Grothendieck enriched categories

TL;DR

This paper introduces Grothendieck enriched categories over certain monoidal categories, generalizes classical notions, and proves key theorems including a Gabriel-Popescu type theorem, with applications to dg categories of sheaves.

Contribution

It defines Grothendieck enriched categories, extends classical results to this setting, and shows their stability under base change, with applications to dg categories of sheaves.

Findings

Established the Gabriel-Popescu type theorem for Grothendieck enriched categories.

Proved stability of Grothendieck property under change of base monoidal categories.

Identified the dg category of complexes of quasi-coherent sheaves as a Grothendieck dg category.

Abstract

In this paper, we introduce the notion of Grothendieck enriched categories for categories enriched over a sufficiently nice Grothendieck monoidal category $\mathcal{V}$, generalizing the classical notion of Grothendieck categories. Then we establish the Gabriel-Popescu type theorem for Grothendieck enriched categories. We also prove that the property of being Grothendieck enriched categories is preserved under the change of the base monoidal categories by a monoidal right adjoint functor. In particular, if we take as $\mathcal{V}$ the monoidal category of complexes of abelian groups, we obtain the notion of Grothendieck dg categories. As an application of the main results, we see that the dg category of complexes of quasi-coherent sheaves on a quasi-compact and quasi-separated scheme is an example of Grothendieck dg categories.