Pseudolimits for Tangent Categories with Applications to Equivariant Algebraic and Differential Geometry

TL;DR

This paper develops a framework for constructing tangent structures on pseudolimits of categories, with applications to equivariant descent in algebraic and differential geometry.

Contribution

It introduces a method to induce tangent structures on pseudolimits of categories with tangent structures, extending the understanding of tangent categories in complex settings.

Findings

Pseudolimits of tangent categories inherit tangent structures.

The forgetful 2-functor preserves and creates pseudolimits indexed by 1-categories.

Application to equivariant descent in smooth and algebraic geometry.

Abstract

In this paper we show that if $\mathscr{C}$ is a category and if $F\colon\mathscr{C}^{\operatorname{op}} \to \mathfrak{Cat}$ is a pseudofunctor such that for each object $X$ of $\mathscr{C}$ the category $F(X)$ is a tangent category and for each morphism $f$ of $\mathscr{C}$ the functor $F(f)$ is part of a strong tangent morphism $(F(f),{}_{f}{\alpha})$ and that furthermore the natural transformations ${}_{f}{\alpha}$ vary pseudonaturally in $\mathscr{C}^{\operatorname{op}}$, then there is a tangent structure on the pseudolimit $\mathbf{PC}(F)$ which is induced by the tangent structures on the categories $F(X)$ together with how they vary through the functors $F(f)$. We use this observation to show that the forgetful $2$-functor $\operatorname{Forget}:\mathfrak{Tan} \to \mathfrak{Cat}$ creates and preserves pseudolimits indexed by $1$-categories. As an application, this allows us to describe how equivariant descent interacts with the tangent structures on the category of smooth (real) manifolds and on various categories of (algebraic) varieties over a field.