diff --git "a/DdE5T4oBgHgl3EQfUQ9g/content/tmp_files/load_file.txt" "b/DdE5T4oBgHgl3EQfUQ9g/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/DdE5T4oBgHgl3EQfUQ9g/content/tmp_files/load_file.txt" @@ -0,0 +1,2221 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf,len=2220 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='05542v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='CT] 13 Jan 2023 Tangent categories as a bridge between differential geometry and algebraic geometry G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In this paper, we use the abstract notion of a tangent category to make some of these commonalities precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Tangent Categories 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 Basics of Tangent Categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2 Commutative rings as a Tangent Category .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3 Affine schemes as a Tangent Category .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 14 3 Differential Bundles 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 Differential Bundles and Differential Objects .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2 Differential Bundles as Pre-Differential Bundles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3 Morphisms and Categories of Differential Bundles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 24 4 Differential Bundles for Commutative Rings 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 From Differential Bundles to Modules .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2 From Modules to Differential Bundles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3 Equivalence .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 29 5 Differential Bundles for (Affine) Schemes 34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 From Differential Bundles to Modules .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2 From Modules to Differential Bundles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3 Equivalence .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4 Differential bundles in schemes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 42 6 Future work 44 ∗Partially supported by an NSERC Discovery grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' †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?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' While there are many differences between these two subjects, one common thread is the use of “differential” methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Indeed, discussions of tangent vectors, tangent spaces, and differentials are important in both subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' A natural question to ask then is: can we precisely relate and contrast how differential geometry and algebraic geometry use these ideas?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Tangent categories were first introduced by Rosick´y in [25], and later generalized and further developed by Cockett and Cruttwell in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' A tangent category (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The canonical example of a tangent category is the category of smooth manifolds, where the endofunctor is the tangent bundle functor (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' But there are many other interesting examples of tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In fact, almost any category which has some form of “differentiation” for its morphisms can be given the structure of a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Examples of tangent categories include: Most generalizations of smooth manifolds form tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Any Cartesian differential category [3], which formalizes differential calculus over Euclidean spaces, gives a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In particular, there are many examples of Cartesian differential categories from computer science, such as models of the differential lambda-calculus [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The category of commutative rings and the category of commutative algebras are tangent categories, with a particularly simple tangent structure induced by dual numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' This example will be discussed in more detail below (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' “Tangent infinity” categories model ideas in Goodwillie functor calculus [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The theory of tangent categories is now well-established with a rich literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' There are also many things one can do in an arbitrary tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In particular, one can discuss: Vector fields (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='10), and prove important ideas such as the Jacobi identity [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The analogue of vector bundles, known as differential bundles [8], which is one of the main structures of focus in this paper (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Connections on differential bundles and corresponding results such as the Bianchi identities and the existence of geodesics [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Solutions to differential equations and dynamical systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Differential forms and de Rham cohomology [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' It is worth noting that translating these ideas from the category of smooth manifolds to an arbitrary tangent category is not a trivial process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' A central question of tangent category theory 2 is to understand what these notions look like in the various examples of tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' But in other areas, what these notions give is less obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In particular, categories of (affine) schemes (potentially over some fixed ring or scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' While this example was mentioned in [5], as a corollary to a more general result, the tangent structure was not explored explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' One of the contributions of this paper is an explicit description of the natural transformations for this tangent structure (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' This is a necessary component to further understand how the theory of tangent categories applies to algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Do they recreate existing notions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Do they give us new perspectives on existing ideas?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The main focus of this paper is on differential bundles (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1) in the examples of tangent cate- gories in algebra and algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Differential bundles are a central structure in tangent categories, as they generalize smooth vector bundles in the category of smooth manifolds (Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, intriguingly, they are defined quite differently than vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The definition of a differential bundle contains no mention of either vector spaces, a base field, or local triviality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' But what are differential bundles in the tangent categories of (affine) schemes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' It is not immediately obvious what they should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In summary, the main results of this paper are that: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' These results are fascinating for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For one, they show how diverse differential bundles can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In the canonical tangent category example of smooth manifolds, differential bundles are exactly smooth vector bundles, which includes the strict condition of local triviality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Independently, these results are also interesting as they give a new characterization of these categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In particular, differential bundles provide a novel characterization of the opposite of the category of (quasicoherent sheaves of) modules over a commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' These results are thus interesting in and of themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Even more promising than the results themselves is what future results and ideas they can lead to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' As described above, in any tangent category one can define and prove results about connections on such bundles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' again, when applied to the tangent category of smooth manifolds, this recreates the usual notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' But now via tangent categories, we get a notion of connection on modules - what do these look like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' What examples of them are there?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Do they recreate existing notions of connections in algebraic geometry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Outline: In section 2, we review the definition of tangent categories and explore some of their basic examples and theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We also explicitly describe the tangent structure of the category of (affine) schemes, which as noted above, has not previously been given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In Section 4, we give our first major result: a characterization of differential bundles in the tangent category of commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Section 5 contains our most important results: characterizations of differential bundles in the tangent categories of affine schemes and schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' As mentioned above, as far as we know, these results provide new characterizations of the opposites of categories of (quasicoherent sheaves of) modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lastly, in Section 6, we describe future work that we hope to pursue that builds on the ideas presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Tangent categories were first defined by Rosick´y [25], then later generalized by Cockett and Cruttwell [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We begin by providing the full definition of a tangent category, both the Cockett and Cruttwell version without negatives (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1), and the Rosick´y version with negatives (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Afterwards, we provide a detailed description of the tangent categories of commutative rings (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2) and (affine) schemes (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3), the latter of which has not been previously done in full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We also discuss some basic, but important, concepts in these tangent categories, like tangent spaces (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8) and vector fields (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 Basics of Tangent Categories The following definition of a tangent category is the one provided in [8, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1], which when compared to the original definition provided in [5, Definion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3] is the same except for the universality of the vertical lift which is presented as a pullback instead of an equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' That said, these two axiomatizations are indeed equivalent [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 [5, Definion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (iii) A natural transformation1 +A : T2(A) −→ T(A), called the sum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (iv) A natural transformation 0A : A −→ T(A), called the zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (v) A natural transformation ℓA : T(A) −→ T2(A), called the vertical lift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (vi) A natural transformation cA : T2(A) −→ T2(A), called the canonical flip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and such that: [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1] (pA, +A, 0A) is an additive bundle over A [5, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='π2⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='π3⟩ � ⟨π1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='+A◦⟨π2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='π3⟩⟩ � T2(A) +A � T(A) ⟨0A◦pA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1T(A)⟩ � ⟨1T(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='0A◦pA⟩ � ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ T2(A) +A � T2(A) +A � T(A) T2(A) +A � T(A) T2(A) +A �◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ⟨π2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='π1⟩ � T2(A) +A � T(A) (1) [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 5 [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' A tangent category is a pair (X, T) consisting of a category X equipped with a tangent structure T on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Tangent categories formalize the properties of the tangent bundle on smooth manifolds from classical differential geometry, as we will review in Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' An object A should be interpreted as a base space, and T(A) as its abstract tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To make this more precise, this notion is captured by the concept of an additive bundle [5, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1], which can be defined in any arbitrary category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Briefly, additive bundles are commutative monoid objects in slice categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The top two diagrams in [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus, the vertical lift ℓA is an analogue of the embedding of the tangent bundle into the double tangent bundle via the vertical bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The canonical flip cA is an analogue of the natural canonical flip, which is a smooth involution on the double tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The diagrams in [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2] and [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3] say respectively that ℓA and cA are additive bundle morphisms [5, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2], that is, monoid morphisms in the slice category over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The diagrams in [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The diagrams in [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5] are compatibility relations between the vertical lift and canonical flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The universality of the vertical lift in [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='6] is essential for generalizing desired important properties of the tangent bundle from differential geometry, see [5, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5] for more details on this axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lastly, for maps, T(f) is interpreted as the differential of f, and so the functoriality of T represents the chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In [5, 22], such a setting was simply called a tangent category with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Here, we introduce new terminology and call such a setting a Rosick´y tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2 [5, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' such that: [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 7 In a Rosick´y tangent category, the negative nA makes each fiber an Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The left diagram of [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='13]: [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Indeed, recall that in classical group theory that morphisms which preserve the group’s addition also preserve inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The same is true for Abelian group bundles in the sense that additive bundle morphisms between Abelian group bundles automatically also preserve inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Explicitly: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4 [5, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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(∗) ∼= ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' This example provides a direct link between tangent categories and differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Here we review in full the tangent structure on the subcategory of Euclidean spaces, as it is simpler to describe in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For lists of other examples of tangent categories see [8, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2] and [14, Example 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Let SMOOTH be the category whose objects are Euclidean spaces Rn and whose maps are smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To define the tangent structure, recall that for a smooth 8 function F : Rm −→ Rn, which is actually an n-tuple F = ⟨f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , m � j=1 ∂fn ∂uj (⃗x)yj � The total derivative D[F] can also be expressed in terms of the Jacobian of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗y) = � F(⃗x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' D[F](⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗y) � (ii) The projection pRn : Rn × Rn −→ Rn is defined as the projection of the first component: pRn(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗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 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' × Rn � �� � m+1 times and where the projection πj : Tm(Rn) −→ Rn × Rn projects out the first and j-th components: πj(⃗x, ⃗y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗ ym) = (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗yj) (iv) The sum +Rn : Rn × Rn × Rn −→ Rn × Rn adds the second and third components: +Rn(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='⃗z) = (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗y + ⃗z) (v) The zero 0Rn : Rn −→ Rn × Rn inserts the zero vector into the second component: 0Rn(⃗x) = (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='⃗0) (vi) The vertical lift ℓRn : Rn × Rn −→ Rn × Rn × Rn × Rn inserts zero vectors into the middle components: ℓRn(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗y) = (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='⃗0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='⃗0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗y) (vii) The canonical flip cRn : Rn × Rn × Rn × Rn −→ Rn × Rn × Rn × Rn flips the middle two components: cRn(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='⃗z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗w) = (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='⃗z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗w) (viii) The negative −Rn : Rn × Rn −→ Rn × Rn makes the second component negative: −Rn(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⃗y) = (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' −⃗y) So T = (T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' n) is a tangent structure with negatives on SMOOTH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lastly, we also have that Rn × Rn × Rm × Rm ∼= Rn × Rm × Rn × Rm and R0 × R0 ∼= R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So (SMOOTH, T) is a Cartesian Rosick´y tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 9 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='9] and allow our manifolds to have different dimensions in different connected components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Let SMAN be the category whose objects are (finite-dimensional real) smooth manifolds M and whose maps are smooth functions between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For a smooth manifold M, for each point x ∈ M let Tx(M) be the tangent space at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So (SMAN, T) is a Cartesian Rosick´y tangent category, for which (SMOOTH, T) is a sub-Cartesian Rosick´y tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We will use this construction, in particular, to construct tangent categories of algebras from tangent categories of rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7 [5, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5] Suppose that (X, T) is a tangent category, and A is an object of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In the examples above, vector fields and tangent spaces correspond precisely to their namesakes from classical differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Below, we will also discuss these tangent spaces and vector fields for the tangent categories of commutative rings and (affine) schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Recall that in a category with finite products, a point of an object A is a map from the terminal object to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8 [5, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='9 In SMOOTH, a point of Rn in the categorical sense corresponds precisely to elements of ⃗x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So in (SMOOTH, T), the tangent space of Rn at ⃗x ∈ Rn is T��x(Rn) = Rn and π⃗x : Rn −→ Rn × Rn is defined as the injection π⃗x(⃗y) = (⃗x, ⃗y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Similarly, in SMAN, a point of a smooth manifold M in the categorical sense is precisely a point x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So in (SMAN, T), the tangent space of M at x ∈ M is the classical tangent space Tx(M) = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 10 Tangent spaces are commutative monoid objects, where monoid structure is induced by the tangent bundle [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='15], and in a Cartesian Rosick´y tangent category, tangent spaces are also Abelian group objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The category of tangent spaces is a Cartesian differential category [5, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Also observe that, trivially, for the terminal object ∗ the only point is the identity 1∗ : ∗ −→ ∗ and that T1∗(∗) = ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We now turn our attention to vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In a tangent category, a vector field is simply a section of the tangent bundle’s projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='10 [5, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, vector fields in (SMOOTH, T) correspond precisely to endomorphisms in SMOOTH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In (SMAN, T), vector fields in the tangent category sense correspond precisely to vector fields in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In any tangent category, the zero 0A : A −→ T(A) is a vector field, and the map νA : T2(A) −→ T2(A) from [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' One can also define the sum of vector fields using the sum + [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8], and said category of vector fields is also a tangent category [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In a Rosick´y tangent category, it is possible to define the Lie Bracket of vector fields [5, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='14], which in particular also satisfies the Jacobi identity [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Vector fields can also be used to describe differential equations, dynamical systems, and their solutions in a tangent category [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4] or the notion of a representable tangent category [5, Section 5], which provides a link to synthetic differential geometry (SDG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' This was one of the main examples in Rosick´y’s original paper [25, Example 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' By a commutative ring, we mean a commutative, unital, and associative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Let CRING be the category whose objects are commutative rings and whose maps are ring morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Using the ring of dual numbers, we define a tangent structure with negatives T = ( T , p, +, 0, ℓ, c, −) on CRING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and so is defined as projecting out the first component: pR(a + bε) = a To describe the pullbacks of the projection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' first recall that CRING is a complete category,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and therefore all pullbacks exist in CRING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , xn)| xj ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' f(xi) = f(xj) for all 1 ≤ i, j ≤ n} and whose ring structure is given coordinate-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However for the projection of the ring of dual numbers, one can instead describe these pullbacks in terms of multivariable dual numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So for a commutative ring R, define R[ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , εn] as follows: R[ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , εn] = {a + b1ε1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' + bnεn| ∀a, bi ∈ R and εiεj = 0} Then R[ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , εn] is a commutative ring whose structure is defined in the obvious way, so in particular the multiplication is induced by εiεj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We leave it as an exercise for the reader to check for themselves that R[ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , εn] is indeed isomorphic to the pullback of n copies of pR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , εn] and where πj : R[ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , 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 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' + 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We also have that (R×S)[ε] ∼= R[ε]×S[ε] and 0[ε] ∼= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we have that: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='12 (CRING, T ) is a Cartesian Rosick´y tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For any commutative (semi)ring R, the category of commutative R-algebras will also be a Cartesian tangent category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' this follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In fact, these tangent categories of commutative (semi)rings/algebras are precisely the Eilenberg-Moore categories of the appropriate polynomial models of codifferential categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We conclude this section by discussing tangent spaces and vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We first explain how in (CRING, T ), there are no non-trivial tangent spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' But the zero ring is the only ring for which the zero and unit are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Next we discuss vector fields in (CRING, T ) and explain how they correspond precisely to derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' A commutative differential ring is a pair (R, D) consisting of a commutative ring R and derivation D on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' It is a well-known result that ring morphisms v : R −→ R[ε] such that pR ◦ v = 1R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' vector fields, correspond precisely to derivations on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Indeed, if v is a vector field, then since it preserves the multiplication, it follows that Dv satisfies the product rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus Dv is a derivation and (R, Dv) is a commutative differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Conversely, given a derivation D : R −→ R on R, define the vector field vD : R −→ R[ε] as vD(a) = a + D(a)ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Since these constructions are inverses of each other, we obtain the desired equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='15 For a commutative ring R, vector fields on R in (CRING, T ) are in bijective correspondence with derivations on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 13 Therefore, it follows that the category of vector fields of (CRING, T ) is equivalent to the category of commutative differential rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus by [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lastly, the tangent category Lie bracket corresponds precisely to the standard Lie bracket of derivations, [D1, D2] = D1 ◦ D2 − D2 ◦ D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In this paper, by the category of affine schemes we mean the opposite category of commutative rings CRINGop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So in this section, we provide a full description of the tangent structure on CRINGop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To give a tangent structure with negatives on CRINGop, we must give a “co-tangent structure with nega- tives” on CRING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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) � ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' where ⊗s R is the symmetrized tensor product over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In [15, Definition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='I], Grothendieck calls T(R) the “fibr´e tangente” (French for tangent bundle) of R, while in [17, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='6], Jubin calls T(R) the tangent algebra of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For the story of this paper, it will be useful to have a more explicit description of T(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus, an arbitrary element of T(R) is a finite sum of monomials of the form ad(b1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' d(bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So the ring structure of T(R) is essentially the same as polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Furthermore, T(R) also has a universal property similar to that of the module of K¨ahler differentials, but instead for algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , xn], its tangent bundle is the 2n-variable polynomial ring: T(Z[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , xn]) = Z[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , xn, d(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , d(xn)], with no added assumptions on the variables d(xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (iii) For coordinate rings of varieties, that is, the polynomial rings quotiented by some finitely generated ideal Z[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , xn]/⟨p(⃗x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , xn]/⟨p(⃗x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , q(⃗x)⟩ � =Z[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , xn, d(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , d(xn)]/⟨p(⃗x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , q(⃗x), d(p)(⃗x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, to define ring morphisms with domain T(R), it suffices to define them on generators a and d(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Using this to our advantage, we can define a tangent structure on CRINGop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Recall that CRING is cocomplete, and therefore admits all pushouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⊗R R′ � �� � n times , where ⊗R is the tensor product over R of R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The induced R-algebra structure on T(R) via pR is precisely given by multiplication, as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (iii) The pushouts of n copies of pR is given by Tn(R) := T(R) ⊗R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⊗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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⊗R 1 ⊗R w ⊗R 1 ⊗R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⊗R 1 To describe the sum, zero, and negative, let us first explain what additive bundles and Abelian group bundles are in CRINGop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' An additive bundle over R in CRINGop corresponds precisely to a commutative R- bialgebra over the tensor product ⊗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The sum and zero of the additive bundle are the comultiplicaiton and counit respectively of the R-coalgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' An Abelian group bundle over R in CRINGop corresponds precisely to a commutative R-Hopf algebra over the tensor product ⊗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The negative of the Abelian group bundle gives the antipode for the R-Hopf algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Luckily, free symmetric R-algebras have a canonical commutative R-Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' defined on generators as follows: −R(a) = a −R(d(a)) = −d(a) To describe the vertical lift and the canonical flip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and that d′d is the composite of derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus CRINGop has finite products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we have that: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='17 (CRINGop, T) is a Cartesian Rosick´y tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Per [5, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' This is precisely what is happening between the tangent categories (CRING, T ) and (CRINGop, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Furthermore, (CRINGop, T) is not only a Cartesian Rosick´y tangent category but also a representable tangent category [5, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The object D is called the infinitesimal object [5, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (CRINGop, T) is a representable tangent category where the infinitesimal object is the ring dual of numbers for the integers, Z[ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we have that T(R) ∼= RZ[ε] in CRINGop, and T (R) ∼= R ⊗ Z[ε] in CRING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 16 Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7, we then get that for each commutative ring R, the slice category CRINGop/R is also a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' But as is well-known, this slice category is equal to the (opposite of) the category of commutative R-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus we have: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The category of schemes SCH is also a Cartesian Rosick´y tangent category in a similar fashion to the category of affine schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Indeed, recall that a scheme is by definition the gluing of affine schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So the tangent bundle of a scheme is defined as the gluing of the tangent bundles of each affine piece of the said scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Full details can be found in [14, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (iii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='21 (SCH, T) is a Cartesian Rosick´y tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' As with affine schemes, we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7 to tangent structure on each category of relative schemes: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='22 For each scheme A, the slice category SCH/A has the structure of a Cartesian Rosick´y tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We now discuss tangent spaces in (CRINGop, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The terminal object in CRINGop is the initial object in CRING, which is the integers Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus a tangent space of R at a point r in (CRINGop, T) corresponds to the pushout of r and pR in CRING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' This essentially amounts to applying r on the R parts of the tangent bundle T(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So let im(r) = {r(a)| ∀a ∈ R} be the image of r, which is a sub-ring of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' d(bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='23 Here are some examples of tangent spaces: (i) The only point for Z is the identity, and T1Z(Z) = Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (ii) For the polynomial ring in n-variables Z[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , xn], points correspond precisely to evaluating polyno- mials at a point ⃗a ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, for any point, the tangent space at that point is the polynomial ring in n-variables, T⃗a(Z[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , xn]) = Z[d(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , d(xn)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 17 (iii) For a polynomial ring quotiented by a finitely generated ideal Z[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , xn]/⟨p(⃗x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , q(⃗x)⟩, points correspond to points ⃗a ∈ Zn which are solutions to p(⃗a) = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=', q(⃗a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , d(q)(⃗x) in the xi variables at ⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' This is not true in the tangent category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Next, we discuss vector fields in (CRINGop, T) and explain how, like in the commutative ring case, they correspond precisely to derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Then define Dv : R −→ R as Dv(a) = v(d(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' It follows that Dv is a derivation and (R, Dv) is a commutative differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Since these constructions are inverses of each other, we obtain the desired equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='24 For a commutative ring R, vector fields on R in (CRINGop, T) are in bijective correspondence with derivations on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, it follows that the category of vector fields of (CRINGop, T) is equivalent to the opposite category of commutative differential rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus by [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 3 Differential Bundles In this section, we review differential bundles, as introduced by Cockett and Cruttwell in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Differential bundles generalize the notion of smooth vector bundles to an arbitrary tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We provide the full definition (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1), review how smooth vector bundles do indeed correspond to differential bundles (Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3, as shown by MacAdam in [22]), and also consider differential object (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5), which are differential bundles over the terminal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We then discuss morphisms between differential bundles and categories of differential bundles (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 [8, Definion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (iii) A map σ : E2 −→ E of X, called the sum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (iv) A map z : A −→ E of X, called the zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (v) A map λ : E −→ T(E) of X, called the lift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and such that: [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1] (q, σ, z) is an additive bundle over A [5, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4] The following diagram commutes: E λ � λ � T(E) T(λ) � T(E) ℓE � T2(E) (14) [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Differential bundles generalize smooth vector bundles over smooth manifolds in the context of a tangent category, as we will review in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1], which recall is a commutative monoid in the slice category over A, which is what the diagrams of [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1] state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The lift and its universal property [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5] is related to local triviality for smooth vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Then [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2] and [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3] say the lift λ preserves both of these monoid structures, or more precisely, that λ is an additive bundle morphism [5, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2], which recall is a monoid morphism in the slice category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lastly, [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4] is the compatibility between the lift of a smooth vector bundle and the vertical lift of the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For more properties of differential bundles, see [8, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We now define differential bundles with negatives, which are differential bundles with an added structure map which makes each fiber into an Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (ii) A map ι : E −→ E of X, called the negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' such that: [D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Note that a differential bundle can have negatives in any arbitrary tangent category and that the negative is necessarily unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore in a Cartesian Rosick´y tangent category, differential bundles are the same as differential bundles with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The result is somewhat surprising, as the definition of differential bundles contains no mention of local triviality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2 of [22] shows that this indeed gives a differential bundle in the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 21 The following result about differential bundles in slice tangent categories is easy to check: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4 If (X, T) is a tangent category with an object A which satisfies the requirements of Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We conclude this section by briefly discussing differential objects, which are the differential bundles over the terminal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Differential objects were first defined in [5, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8], before the introduction of differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Differential objects are quite important since they provide the link from tangent categories to Cartesian differential categories [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Indeed, the subcategory of differential objects (and all maps between them) is a Cartesian differential category [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Conversely, every Cartesian differential category is a tangent category in which every object is a differential object [8, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Later, it was shown in [8, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4] that differential objects were precisely the same thing as differential bundles over the terminal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Since the focus of this paper is on differential bundles, we take this approach to defining differential objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5 [8, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4] In a Cartesian tangent category (X, T), a differential object is a differential bundle over the terminal object ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8] commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In a Cartesian Rosick´y tangent category, every differential object is automatically an Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='6 In (SMAN, T), the differential objects are precisely the Euclidean spaces since in particular T(Rn) = Rn × Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore (SMOOTH, T) is equivalent to the resulting Cartesian differential category of differential objects of (SMAN, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2 Differential Bundles as Pre-Differential Bundles In this section, we review MacAdam’s pre-differential bundles as introduced in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' These allow for an alternative characterization of differential bundles, which in particular requires less data and axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' From this special pullback, the sum (and negative) for the differential bundle can be constructed from the sum (and negative) of the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' This will be particularly useful when we will characterize differential bundles for commutative rings and (affine) schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7 [21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 10] In a tangent category (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' a pre-differential bundle is a triple (q : E −→ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z : A −→ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λ : E −→ T(E)) consisting of objects A and E of X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and maps q : E −→ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z : A −→ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z : A −→ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λ : E −→ T(E)) is a pre-differential bundle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' we say that it is a pre-differential bundle over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' By definition, the projection, zero, and lift of a differential bundle gives a pre-differential bundle, since the diagrams in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7 all appear in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Furthermore, the induced sum as constructed in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8 is precisely the sum σ one started with, and so (q, σ, z, λ, ι) is a differential bundle with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Similarly, if (q : E −→ A, σ, z, λ, ι) is a differential bundle with negatives, then the induced negative as constructed in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8 is precisely the negative ι one started with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, in a Cartesian Rosick´y tangent category, every differential bundle is in fact a differential bundle with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In conclusion, we have the following equivalence: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (ii) Differential bundles with negatives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (iii) Pre-differential bundles that satisfy (i) and (ii) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3 Morphisms and Categories of Differential Bundles In this section, we discuss morphisms between differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' There are two possible kinds: one where the base objects can vary and one where the base object is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In either case, a differential bundle morphism is asked to preserve the projections and the lifts of the differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='10 [8, Definion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3] Let (X, T) be a (Rosick´y) tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (i) Let E = (q : E −→ A, σ, z, λ) and E′ = (q′ : E′ −→ A′, σ′, z′, λ′) be differential bundles in (X, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' A differential bundle morphism (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' T) � be the category whose objects are differential bundles in (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' maps are differential bundle morphisms between them,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' identity maps are pairs of identity maps (1E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 1A) : E −→ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and composition is defined point-wise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g) ◦ (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' k) = (f ◦ h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g ◦ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Differential bundle morphisms automatically preserve the sum and zero, and negatives if they exist: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='11 [8, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Then (f, g) is an additive bundle morphism [5, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore for a Rosick´y tangent category, it follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='9 that its category of differential bundles is the same as its category of differential bundles with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For a Cartesian (Rosick´y) tangent category, its category of differential objects is the category of differential bundles over the terminal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Note that this is not the same as the Cartesian differential category of differential objects [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='11], since in that category the morphisms are not required to preserve the lift, sum, zero, or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='12 Let (X, T) be a Cartesian (Rosick´y) tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Define DIFF � (X, T) � to be the cate- gory of differential objects and differential bundle morphisms over ∗ between them, DIFF � (X, T) � = DBunT[∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We conclude this section by discussing differential bundle isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' If (X, T) is a (Rosick´y) tangent category, then a differential bundle isomorphism is an isomorphism in the category DBun � (X, T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' By definition of the composition in DBun � (X, T) � , this is precisely the same as requiring that f and g are isomorphisms in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' This will allow us to reduce the number of things to check when characterizing differential bundles in various tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='13 Let (X, T) be a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' If f : E −→ E′ and g : A −→ A′ are isomorphisms in X, then (f −1, g−1) : E′ −→ E is a differential bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, (f, g) is a differential bundle isomorphism with inverse (f −1, g−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' If f : E −→ E′ is an isomorphism in X, then f −1 : E′ −→ E is a differential bundle morphism over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore f is a differential bundle isomorphism with inverse f −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='ii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For (ii), if f is a differential bundle morphism over A, then (f, 1A) is a differential bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To go from a differential bundle to a module, we take the kernel of the projection (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We then obtain that the category of differential bundles is equivalent to the category of modules (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We will also explain how the only differential object is the zero ring (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 From Differential Bundles to Modules We begin by unpacking what a differential bundle with negatives would consist of in (CRING, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' First recall that (CRING, T ) is a Cartesian Rosick´y tangent category, so by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='9, differential bundles are the same thing as differential bundles with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Also, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , xn)| xj ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' q(xi) = q(xj) for all 1 ≤ i, j ≤ n} and whose ring structure is given coordinate-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In particular, E2 = {(x, y)| x, y ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' q(x) = q(y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' These also need to satisfy the equalities and properties of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1, many of which we will expand further in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To obtain an R-module, we take the kernel of the projection q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Then the kernel of the projection ker(q) = {x| q(x) = 0} is an R-module with action a · x = z(a)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proof: Since q : E −→ R is a ring morphism, this induces an R-module structure on E with action a · e = z(a)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2 From Modules to Differential Bundles We now construct a differential bundle from a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We now define a differential bundle over R structure on M[ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So define M[ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , εn] as follows: M[ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , εn] = {a + m1ε1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' + mnεn| ∀a ∈ R, mj ∈ M and εiεj = 0} Then M[ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , εn] is a commutative ring whose structure is defined in the obvious way, so in particular the multiplication is induced by εiεj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We leave it as an exercise for the reader to check for themselves that M[ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , εn] is the pullback of n copies of pR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , εn] and where the pullback projection πj : M[ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , 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 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' + mnεn) = a + mjε (iii) The sum σ : M[ε1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ε2] −→ M[ε] maps both ε1 and ε2 to ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' which results in making the M component negative: ι(a + mε) = a − mε To describe the lift,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' let us describe T � M[ε] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' b ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' n ∈ M and ε2 = ε′2 = 0} where the multiplication is induced by ε2 = ε′2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To show that (qM, zM, λM) is a pre-differential bundle, we must show that the four equalities from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7 hold, but these all follow from straightforward computation, which we leave to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Next, we must show that this pre-differential bundle also satisfies the extra assumptions required to make it a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Firstly, it is straightforward to observe that M[ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' , εn] is indeed the pullback of n copies of the projection q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Also, since T is a right adjoint, it preserves all limits, and therefore all T n preserves these pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So (qM, zM, λM) satisfies assumption (i) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' we must show that the following square is a pullback: M[ε] λM � qM � M[ε][ε′] ⟨ T (qM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='pM[ε]⟩ � R ⟨0R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='zM⟩ � R[ε] × M[ε] (25) So suppose S is a commutative ring,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and we have ring morphisms f : S −→ M[ε][ε′] and g : S −→ R such that ⟨ T (qM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' pM[ε]⟩ ◦ f = ⟨0R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' zM⟩ ◦ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' for every x ∈ S the following equality holds: � T (qM)(f(x)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' pM[ε](f(x)) � = � g(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' f3(x) ∈ R and f2(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' f4(x) ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, in both equalities, the left-hand side has no nilpotent component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, we have that g(x) = f1(x), f2(x) = 0, and f3(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So f(x) = g(x) + f4(x)εε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lastly, it remains to show that ⟨f, g⟩ is the unique such ring morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So suppose we have a ring morphism h : S −→ M[ε] such that λM ◦ h = f and qM ◦ h = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, h(x) = g(x) + f4(x)ε = ⟨f, g⟩(x), and so ⟨f, g⟩ is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that the above square is a pullback 28 diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Furthermore, since T is a right adjoint, we also have that T n preserves these pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus (qM, zM, λM) satisfies assumption (ii) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, (qM, zM, λM) will induce a differential bundle with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' It remains to construct the sum and the negative as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8, and show that these are the same as the proposed σ and ι above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that M R(M) = (qM, σM, zM, λM, ιM) is a differential bundle with negatives over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3 Equivalence We will now show that the constructions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2 are inverses of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Beginning from the module side of things, let R be a commutative ring and M be an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Consider ker(qM), the kernel of the projection of the induced differential bundle M R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, qM(a + mε) = 0 implies that a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, we must explain why αM and α−1 M are also R-linear morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Clearly, they are both linear, so we must show that they preserve the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Since αM and α−1 M are inverses as functions, it then follows that α−1 M will also be an R-module morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that αM and α−1 M are inverse R-linear isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ Let’s now start from the differential bundle side of the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Let E = (q : E −→ R, σ, z, λ, ι) be a differential bundle with negatives over a commutative ring R in (CRING, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To do so, we must first take a closer look at the lift λ : E −→ E[ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Now by the first diagram of [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3], we have that pE ◦ λ = z ◦ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' First, define the ring morphism ζE : ker(q)[ε] −→ E[ε] as ζE(a + xε) = z(a) + xε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proof: We first explain why βE and β−1 E are well-defined ring morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Starting with βE, we must first explain why Dλ(x) is in the kernel of the projection q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' By the first diagram of [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2], we have that T (q)◦ λ = 0R ◦ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, for all x ∈ E, we have that q(z(q(x)))+ q(Dλ(x))ε = q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Since the right-hand side has no nilpotent component, this implies that q(Dλ(x)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So for all x ∈ E, Dλ(x) ∈ ker(q), and so βE is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We leave it to the reader to check for themselves that βE is indeed a ring morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Next we explain why β−1 E is a well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To do so, we must show that the outer diagram of (28) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' First, note that by the second diagram of [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1], q ◦ z = 1R, so q(z(a)) = a for all a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So βE and β−1 E are inverse ring isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lastly, we must show that βE and β−1 E are also differential bundle morphisms over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To do so, we will need to know a bit more about Dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The third diagram of [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3] is 0E ◦ z = λ ◦ z, which implies that for all a ∈ R, z(a) = z(a) + Dλ(z(a))ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Since the left-hand side has no nilpotent component, it follows that Dλ(z(a)) = 0 for all a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' On the other hand, the diagram of [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4] says that T (λ)◦λ = ℓE ◦λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Then using that q(Dλ(x)) = 0, q(z(a)) = a, and Dλ(z(a)) = 0, [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4] explicitly states that q(z(x)) + Dλ(Dλ(x))εε′ = q(z(x)) + Dλ(x)εε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' This implies that Dλ(Dλ(x)) = Dλ(x) for all x ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' With these identities, we can now show that βE is a differential bundle morphism over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Since βE is a ring isomorphism, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='13 it then follows that β−1 E is also a differential bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In particular, note that this implies that λ◦β−1 E = T (β−1 E )◦λker(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' But by definition, we have that λ◦β−1 E = ζE, and so we also have that T (β−1 E )◦λker(q) = ζE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, β−1 E (a + xε) = z(a) + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that βE and β−1 E are differential bundle isomorphisms over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ Therefore, the construction from a module to a differential bundle is the inverse of the construction from a differential bundle to a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5 For a commutative ring R, there is a bijective correspondence between R-modules and differential bundles (with negatives) over R in (CRING, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In CRING, recall that the terminal object is the zero ring 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So differential objects in (CRING, T ) corre- spond precisely to 0-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, the only 0-module is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, there are no non-trivial differential objects in (CRING, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='6 The only differential object in (CRING, T ) is the zero ring 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We now extend Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5 to an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For a commutative ring R, let MODR be the category of R-modules and R-linear morphisms between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' E = (q : E −→ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ι) to the R-module M R(E) = ker(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and sends a differential bundle morphism f : E = (q : E −→ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ι) −→ E′ = (q′ : E′ −→ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7 For a commutative ring R, we have an equivalence of categories: MODR ≃ DBUN T [R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proof: We must first explain why M R and M R are well-defined on morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So given an R-linear morphism f : M −→ M ′, we must show that M R(f) is a differential bundle morphism over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We leave it to the reader to check for themselves that M R(f) is a ring morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus M R(f) : ker(q) −→ ker(q′) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Since f is a differential bundle morphisms over R, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='11, we have that f preserves the zero, that is, f(z(a)) = z′(a) for all a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Next we explain why α, α−1, β, and β−1 are natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, α and α−1 are inverse natural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, β and β−1 are inverse natural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that we have an equivalence of categories, and so MODR ≃ DBUN T [R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ We also obtain an equivalence of categories between the category of all differential bundles and the category of modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Composition is defined as (g′, f ′)◦(g, f) = (g′ ◦g, f ′ ◦f) and identities are (1R, 1M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We define an equivalence of categories between MOD and DBUN � (CRING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' T ) � as follows: (i) Define the functor M : MOD −→ DBUN � (CRING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' T ) � which sends an object (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' M) to the differential bundle M (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' M) = M R(M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and sends a map (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' f) : (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' M) −→ (R′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' M ′) to the differential bundle morphism M (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' f) : M R(M) −→ M R′(M ′) defined as: M (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' f)(a + mε) = g(a) + f(m)ε (ii) Define the functor M : DBUN � (CRING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' T ) � −→ MOD which sends a differential bundle with negatives E = (q : E −→ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ι) to the pair M (E) = (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ker(q)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and sends a differential bundle morphism (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g) : E = (q : E −→ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ι) −→ E′ = (q′ : E′ −→ R′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ι′) to the pair M (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g) = (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' M R(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8 We have an equivalence of categories: MOD ≃ DBUN � (CRING, T ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So M and M are indeed functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that we have an equivalence of categories: MOD ≃ DBUN � (CRING, T ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='9 The equivalence between modules and differential bundles is also true in more general settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, in a setting where one does not have negatives, we would have also needed to prove [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1] and a generalization of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' If a codifferential category has kernels, then every differential bundle induces a module by generalizing Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1, and so in the presence of kernels, differential bundles in the Eilenberg-Moore category also correspond precisely to modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, since not all codifferential categories have all kernels, there may be differential bundles which are not induced by modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, the constructions are quite different in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To go from a module to a differential bundle, we take the free symmetric algebra over said module (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To go from a differential bundle to a module, we take the image of the derivation induced by the lift (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 From Differential Bundles to Modules Let us begin by unravelling what a differential bundle with negatives would be in (CRINGop, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' First recall that (CRINGop, T) is a Cartesian Rosick´y tangent category, so by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='9, differential bundles are the same thing as differential bundles with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Also, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3, CRINGop admits all pullbacks, since CRING admits all pushouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⊗R E � �� � n times .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' These must also satisfy the dual equalities and properties of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In particular, note that [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1] and [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, to prove that im(Dλ) is an R-module, it suffices to show that Dλ is an R-linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Clearly Dλ is linear, so it remains to show that Dλ also preserves the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' First note that by the dual of the first diagram of [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2], that λ ◦ T(q) = q ◦ 0R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In particular this implies that λ(q(a)) = q(a) and λ(d(q(a))) = 0 for all a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Note that the second equality can be rewritten as Dλ(q(a)) = 0 for all a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2 From Modules to Differential Bundles We now construct a differential bundle from a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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) ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 34 where ⊗s R is the symmetrized tensor product over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Note that as a commutative ring, SymR(M) is generated by all a ∈ R and m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, to define ring morphisms with domain SymR(M), it suffices to define them on generators a and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⊗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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⊗R 1 ⊗R w ⊗R 1 ⊗R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ⊗R 1 (iii) The sum σM : SymR(M) −→ SymR(M) ⊗R SymR(M) is the canonical comultiplication of the free symmetric R-algebras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' defined on generators as follows: ιM(a) = a ιM(m) = −m To describe the lift,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' note that T(SymR(M)) as a commutative ring is generated by a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' d(a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and d(m) for all a ∈ R and m ∈ M (and modulo the appropriate equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7 hold in CRING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To do so, we show that these hold on the generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λM(T(λM)(a)) = λM(λM(a)) = λM(a) = λM(ℓSymR(M)(a)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λM(T(λM)(m)) = λM(λM(m)) = λM(0) = 0 = λM(m) = λM(ℓSymR(M)(m)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λM(T(λM)(d(a))) = λM(λM(d(a))) = λM(0) = λM(ℓSymR(M)(d(a))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λM(T(λM)(d(m))) = λM(λM(d(m))) = λM(0) = λM(ℓSymR(M)(d(m))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λM(T(λM)(d′(a))) = λM(d(λM(a))) = λM(d(a)) = 0 = λM(0) = λM(ℓSymR(M)(d′(a))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λM(T(λM)(d′(m))) = λM(d(λM(m))) = λM(d(0)) = λM(0) = λM(ℓSymR(M)(d′(m))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λM(T(λM)(d′d(m))) = λM(d(λM(d(m)))) = λM(d(m)) = λM(ℓSymR(M)(d′d(m))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='So the desired equalities hold and we conclude that (qM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' zM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λM) is a pre-differential bundle in (CRINGop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Dualizing this, we conclude that (qM, zM, λM) satisfies assumption (i) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8 in CRINGop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Next, we must show that the dual of (ii) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' this implies that for every a ∈ R and m ∈ M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='the following equalities hold: f(a) = g(a) f(d(a)) = 0 f(m) = 0 Then define the map [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g] : SymR(M) −→ S as the ring morphism defined on generators as follows: [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](a) = g(a) [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](m) = f(d(m)) (30) Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' we compute the following on generators: [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](qM(a)) = [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](a) = g(a) [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](λM(a)) = [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](a) = g(a) = f(a) 36 [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](λM(m)) = [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](0) = 0 = f(m) [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](λM(d(a))) = [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](0) = 0 = f(d(a)) [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](λM(d(m))) = [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g](m) = f(d(m)) Thus it follows that [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g]◦ λM = f and [f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' g]◦ qM = g as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lastly, it remains to show that [f, g] is the unique such ring morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So suppose we have a ring morphism h : SymR(M) −→ S such that h ◦ λM = f and h ◦ qM = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus we conclude the above diagram is a pushout in CRING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Furthermore, since T is a left adjoint in CRING, we also have that Tn preserves these pushouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Dualizing this, it follows that (qM, zM, λM) satisfies assumption (ii) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8 in CRINGop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8, the pre-differential bundle (qM, zM, λM) will induce a differential bundle with negatives in (CRINGop, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' It remains to construct the sum and the negative as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8, and show that these are the same as the proposed σ and ι above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' By dualizing the construction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' the sum σ is: σM = � [π1 ◦ λM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' π2 ◦ λM] ◦ +SymR(M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' πj ◦ qM � On generators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' we compute: σM(a) = � [π1 ◦ λM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' π2 ◦ λM] ◦ +SymR(M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' πj ◦ qM � (a) = πj(qM(a)) = πj(a) = a ⊗R 1 = 1 ⊗R a σM(m) = � [π1 ◦ λM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' π2 ◦ λM] ◦ +SymR(M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' πj ◦ qM � (m) = [π1 ◦ λM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' π2 ◦ λM](+SymR(M)(d(m))) = [π1 ◦ λM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' π2 ◦ λM](d(m) ⊗R 1) + [π1 ◦ λM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' σM(a) = a ⊗R 1 = 1 ⊗R a and σ(m) = m ⊗R 1 + 1 ⊗R m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that (qM, σM, zM, λM, ιM) is a differential bundle with negatives over R in (CRINGop, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3 Equivalence We will now show that the constructions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2 are inverses of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We will show that the image of the derivation is precisely M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3 For every commutative ring R and R-module M, im(DλM ) = M as R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 37 Proof: Let us compute what this derivation does on pure symmetrized tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For degree 0, that is, for a ∈ R we have that: DλM (a) = λM(d(a)) = 0 so DλM (a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For degree 1, that is, for m ∈ M we have that: DλM (m) = λM(d(m)) = m so DλM (m) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' mn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So it follows that im(DλM ) = {m| ∀m ∈ M}, so im(DλM ) = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Furthermore, note that the multiplication of a and m in SymR(M) is precisely the module action, am = a · m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus the induced action on im(DλM ) from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To define the inverse we will need to use the dual of the Rosick´y’s universality diagram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' which in this case asks that the following diagram be a pushout: T(R) ⊗ E [T(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='pE] � [0R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' define ψ−1 E : ker(q)[ε] −→ E as the unique ring morphism which makes the following diagram commute: T(R) ⊗ E [T(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='pE] � [0R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='z] � T(E) λ � δE � R q � qim(Dλ) � E ψ−1 E �❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ SymR � im(Dλ) � (34) so ψ−1 E = � qim(Dλ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' δE � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 38 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proof: We first explain why ψE and ψ−1 E are well-defined ring morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Clearly, ψE is well-defined by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' On the other hand, to explain why ψ−1 E is well-defined, we must show that the outer diagram of (34) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' First, note that by the dual of the second diagram of [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1], z(q(a)) = a for all a ∈ R, and recall that Dλ(q(a)) = 0 for all a ∈ R as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Then on generators a ∈ R and x ∈ E we compute: δE � [T(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z](a ⊗ x) � and δE � [T(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z](d(a) ⊗ x) � So δE ◦ [T(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' pE] = qim(Dλ) ◦ [0R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So ψE and ψ−1 E are inverse ring isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='10 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We will first show that ψE is a differential bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' To do so, first recall that λ(q(a)) = q(a) and Dλ(q(a)) = 0, and that the dual of [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4] states that λ ◦ T(λ) = λ ◦ ℓE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' on a: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='T(βE)(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='βE(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ(q(a)) = q(a) = ψE(a) = ψE(λim(Dλ)(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='on Dλ(x): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='T(βE) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='Dλ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='βE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='Dλ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='Dλ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λ(d(x)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='T(λ)(d(x)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='ℓE(d(x)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ(0) = 0 = ψE(0) = ψE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λim(Dλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='Dλ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='on d(a): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='T(βE) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='d(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='βE(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='q(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= Dλ(q(a)) = 0 = ψE(0) = ψE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λim(Dλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='d(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='and finally on d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='Dλ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=': ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='T(βE) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='Dλ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='βE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='Dλ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='Dλ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λ(d(x)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='T(λ)(d′d(x)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='ℓE(d′d(x)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='d(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= Dλ(x) = βE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='Dλ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='= ψE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='λim(Dλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='Dλ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='So it follows that ψE is a differential bundle morphism over R in (CRINGop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='13 it then follows that ψ−1 E is also a differential bundle morphism over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that ψE and ψ−1 E are differential bundle isomorphisms over R in (CRINGop, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ Thus, the construction from a module to a differential bundle is the inverse of the construction from a differential bundle to a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5 For a commutative ring R, there is a bijective correspondence between R-modules and differential bundles (with negatives) over R in (CRINGop, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In CRING, recall that initial object is Z, which means that Z is the terminal object in CRINGop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So differential objects in (CRINGop, T) correspond precisely to Z-modules, which are precisely Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='6 There is a bijective correspondence between Z-modules/Abelian groups and differential objects in (CRINGop, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' We now extend Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5 to an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' For a commutative ring R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' E = (q : E −→ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ι) to the R-module M◦ R(E) = im(Dλ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' and sends a differential bundle morphism f : E = (q : E −→ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ι) −→ E′ = (q′ : E′ −→ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' z′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' λ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7 For a commutative ring R, we have an equivalence of categories: MODop R ≃ DBUNT [R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proof: We must first explain why MR and M◦ R are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Clearly, MR is well-defined on objects and maps and preserves composition and identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So MR is indeed a functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' On the other hand, let f : E −→ E′ be a differential bundle morphism over R in (CRINGop, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' This implies that f : E′ −→ E is a ring morphism and also that f(q′(a)) = q(a) for all a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Since MR(f) is clearly linear, we show that it also preserves the action: M◦ R(f) � a · Dλ′(x) � = M◦ R(f) � D��′(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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Clearly, M◦ R also preserves composition and identities, so M◦ R is also a functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Furthermore, we also have that M◦ R ◦ MR = 1MODop R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Next, ψ and ψ−1 are well-defined component-wise and are inverses at each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore, it suffices to show that ψ is natural and then it will follow that ψ−1 is also natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' If f : E −→ E′ is a differential bundle morphism over R in (CRINGop, T), then f ◦λ′ = λ◦T(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In particular, this means that f(λ′(d(x))) = λ(d(f(x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, we can rewrite this as f � Dλ′(x) � = Dλ � f(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Therefore ψ is a natural transformation, and it follows that so is ψ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus, ψ and ψ−1 are natural isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that we have an equivalence of categories: MODop R ≃ DBUNT [R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ It then follows that we have an equivalence between the category of differential objects and the opposite category of Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So let Ab be the category whose objects are Abelian groups and whose morphisms are group morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8 There is an equivalence of categories: DBUN � (CRINGop, T) � ≃ MODZ ≃ Abop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (iii) Observe that M◦ ◦ M = 1MODop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (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 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 41 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='9 We have an equivalence of categories: MODop ≃ DBUN � (CRINGop, T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Furthermore, it also follows that M◦ ◦ M = 1MODop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lastly, that ψ, and ψ −1 are natural isomorphisms follows directly from the fact that ψ, and ψ−1 are natural isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So we conclude that we indeed have an equivalence of categories: MODop ≃ DBUN � (CRINGop, T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='10 The equivalence between modules and differential bundles is also true in more general set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='19 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' As explained before, in a setting where one does not have negatives, we would also need to prove [DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' On the other hand, it is unclear if this result always generalizes to the coEilenberg-Moore category of a differential category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' If the differential category has enough limits and colimits, then it is possible to generalize the constructions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2, and then we obtain a bijective correspondence between differential bundles and comodules of the colagebras of the comonad of said differential category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, in general, a differential category need not have all limits or colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Our first goal is to show that for any differential bundle q : E −→ A in schemes, the projection q is an affine map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Let us first quickly recall the definition of affine morphisms and equivalent characterizations [26, Section 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='11 [26, Definition 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='12 [26, Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='3] For a scheme morphism f : X −→ Y , the following are equivalent: (i) f is affine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (ii) Y has a covering by affine opens {Ui}i∈I such that for all i ∈ I, f −1(Ui) is affine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' (iii) X = Spec(A) for some quasicoherent sheaf of algebras A on the sheaf OY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The following is a general result about affine morphisms which will be useful below: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proof: Let U be an affine open subset of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' So f −1 1 (U) is itself representable, and so by definition f1 is affine, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ We may now prove that for a differential bundle in the category of schemes, the projection is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='14 In the category of schemes, if q : E −→ A is a differential bundle, then q is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proof: By [22, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4], q is a retract of a pullback of a tangent bundle projection pA : T(A) −→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' By definition, T(A) is Spec of a quasicoherent sheaf of algebras on OA, so by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='12, pA is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' But affines are closed under pullback [26, Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='8] and retracts (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='13), so q is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ We may now prove that every differential bundle is a Spec of Sym of a quasicoherent sheaf of modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Moreover, by [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='7], differential bundles are closed under pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus each map q−1(Ui) −→ Ui is a differential bundle in the category of affine schemes, and hence by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='5, each q−1(Ui) is Sym of a module on Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus as E is the gluing of these, E is itself Spec of Sym of a quasicoherent sheaf of modules [28, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 379].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ Conversely, we now prove that every quasicoherent sheaf of modules induces a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 43 Proof: Suppose that A is covered by affines Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Then by [28, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Thus it suffices to show that such a gluing is a differential bundle over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' But by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2, Spec of Sym of each Mi is a differential bundle over Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' It then follows that since the tangent functor on schemes preserves gluings (for an abstract proof of this, see [5, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ✷ The results on morphisms follow similarly, and therefore we obtain: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='18 By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='22 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='4, for any scheme A, there is a similar result for the tangent category of schemes over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' They seem closely related to connections on modules [23, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='2], but more work needs to be done to understand the precise relationship between the two notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Tangent categories have a notion of differential forms and de Rham cohomology [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' However, again more investigation is required to prove this completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Moreover, [11] also develops a second notion of cohomology: sector form cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' It is not clear what this should give in the algebraic geometry setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In [16], Dominic Joyce develops algebraic geometry in the setting of C∞-rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' A key idea in algebraic geometry is that of a smooth morphism or object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Finally, the Serre-Swan theorem provides a very different way to compare vector bundles to modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' It would be interesting to see a proof for the Serre-Swan theorem based on some of the results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Bauer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Burke, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Ching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Tangent infinity-categories and Goodwillie calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='org:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='07819, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 44 [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Beck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Triples, algebras and cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Columbia University, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Blute, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cockett, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Seely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cartesian differential categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Theory and Applications of Categories, 22(23):622–672, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Blute, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lucyshyn-Wright, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' O’Neill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Derivations in codifferential categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques, 57:243–280, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cockett and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cruttwell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Differential structure, tangent structure, and SDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Applied Categorical Structures, 22(2):331–417, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cockett and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cruttwell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The Jacobi identity for tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques, LVI(4):301–316, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cockett and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cruttwell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Connections in tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Theory and applications of categories, 32(26):835–888, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cockett and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cruttwell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Differential bundles and fibrations for tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques, LIX:10–92, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cockett, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cruttwell, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Differential equations in a tangent category I: Complete vector fields, flows, and exponentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Applied Categorical Structures, 29(5):773–825, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cockett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lemay, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lucyshyn-Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Tangent Categories from the Coal- gebras of Differential Categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' In M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Fern´andez and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Muscholl, editors, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020), volume 152 of Leibniz International Proceedings in Informatics (LIPIcs), pages 17:1–17:17, Dagstuhl, Germany, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cruttwell and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lucyshyn-Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' A simplicial foundation for differential and sector forms in tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Journal of Homotopy and Related Structures volume, 13:867–925, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Ehrhard and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Regnier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The differential lambda-calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Theoretical Computer Science, 309(1):1– 41, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [13] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Francis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Handbook of Categorical Algebra, Volume I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cambridge University Press, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Garner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' An embedding theorem for tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Advances in Mathematics, 323:668–687, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Grothendieck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' ´El´ements de g´eom´etrie alg´ebrique IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Publications Math´ematiques de l’IH´ES, 28:5– 255, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Joyce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' An introduction to C-infinity schemes and C-infinity algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Surveys in Differ- ential Geometry, 79:299–325, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [17] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Jubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The tangent functor monad and foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' arXiv preprint arXiv:1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='0940, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Kock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Synthetic differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Cambridge University Press, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Kriegl and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Michor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The convenient setting of global analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' American Mathematical Society, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [20] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Lawevere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Categorical Dynamics, pages 1–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Aarhus University Press, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Mac Lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Categories for the working mathematician, volume 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Springer Science & Business Media, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' MacAdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Vector bundles and differential bundles in the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Applied categorical structures, 29(2):285–310, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 45 [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Mangiarotti and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Sardanashvily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Connections In Classical And Quantum Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' World Scientific Publishing Company, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [24] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Michor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The Jacobi flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Torino, 54(4):365–372, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Rosick`y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Abstract tangent functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Diagrammes, 12:JR1–JR11, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [26] The Stacks Project Authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Stacks Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' https://stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content='edu, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [27] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' An introduction to manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Springer, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' [28] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Vakil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' The rising sea: Foundations of algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' Online textbook, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'} +page_content=' 46' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE5T4oBgHgl3EQfUQ9g/content/2301.05542v1.pdf'}