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.
Categories: math.CT math.AG