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Grothendieck Categories as a Bilocalization of Linear Sites

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Abstract

Let k be a commutative ring. We prove that the 2-category \(\mathsf {Grt}_k\) of Grothendieck abelian k-linear categories with colimit preserving k-linear functors and k-linear natural transformations is a bicategory of fractions in the sense of Pronk [17] of the 2-category \(\mathsf {Site}_{k,\mathsf {cont}}\) of k-linear sites with k-linear continuous functors and k-linear natural transformations. In complete analogy, we prove that the conjugate-opposite 2-category of the 2-category \(\mathsf {Topoi}_k\) of Grothendieck abelian k-linear categories with k-linear geometric morphisms and k-linear morphisms between them is a bicategory of fractions of the 2-category \(\mathsf {Site}_k\) of k-linear sites with k-linear morphisms of sites and k-linear natural transformations. In addition, we show how the first statement can potentially be used to make the tensor product of Grothendieck categories from [14] into a bi-monoidal structure on \(\mathsf {Grt}_k\).

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Acknowledgements

I am very grateful to Wendy Lowen for many interesting discussions, the careful reading of this manuscript and her valuable suggestions. I would also like to thank Ivo Dell’Ambrogio, Boris Shoikhet and Enrico Vitale for useful comments on bicategories of fractions and Matteo Tommasini, whose explanations on the behaviour of 2-morphisms after localization have been essential in order to obtain the main result of this paper. I am also very grateful to an anonymous referee for the careful reading of the paper, for pointing out reference [15] and for the useful comments and suggestions which have led to the parallel analysis of the two different settings considered in the current version of the paper.

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  1. Departement Wiskunde-Informatica, Universiteit Antwerpen, Middelheim Campus G.228, Middelheimlaan 1, Antwerp, 2020, Belgium

    Julia Ramos González

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  1. Julia Ramos González
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Correspondence to Julia Ramos González.

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Communicated by R. Street.

The author acknowledges the support of the Research Foundation Flanders (FWO) under Grant No. G.0112.13N.

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Ramos González, J. Grothendieck Categories as a Bilocalization of Linear Sites. Appl Categor Struct 26, 717–745 (2018). https://doi.org/10.1007/s10485-017-9511-1

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