Tangent Ind-Categories

TL;DR

This paper extends tangent category structures to Ind-categories, explores their properties, and computes examples including schemes, polynomial categories, and smooth spaces, introducing the concept of formal tangent schemes.

Contribution

It demonstrates that Ind-categories of tangent categories inherit tangent structures, characterizes differential bundles in Ind-categories, and computes explicit examples like schemes and polynomial categories.

Findings

Ind-category of a tangent category is itself a tangent category.

The Ind-tangent structure locally resembles the original tangent structure.

Explicit descriptions of Ind-tangent categories for schemes, polynomial categories, and smooth spaces.

Abstract

In this paper we show that if $\mathscr{C}$ is a tangent category then the Ind-category $\operatorname{Ind}(\mathscr{C})$ is a tangent category as well with a tangent structure which locally looks like the tangent structure on $\mathscr{C}$. Afterwards we give a pseudolimit description of $\operatorname{Ind}(\mathscr{C})_{/X}$ when $\mathscr{C}$ admits finite products, show that the $\operatorname{Ind}$-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 $\operatorname{Ind}(\mathscr{C})$ when $\mathscr{C}$ is a Cartesian differential category. Finally we compute the $\operatorname{Ind}$-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 $\operatorname{Ind}(\mathbf{Sch}_{/S})$ we give a definition of what it means to have a formal tangent scheme over a base scheme $S$.