Exponentiable Grothendieck categories in flat algebraic geometry

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

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Bibliographic Details
Published in:Journal of algebra 2022-08, Vol.604, p.362-405
Main Authors: Di Liberti, Ivan, Ramos González, Julia
Format: Article
Language:English
Subjects:
Continuous categories
Exponentiability
Flatness
Grothendieck categories
Monoidal structures
Noncommutative algebraic geometry
Quasi-coherent sheaves
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Summary:We introduce and describe the 2-category Grt♭ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories ⊠ restricts nicely to Grt♭. Then, we characterize exponentiable objects with respect to ⊠: these are the 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 Qcoh(X) is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2022.03.040