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Grothendieck Enriched Categories

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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.

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Notes

  1. A subcategory is called reflective if its inclusion functor has a left adjoint.

  2. A right enriched adjoint is fully faithful if and only if its underlying functor is so.

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Acknowledgements

The author would like to express his sincere gratitude to his supervisor, Shinnosuke Okawa, for a lot of advice and helpful inspiration through regular seminars. The author is also very grateful to Ivan Di Liberti for informing the author of the references on localizations and \(\mathcal {V}\)-topoi (see Remark 3.16). The author would like to thank the anonymous referee for a careful reading of the manuscript and for many helpful comments.

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  1. Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka, 560-0043, Japan

    Yuki Imamura

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  1. Yuki Imamura
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Correspondence to Yuki Imamura.

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Communicated by Ross Street.

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Imamura, Y. Grothendieck Enriched Categories. Appl Categor Struct 30, 1017–1041 (2022). https://doi.org/10.1007/s10485-022-09681-1

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