equations).
+(vii) The lift λM : T(SymR(M)) −→ SymR(M) is defined on generators as follows:
+λM(a) = a
+λM(m) = 0
+λM(d(a)) = 0
+λM(d(m)) = m
+Lemma 5.2 For every commutative ring R and R-module M, MR(M) := (qM, σM, zM, λM, ιM) is a differ-
+ential bundle with negatives over R in (CRINGop, T).
+Proof: To show that we have a differential bundle, we will instead show that we have a pre-differential
+bundle which satisfies (i) and (ii) in Proposition 3.8. So to show that (qM, zM, λM) is a pre-differential
+bundle in (CRINGop, T), we must show that the dual of the four equalities from Definition 3.7 hold in CRING.
+To do so, we show that these hold on the generators.
+(i) zM ◦ qM = 1R
+zM(qM(a)) = zM(a) = a
+(ii) λM ◦ pSymR(M) = qM ◦ zM
+λM(pSymR(M)(a)) = λM(a) = a = qM(a) = qM(zM(a))
+λM(pSymR(M)(m)) = λM(m) = 0 = qM(0) = qM(zM(m))
+35
+
+(iii) zM ◦ 0SymR(M) = zM ◦ λM
+zM
+�
+0SymR(M)(a)
+�
+= zM(a) = zM(λM(a))
+zM
+�
+0SymR(M)(m)
+�
+= zM(m) = 0 = zM(0) = zM(λM(0))
+(iv) λM ◦T(λM) = λM ◦ℓSymR(M): Note that T2(SymR(M)) has eight kinds of generators, a, m, d(a), d(m),
+d′(a), d′(m), d′d(a), and d′d(m) for all a ∈ R and m ∈ M.
+λM(T(λM)(a)) = λM(λM(a)) = λM(a) = λM(ℓSymR(M)(a))
+λM(T(λM)(m)) = λM(λM(m)) = λM(0) = 0 = λM(m) = λM(ℓSymR(M)(m))
+λM(T(λM)(d(a))) = λM(λM(d(a))) = λM(0) = λM(ℓSymR(M)(d(a)))
+λM(T(λM)(d(m))) = λM(λM(d(m))) = λM(0) = λM(ℓSymR(M)(d(m)))
+λM(T(λM)(d′(a))) = λM(d(λM(a))) = λM(d(a)) = 0 = λM(0) = λM(ℓSymR(M)(d′(a)))
+λM(T(λM)(d′(m))) = λM(d(λM(m))) = λM(d(0)) = λM(0) = λM(ℓSymR(M)(d′(m)))
+λM(T(λM)(d′d(a)))=λM(d(λM(d(a)))) = λM(d(0)) = λM(0) = 0 = λM(d(a))=λM(ℓSymR(M)(d′d(a)))
+λM(T(λM)(d′d(m))) = λM(d(λM(d(m)))) = λM(d(m)) = λM(ℓSymR(M)(d′d(m)))
+So the desired equalities hold and we conclude that (qM, zM, λM) is a pre-differential bundle in (CRINGop, T).
+Next, we must show that this pre-differential bundle also satisfies the extra assumptions required to make
+it a differential bundle, or rather that the dual of the assumptions hold in CRING. As explained above, the
+pushout of n copies of the projection qM exists, chosen here to be SymR(M)n, and since T is a left adjoint,
+it preserves all colimits, so Tn preserves these pushouts. Dualizing this, we conclude that (qM, zM, λM)
+satisfies assumption (i) of Proposition 3.8 in CRINGop.
+Next, we must show that the dual of (ii) of Proposition 3.8 also holds, that is, we must show that the
+following square is a pushout in CRING:
+T(R) ⊗ SymR(M)
+[T(qM),pSymR(M)]
+�
+[0R,zM]
+� R
+qM
+�
+T(SymR(M))
+λM
+� SymR(M)
+(29)
+where [−, −] is the copairing operation of the coproduct, which recall in CRING is given by the tensor
+product. Now suppose that S is a commutative ring, and we have ring morphisms f : T(SymR(M)) −→ S
+and g : R −→ S such that f ◦[T(qM), pSymR(M)] = g ◦[0R, zM]. In particular, this implies that for every a ∈ R
+and m ∈ M,the following equalities hold:
+f(a) = g(a)
+f(d(a)) = 0
+f(m) = 0
+Then define the map [f, g] : SymR(M) −→ S as the ring morphism defined on generators as follows:
+[f, g](a) = g(a)
+[f, g](m) = f(d(m))
+(30)
+Next, we compute the following on generators:
+[f, g](qM(a)) = [f, g](a) = g(a)
+[f, g](λM(a)) = [f, g](a) = g(a) = f(a)
+36
+
+[f, g](λM(m)) = [f, g](0) = 0 = f(m)
+[f, g](λM(d(a))) = [f, g](0) = 0 = f(d(a))
+[f, g](λM(d(m))) = [f, g](m) = f(d(m))
+Thus it follows that [f, g]◦ λM = f and [f, g]◦ qM = g as desired. Lastly, it remains to show that [f, g] is the
+unique such ring morphism. So suppose we have a ring morphism h : SymR(M) −→ S such that h ◦ λM = f
+and h ◦ qM = g. Then on generators, we compute that:
+h(a) = h(qM(a)) = g(a) = [f, g](a)
+h(m) = h(λM(d(m))) = f(d(m)) = [f, g](m)
+Since h and [f, g] are ring morphisms that are equal on generators, it follows that h = [f, g], thus [f, g] is
+unique. Thus we conclude the above diagram is a pushout in CRING. Furthermore, since T is a left adjoint in
+CRING, we also have that Tn preserves these pushouts. Dualizing this, it follows that (qM, zM, λM) satisfies
+assumption (ii) of Proposition 3.8 in CRINGop. Therefore by Proposition 3.8, the pre-differential bundle
+(qM, zM, λM) will induce a differential bundle with negatives in (CRINGop, T).
+It remains to construct the sum and the negative as in Proposition 3.8, and show that these are the same
+as the proposed σ and ι above. By dualizing the construction, the sum σ is:
+σM =
+�
+[π1 ◦ λM, π2 ◦ λM] ◦ +SymR(M), πj ◦ qM
+�
+On generators, we compute:
+σM(a) =
+�
+[π1 ◦ λM, π2 ◦ λM] ◦ +SymR(M), πj ◦ qM
+�
+(a) = πj(qM(a)) = πj(a) = a ⊗R 1 = 1 ⊗R a
+σM(m) =
+�
+[π1 ◦ λM, π2 ◦ λM] ◦ +SymR(M), πj ◦ qM
+�
+(m) = [π1 ◦ λM, π2 ◦ λM](+SymR(M)(d(m)))
+= [π1 ◦ λM, π2 ◦ λM](d(m) ⊗R 1) + [π1 ◦ λM, π2 ◦ λM](1 ⊗R d(m))
+= π1(λM(d(m))) + π2(λM(d(m))) = π1(m) + π2(m) = m ⊗R 1 + 1 ⊗R m
+Thus on generators, σM(a) = a ⊗R 1 = 1 ⊗R a and σ(m) = m ⊗R 1 + 1 ⊗R m, as defined above. On the
+other hand, the negative ι is:
+ιM =
+�
+λM ◦ −SymR(M), qM
+�
+On generators, we compute:
+ιM(a) =
+�
+λM ◦ −SymR(M), qM
+�
+(a) = qM(a) = a
+ιM(m) =
+�
+λM ◦ −SymR(M), qM
+�
+(m) = λM(−SymR(M)(d(m)) = λM(−d(m)) = −λM(d(m)) = −m
+So on generators ιM(a) = a, and ιM(m) = −m as desired. So we conclude that (qM, σM, zM, λM, ιM) is a
+differential bundle with negatives over R in (CRINGop, T).
+✷
+5.3
+Equivalence
+We will now show that the constructions of Lemma 5.1 and Lemma 5.2 are inverses of each other. Starting
+from the module side of things, let R be a commutative ring, M an R-module, and consider the induced
+derivation DλM : SymR(M) −→ SymR(M). We will show that the image of the derivation is precisely M.
+Lemma 5.3 For every commutative ring R and R-module M, im(DλM ) = M as R-modules.
+37
+
+Proof: Let us compute what this derivation does on pure symmetrized tensors. For degree 0, that is, for
+a ∈ R we have that:
+DλM (a) = λM(d(a)) = 0
+so DλM (a) = 0. For degree 1, that is, for m ∈ M we have that:
+DλM (m) = λM(d(m)) = m
+so DλM (m) = m. For degree 2, that is, for m, n ∈ M using the product rule, we have that:
+DλM (mn) = λM(m)DλM (n) + λM(n)DλM (m) = 0 + 0 = 0
+And similarly for degree n ≥ 2, again by using the product rule, we have that DλM (m1m2 . . . mn) = 0. So
+it follows that im(DλM ) = {m| ∀m ∈ M}, so im(DλM ) = M. Furthermore, note that the multiplication of a
+and m in SymR(M) is precisely the module action, am = a · m. Thus the induced action on im(DλM ) from
+Lemma 5.1 is given by:
+a · DλM (m) = DλM (q(a)m) = DλM (am) = DλM (a · m) = a · m
+So im(DλM ) = M as R-modules.
+✷
+Conversely, let us start from a differential bundle, so let E = (q : E −→ R, σ, z, λ, ι) be a differential bundle
+with negatives over a commutative ring R in (CRINGop, T). To define a differential bundle isomorphism
+between E and M(im(Dλ), we will first need to define ring isomorphisms between E and SymR
+�
+im(Dλ)
+�
+.
+Define the ring morphism ψE : SymR
+�
+im(Dλ)
+�
+−→ E on generators a ∈ R and x ∈ E as follows:
+ψE(a) = q(a)
+ψE
+�
+Dλ(x)
+�
+= Dλ(x)
+(31)
+Note that ψE can also be defined by the universal property of the free symmetric R-algebra, that is, it is
+the unique R-algebra morphism induced by the inclusion im(Dλ) −→ E. To define the inverse we will need
+to use the dual of the Rosick´y’s universality diagram, which in this case asks that the following diagram be
+a pushout:
+T(R) ⊗ E
+[T(q),pE]
+�
+[0R,z]
+�
+T(E)
+λ
+�
+R
+q
+� E
+(32)
+So define the ring morphism δE : T(E) −→ SymR
+�
+im(Dλ)
+�
+on generators x ∈ E as follows:
+δE(x) = z(x)
+δE(d(x)) = Dλ(x)
+(33)
+By universality of the pushout, define ψ−1
+E
+: ker(q)[ε] −→ E as the unique ring morphism which makes the
+following diagram commute:
+T(R) ⊗ E
+[T(q),pE]
+�
+[0R,z]
+�
+T(E)
+λ
+�
+δE
+�
+R
+q
+�
+qim(Dλ)
+�
+E
+ψ−1
+E
+�❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+❚
+SymR
+�
+im(Dλ)
+�
+(34)
+so ψ−1
+E
+=
+�
+qim(Dλ), δE
+�
+.
+38
+
+Lemma 5.4 For a commutative ring R and a differential bundle with negatives E = (q : E −→ R, σ, z, λ, ι)
+over R in (CRINGop, T), ψE : E −→ M(im(Dλ)) is a differential bundle isomorphism over R in (CRINGop, T)
+with inverse ψ−1
+E
+: M(im(Dλ)) −→ E.
+Proof: We first explain why ψE and ψ−1
+E
+are well-defined ring morphisms. Clearly, ψE is well-defined by
+construction. On the other hand, to explain why ψ−1
+E
+is well-defined, we must show that the outer diagram
+of (34) commutes. First, note that by the dual of the second diagram of [DB.1], z(q(a)) = a for all a ∈ R,
+and recall that Dλ(q(a)) = 0 for all a ∈ R as well. Then on generators a ∈ R and x ∈ E we compute:
+δE
+�
+[T(q), pE](a ⊗ x)
+�
+= δE
+�
+(T(q)(a)pE(x))
+�
+= δE
+�
+q(a)x
+�
+= δE
+�
+q(a)
+�
+δE (x)= qim(Dλ)
+�
+z(q(a))
+�
+qim(Dλ)
+�
+z(x)
+�
+= qim(Dλ) (a) qim(Dλ)
+�
+z(x)
+�
+= qim(Dλ)(az(x)) = qim(Dλ)
+�
+0R(a)z(x)
+�
+= qim(Dλ)
+�
+[0R, z](a ⊗ x)
+�
+and
+δE
+�
+[T(q), pE](d(a) ⊗ x)
+�
+= δE
+�
+T(q)(d(a))pE(x)
+�
+= δE
+�
+d(q(a))x
+�
+= δE
+�
+d(q(a))
+�
+δE (x) = Dλ(q(a))x = 0
+= qim(Dλ)(0) = qim(Dλ)(0z(x)) = qim(Dλ)
+�
+0R(d(a))z(x)
+�
+= qim(Dλ)
+�
+[0R, z](d(a) ⊗ x)
+�
+So δE ◦ [T(q), pE] = qim(Dλ) ◦ [0R, z]. Therefore, by the universal property of the pushout square, there exists
+a unique ring morphism ψ−1
+E
+: E −→ SymR
+�
+im(Dλ)
+�
+such that ψ−1
+E
+◦ λ = δE and ψ−1
+E
+◦ q = qim(Dλ). In
+particular, these imply that for every a ��� R and x ∈ E the following equalities hold:
+ψ−1
+E (q(a)) = a
+ψ−1
+E (Dλ(x)) = Dλ(x)
+Next we show that ψE and ψ−1
+E
+are inverses of each other. To show that ψ−1
+E
+◦ ψE = 1SymR(im(Dλ)), we use
+the above identities and compute the following on generators a ∈ R and x ∈ E:
+ψ−1
+E (ψE(a)) = ψ−1
+E (q(a)) = a
+ψ−1
+E
+�
+ψE
+�
+Dλ(x)
+��
+= ψ−1
+E (Dλ(x)) = Dλ(x)
+So ψ−1
+E
+◦ ψE = 1SymR(im(Dλ)). On the other hand, to show that ψE ◦ ψ−1
+E
+= 1E, we will first show that
+ψE ◦ ψ−1
+E
+◦ q = q and ψE ◦ ψ−1
+E
+◦ λ = λ. So on generators a ∈ R and x ∈ E, we compute:
+ψE(ψ−1
+E (q(a))) = ψE(a) = q(a)
+ψE(ψ−1
+E (λ(x))) = ψE(δE(x)) = ψE(z(x)) = q(z(x)) = x
+ψE(ψ−1
+E (λ(d(x)))) = ψE(δE(d(x))) = ψE(Dλ(x)) = Dλ(x) = λ(d(x))
+Therefore, by the universal property of the pushout, it follows that ψE ◦ ψ−1
+E
+= 1E. So ψE and ψ−1
+E
+are
+inverse ring isomorphisms.
+Lastly, we must show that ψE and ψ−1
+E
+are also differential bundle morphisms over R in (CRINGop, T),
+that is, we must show the dual of the axioms in Definition 3.10 hold.
+We will first show that ψE is a
+differential bundle morphism. To do so, first recall that λ(q(a)) = q(a) and Dλ(q(a)) = 0, and that the
+dual of [DB.4] states that λ ◦ T(λ) = λ ◦ ℓE. So we show that the desired equalities hold by computing the
+following on generators:
+(i) ψE ◦ qim(Dλ) = q:
+ψE(qim(Dλ)(a)) = ψE(a) = q(a)
+(ii) λ ◦ T(βE) = ψE ◦ λim(Dλ), on a:
+λ
+�
+T(βE)(a)
+�
+= λ
+�
+βE(a)
+�
+= λ(q(a)) = q(a) = ψE(a) = ψE(λim(Dλ)(a)
+39
+
+on Dλ(x):
+λ
+�
+T(βE)
+�
+Dλ(x)
+��
+= λ
+�
+βE
+�
+Dλ(x)
+��
+= λ
+�
+Dλ(x)
+�
+= λ
+�
+λ(d(x))
+�
+= λ
+�
+T(λ)(d(x))
+�
+= λ
+�
+ℓE(d(x))
+�
+= λ(0) = 0 = ψE(0) = ψE
+�
+λim(Dλ)
+�
+Dλ(x)
+��
+on d(a):
+λ
+�
+T(βE)
+�
+d(a)
+��
+= λ
+�
+d
+�
+βE(a)
+��
+= λ
+�
+d
+�
+q(a)
+��
+= Dλ(q(a)) = 0 = ψE(0) = ψE
+�
+λim(Dλ)
+�
+d(a)
+��
+and finally on d
+�
+Dλ(x)
+�
+:
+λ
+�
+T(βE)
+�
+d
+�
+Dλ(x)
+���
+= λ
+�
+d
+�
+βE
+�
+Dλ(x)
+���
+= λ
+�
+d
+�
+Dλ(x)
+��
+= λ
+�
+d
+�
+λ(d(x))
+��
+= λ
+�
+T(λ)(d′d(x))
+�
+= λ
+�
+ℓE(d′d(x))
+�
+= λ
+�
+d(x)
+�
+= Dλ(x) = βE
+�
+Dλ(x)
+�
+= ψE
+�
+λim(Dλ)
+�
+d
+�
+Dλ(x)
+���
+So it follows that ψE is a differential bundle morphism over R in (CRINGop, T). By Lemma 3.13 it then follows
+that ψ−1
+E
+is also a differential bundle morphism over R. So we conclude that ψE and ψ−1
+E
+are differential
+bundle isomorphisms over R in (CRINGop, T).
+✷
+Thus, the construction from a module to a differential bundle is the inverse of the construction from a
+differential bundle to a module. So we conclude that:
+Proposition 5.5 For a commutative ring R, there is a bijective correspondence between R-modules and
+differential bundles (with negatives) over R in (CRINGop, T).
+In CRING, recall that initial object is Z, which means that Z is the terminal object in CRINGop. So
+differential objects in (CRINGop, T) correspond precisely to Z-modules, which are precisely Abelian groups.
+Corollary 5.6 There is a bijective correspondence between Z-modules/Abelian groups and differential objects
+in (CRINGop, T).
+We now extend Proposition 5.5 to an equivalence of categories. For a commutative ring R, we define an
+equivalence of categories between MODop
+R and DBUNT [R] as follows:
+(i) Define the functor MR : MODop
+R −→ DBUNT [R] which sends an R-module M to the differential bundle
+MR(M), and sends an R-linear morphism f : M −→ M ′ to the differential bundle morphism over R
+MR(f) : MR(M ′) −→ MR(M) defined to be the ring morphism MR(f) : SymR(M) −→ SymR(M ′) defined
+on generators as follows:
+MR(f)(a) = a
+MR(f)(m) = f(m)
+(ii) Define the functor M◦
+R : DBUNT [R] −→ MODop
+R which sends a differential bundle with negatives over R,
+E = (q : E −→ R, σ, z, λ, ι) to the R-module M◦
+R(E) = im(Dλ), and sends a differential bundle morphism
+f : E = (q : E −→ R, σ, z, λ, ι) −→ E′ = (q′ : E′ −→ R, σ′, z′, λ′, ι′) over R to the R-linear morphism
+M◦
+R(f) : im(Dλ′) −→ im(Dλ) defined as:
+M◦
+R(f)(Dλ′(x)) = Dλ(f(x))
+(iii) Observe that M◦
+R ◦ MR = 1MODop
+R .
+40
+
+(iv) Define the natural isomorphism ψ : 1DBUNT[R] ⇒ MR ◦ M◦
+R with inverse ψ−1 : MR ◦ M◦
+R ⇒ 1DBUNT[R] as
+ψE and ψ−1
+E
+as defined in Lemma 5.4.
+Theorem 5.7 For a commutative ring R, we have an equivalence of categories: MODop
+R ≃ DBUNT [R].
+Proof: We must first explain why MR and M◦
+R are well-defined. Clearly, MR is well-defined on objects
+and maps and preserves composition and identities. So MR is indeed a functor. On the other hand, let
+f : E −→ E′ be a differential bundle morphism over R in (CRINGop, T). This implies that f : E′ −→ E is a
+ring morphism and also that f(q′(a)) = q(a) for all a ∈ R. Since MR(f) is clearly linear, we show that it
+also preserves the action:
+M◦
+R(f)
+�
+a · Dλ′(x)
+�
+= M◦
+R(f)
+�
+Dλ′(q(a)x)
+�
+= Dλ
+�
+f(q(a)x)
+�
+= Dλ
+�
+f(q′(a))f(x)
+�
+= Dλ
+�
+q(a)f(x)
+�
+= a · Dλ(f(x)) = a · M◦
+R(f)(Dλ′(x))
+So we have that MR(f) is an R-linear morphism, and so M◦
+R is well-defined. Clearly, M◦
+R also preserves
+composition and identities, so M◦
+R is also a functor. Furthermore, we also have that M◦
+R ◦ MR = 1MODop
+R .
+Next, ψ and ψ−1 are well-defined component-wise and are inverses at each component. Therefore, it suffices
+to show that ψ is natural and then it will follow that ψ−1 is also natural. If f : E −→ E′ is a differential
+bundle morphism over R in (CRINGop, T), then f ◦λ′ = λ◦T(f). In particular, this means that f(λ′(d(x))) =
+λ(d(f(x))). However, we can rewrite this as f
+�
+Dλ′(x)
+�
+= Dλ
+�
+f(x)
+�
+. Therefore, we compute on generators
+that:
+ψE
+�
+MR
+�
+M◦
+R(f)
+�
+(a)
+�
+= ψE(a) = q(a) = f(q′(a)) = f(ψE′(a))
+and
+ψE
+�
+MR
+�
+M◦
+R(f)
+� �
+Dλ′(x)
+��
+= ψE
+�
+M◦
+R(f)
+�
+Dλ′ (x)
+��
+= ψE
+�
+Dλ
+�
+f(x)
+��
+= Dλ
+�
+f(x)
+�
+=
+�
+Dλ′(x)
+�
+= f
+�
+ψE′ �
+Dλ′(x)
+��
+So ψE ◦ MR
+�
+M◦
+R(f)
+�
+= f ◦ ψE′ in CRING. Therefore ψ is a natural transformation, and it follows that so is
+ψ−1. Thus, ψ and ψ−1 are natural isomorphisms. So we conclude that we have an equivalence of categories:
+MODop
+R ≃ DBUNT [R].
+✷
+It then follows that we have an equivalence between the category of differential objects and the opposite
+category of Abelian groups. So let Ab be the category whose objects are Abelian groups and whose morphisms
+are group morphisms.
+Corollary 5.8 There is an equivalence of categories: DBUN
+�
+(CRINGop, T)
+�
+≃ MODZ ≃ Abop.
+We now define an equivalence of categories between MODop and DBUN
+�
+(CRINGop, T)
+�
+as follows:
+(i) Define the functor M : MODop −→ DBUNT [R] which sends an object (R, M) to the differential bundle
+M(R, M) = MR(M), and sends a map (g, f) : (R, M) −→ (R′, M ′) in MOD to the differential bundle
+morphism M(f) = (MR(f), g) : MR′(M ′) −→ MR(M).
+(ii) Define the functor M◦ : DBUN
+�
+(CRINGop, T)
+�
+−→ MODop which sends a differential bundle with
+negatives E = (q : E −→ R, σ, z, λ, ι) to the pair M◦(E) = (R, im(Dλ)), and sends a differential bun-
+dle morphism (f, g) : E = (q : E −→ R, σ, z, λ, ι) −→ E′ = (q′ : E′ −→ R′, σ′, z′, λ′, ι′) to the pair
+M◦(f, g) = (g, M◦
+R(f)).
+(iii) Observe that M◦ ◦ M = 1MODop.
+(iv) Define the natural isomorphism ψ : 1DBUN[(CRINGop,T)] ⇒ M ◦ M◦ as ψE = (1, ψE), with inverse natural
+isomrophism ψ
+−1 : M ◦ M◦ ⇒ 1DBUN[(CRINGop,T)] as ψ
+−1
+E
+= (1, ψ−1
+E ).
+41
+
+Theorem 5.9 We have an equivalence of categories: MODop ≃ DBUN
+�
+(CRINGop, T)
+�
+.
+Proof: The proof that M and M◦ are well-defined functors is similar to the proof that MR and M◦
+R in the
+proof of Theorem 5.9. Furthermore, it also follows that M◦ ◦ M = 1MODop. On the other hand, since ψE and
+ψ−1
+E
+are differential bundle morphisms over the base commutative ring, it follows by definition that ψE =
+(1, ψE) and ψ
+−1
+E
+= (1, ψ−1
+E ) are differential bundle morphisms, so ψ and ψ
+−1 are well-defined. Lastly, that ψ,
+and ψ
+−1 are natural isomorphisms follows directly from the fact that ψ, and ψ−1 are natural isomorphisms.
+So we conclude that we indeed have an equivalence of categories: MODop ≃ DBUN
+�
+(CRINGop, T)
+�
+.
+✷
+Remark 5.10 The equivalence between modules and differential bundles is also true in more general set-
+tings. Indeed, both for the opposite category of commutative semirings and the opposite category of com-
+mutative algebras over a (semi)ring, differential bundles correspond precisely to modules via the above
+constructions (where the latter follows from Corollary 2.19 and Proposition 3.4). As explained before, in
+a setting where one does not have negatives, we would also need to prove [DB.5]. On the other hand, it
+is unclear if this result always generalizes to the coEilenberg-Moore category of a differential category. If
+the differential category has enough limits and colimits, then it is possible to generalize the constructions of
+Lemma 5.1 and Lemma 5.2, and then we obtain a bijective correspondence between differential bundles and
+comodules of the colagebras of the comonad of said differential category. However, in general, a differential
+category need not have all limits or colimits. In future work, it would be interesting to characterize differ-
+ential bundles in arbitrary differential categories and understand what assumptions are needed so that they
+correspond to (co)modules.
+5.4
+Differential bundles in schemes
+In this section, we show how we can extend the characterization of differential bundles in affine schemes to
+differential bundles in the larger category of schemes. Since schemes are the gluing of affine schemes, this
+follows relatively straightforwardly from the results of the previous sections, so here we merely sketch the
+proofs. Our first goal is to show that for any differential bundle q : E −→ A in schemes, the projection q is
+an affine map. Let us first quickly recall the definition of affine morphisms and equivalent characterizations
+[26, Section 29.11].
+Definition 5.11 [26, Definition 29.11.1] A morphism of schemes f : X −→ Y is affine if for all affine
+opens U of Y , the inverse image f −1(U), that is, the following pullback:
+f −1(U)
+�
+�
+X
+�
+U
+� Y
+is itself affine.
+Proposition 5.12 [26, Lemma 29.11.3] For a scheme morphism f : X −→ Y , the following are equivalent:
+(i) f is affine;
+(ii) Y has a covering by affine opens {Ui}i∈I such that for all i ∈ I, f −1(Ui) is affine;
+(iii) X = Spec(A) for some quasicoherent sheaf of algebras A on the sheaf OY .
+The following is a general result about affine morphisms which will be useful below:
+Lemma 5.13 Affine morphisms are closed under retract, that is, if we have scheme morphisms
+42
+
+X1
+s
+�
+f1
+�❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+X2
+r
+�
+f2
+�⑥⑥⑥⑥⑥⑥⑥⑥
+Y
+with (s, r) a section/retraction pair in the category of schemes over Y and f2 is affine, then so is f1.
+Proof: Let U be an affine open subset of Y . Then we can define a section/retraction pair (sU, rU) between
+f −1
+1 (U) and f −1
+2 (U) with both defined by pullback. For example, here is the defining diagram for sU:
+f −1
+1 (U)
+�
+sU
+�
+�
+X1
+s
+�❅
+❅
+❅
+❅
+❅
+❅
+❅
+❅
+f −1
+2 (U)
+�
+�
+X2
+f2
+�
+U
+� Y
+Thus f −1
+1 (U) is a retract of a representable element in the presheaf category [CRING, SET] (where SET is the
+category of sets and arbitrary functions between them). But so long as a category X has split idempotents,
+then representables in the functor category [Xop, SET] are closed under retract [13, Lemma 6.5.6]. So f −1
+1 (U)
+is itself representable, and so by definition f1 is affine, as required.
+✷
+We may now prove that for a differential bundle in the category of schemes, the projection is affine.
+Proposition 5.14 In the category of schemes, if q : E −→ A is a differential bundle, then q is affine.
+Proof: By [22, Corollary 3.1.4], q is a retract of a pullback of a tangent bundle projection pA : T(A) −→ A.
+By definition, T(A) is Spec of a quasicoherent sheaf of algebras on OA, so by Lemma 5.12, pA is affine. But
+affines are closed under pullback [26, Lemma 29.11.8] and retracts (Lemma 5.13), so q is affine.
+✷
+We may now prove that every differential bundle is a Spec of Sym of a quasicoherent sheaf of modules.
+Proposition 5.15 If q : E −→ A is a differential bundle in the category of schemes, then E is Spec of Sym
+of a quasicoherent sheaf of modules.
+Proof: Cover A by affines Ui, and since q is affine, each pullback q−1(Ui):
+q−1(U)
+�
+�
+E
+q
+�
+Ui
+� A
+is also affine. Moreover, by [8, Lemma 2.7], differential bundles are closed under pullback. Thus each map
+q−1(Ui) −→ Ui is a differential bundle in the category of affine schemes, and hence by Proposition 4.5, each
+q−1(Ui) is Sym of a module on Ui. Thus as E is the gluing of these, E is itself Spec of Sym of a quasicoherent
+sheaf of modules [28, pg. 379].
+✷
+Conversely, we now prove that every quasicoherent sheaf of modules induces a differential bundle.
+Proposition 5.16 If M is a quasicoherent sheaf of modules on a scheme A, then Spec of Sym of M is a
+differential bundle over A.
+43
+
+Proof: Suppose that A is covered by affines Ui. Then by [28, pg.379], if M is a quasicoherent sheaf of
+modules on A, then M is the gluing of modules Mi over the Ui, and Spec of Sym of M is the gluing of
+Spec of Sym of the Mi’s. Thus it suffices to show that such a gluing is a differential bundle over A. But
+by Lemma 5.2, Spec of Sym of each Mi is a differential bundle over Ai.
+It then follows that since the
+tangent functor on schemes preserves gluings (for an abstract proof of this, see [5, Prop. 6.15.ii]), the lifts
+of each such differential bundle λi glue together to give a lift λ for Spec of Sym of M, and it follows through
+straightforward calculations that this satisfies the required conditions to be a differential bundle.
+✷
+The results on morphisms follow similarly, and therefore we obtain:
+Theorem 5.17 For a scheme A, there is an equivalence of categories between differential bundles over A
+in the tangent category SCH and the opposite category of quasicoherent sheaves of modules over A.
+Remark 5.18 By Corollary 2.22 and Proposition 3.4, for any scheme A, there is a similar result for the
+tangent category of schemes over A.
+6
+Future work
+Understanding differential bundles in the tangent categories of commutative rings, affine schemes, and
+schemes is just the beginning of applying tangent category theory to algebra and algebraic geometry. There
+are many possible future avenues for exploration based on this work, such as:
+• The most immediate next step is to understand how connections in tangent categories apply to these
+examples. They seem closely related to connections on modules [23, Section 8.2], but more work needs
+to be done to understand the precise relationship between the two notions.
+• Tangent categories have a notion of differential forms and de Rham cohomology [11]. Initial inves-
+tigation with this idea suggests that for affine schemes over R, when the coefficient object is taken
+to be the polynomial ring R[x], then this tangent category cohomology recreates algebraic de Rham
+cohomology. However, again more investigation is required to prove this completely. Moreover, [11]
+also develops a second notion of cohomology: sector form cohomology. It is not clear what this should
+give in the algebraic geometry setting.
+• In [16], Dominic Joyce develops algebraic geometry in the setting of C∞-rings. It seems likely that
+the categories involved are tangent categories, and one expects many of the tangent categories theory
+ideas, applied to this example, recreate the corresponding notions Joyce has developed.
+• A key idea in algebraic geometry is that of a smooth morphism or object. It would be interesting to see
+if such a notion could be generalized to arbitrary tangent categories (in such a way, that, for example,
+all objects in the tangent category of smooth manifolds are smooth).
+• Finally, the Serre-Swan theorem provides a very different way to compare vector bundles to modules.
+It would be interesting to see a proof for the Serre-Swan theorem based on some of the results of this
+paper.
+Thus, while the results of this paper are interesting enough on their own, we hope they will also serve as
+inspiration for future work in this area.
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