2023/07/17 by Geoff Vooys, Vooys, Geoff
Mathematics · #18F40 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2307.08183
openalex publication_date 2023/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we show that if \mathscrC is a tangent category then the Ind-category Ind(\mathscrC) is a tangent category as well with a tangent structure which locally looks like the tangent structure on \mathscrC. Afterwards we give a pseudolimit description of Ind(\mathscrC)/X when \mathscrC admits finite products, show that the Ind-tangent category of a representable tangent category remains representable (in the sense that it has a microlinear object), and we characterize the differential bundles in Ind(\mathscrC) when \mathscrC is a Cartesian differential category. Finally we compute the Ind-tangent category for the categories CAlgA of commutative A-algebras, Sch/S of schemes over a base scheme S, A-Poly (the Cartesian differential category of A-valued polynomials), and ℝ-Smooth (the Cartesian differential category of Euclidean spaces). In particular, during the computation of Ind(Sch/S) we give a definition of what it means to have a formal tangent scheme over a base scheme S.