Exponentiable Grothendieck categories in flat Algebraic Geometry

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 exponent...

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Bibliographic Details
Published in:arXiv.org 2022-05
Main Authors: Ivan Di Liberti, Julia Ramos González
Format: Article
Language:English
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Sheaves
Tensors
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2103.07876