Pseudolimits for Tangent Categories and Equivariant Tangents for Varieties and Smooth Manifolds

Abstract: In this paper we show that if $\mathscr{C}$ is a category and if $F\colon\mathscr{C} \to \mathfrak{Cat}$ 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.