Tangent Ind-Categories

Abstract

In this paper we show that if $\mathcal{C}$ is a tangent category then the Ind-category $\mathrm{Ind}(\mathcal{C})$ is a tangent category as well with a tangent structure which locally looks like the tangent structure on $\mathcal{C}$. Afterwards we give a pseudolimit description of $\mathrm{Ind}(\mathcal{C}{)}_{/X}$ when $\mathcal{C}$ admits finite products, show that the $\mathrm{Ind}$-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 $\mathrm{Ind}(\mathcal{C})$ when $\mathcal{C}$ is a Cartesian differential category. Finally we compute the $\mathrm{Ind}$-tangent category for the categories ${\mathbf{CAlg}}_A$ of commutative $A$-algebras, ${\mathbf{Sch}}_{/S}$ of schemes over a base scheme $S$, $A$-$\mathbf{Poly}$ (the Cartesian differential category of $A$-valued polynomials), and $\mathbb{R}$-$\mathbf{Smooth}$ (the Cartesian differential category of Euclidean spaces). In particular, during the computation of $\mathrm{Ind}({\mathbf{Sch}}_{/S})$ we give a definition of what it means to have a formal tangent scheme over a base scheme $S$.

35 pages