Exponentiable Grothendieck categories in flat Algebraic Geometry

TL;DR

This paper develops a framework for understanding Grothendieck categories with flat morphisms, characterizes exponentiable objects as continuous categories, and applies these results to categories of quasi-coherent sheaves on schemes.

Contribution

It introduces the 2-category of Grothendieck categories with flat morphisms, characterizes exponentiable objects, and connects these to categories of quasi-coherent sheaves on schemes.

Findings

Tensor product restricts to Grothendieck categories with flat morphisms

Exponentiable objects are exactly the continuous Grothendieck categories

Categories of quasi-coherent sheaves on quasi-compact, quasi-separated schemes are exponentiable

Abstract

We introduce and describe the $2$-category $\mathsf{Grt}_{\flat}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $\boxtimes$ restricts nicely to $\mathsf{Grt}_{\flat}$. Then, we characterize exponentiable objects with respect to $\boxtimes$: these are continuous Grothendieck categories. In particular, locally finitely presentable Grothendieck categories are exponentiable. Consequently, we have that, for a quasi-compact quasi-separated scheme $X$, the category of quasi-coherent sheaves $\mathsf{Qcoh}(X)$ is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials.