Pseudolimits for tangent categories with applications to equivariant algebraic and differential geometry

In this paper, we show that if C is a category and if F C\;op → Cat is a pseudofunctor such that for each object X of C the category $F(X)$ is a tangent category and for each morphism f of C the functor $F(\,f)$ is part of a strong tangent morphism (F(\,f),\!\,fα ) and that furthermore the natural transformations \!\,fα vary pseudonaturally in C\;op , then there is a tangent structure on the pseudolimit 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 Forget:Tan → 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.