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

Abstract

In this paper we show that if $\mathcal{C}$ is a category and if $F:{\mathcal{C}}^{\mathrm{op}}\to \mathfrak{Cat}$ is a pseudofunctor such that for each object $X$ of $\mathcal{C}$ the category $F(X)$ is a tangent category and for each morphism $f$ of $\mathcal{C}$ 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 ${\mathcal{C}}^{\mathrm{op}}$, 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 $\mathrm{Forget}:\mathfrak{Tan}\to \mathfrak{Cat}$ 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.

66 Pages with changes to reflect publication version. To appear in Mathematical Structures in Computer Science special issue on Differential Structures

References in corpus (3)

• A 2-categories companion
• An embedding theorem for tangent categories
• Enough vector bundles on orbispaces