Grothendieck Enriched Categories

In this paper, we introduce the notion of Grothendieck enriched categories for categories enriched over a sufficiently nice Grothendieck monoidal category V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {V}$$\end{document}, 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 V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {V}$$\end{document} 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.