Exponentiable Grothendieck categories in flat Algebraic Geometry

Abstract

We introduce and describe the $2$-category ${\mathsf{Grt}}_{♭}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $⊠$ restricts nicely to ${\mathsf{Grt}}_{♭}$. Then, we characterize exponentiable objects with respect to $⊠$: 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.

arXiv commentFinal version. Accepted for publication

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