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.

Final version. Accepted for publication

References in corpus (8)

• Derived Algebraic Geometry III: Commutative Algebra
• Tensor categorical foundations of algebraic geometry
• Derived Algebraic Geometry II: Noncommutative Algebra
• General facts on the Scott Adjunction
• Grothendieck categories as a bilocalization of linear sites
• Bicategorical colimits of tensor categories
• Localizations of tensor categories and fiber products of schemes
• The Scott adjunction