Grothendieck categories as a bilocalization of linear sites

TL;DR

This paper demonstrates that the 2-category of Grothendieck abelian categories can be viewed as a bicategory of fractions of the 2-category of linear sites, enabling new insights into their tensor product structure.

Contribution

It establishes a bicategory of fractions framework for Grothendieck categories, connecting them to linear sites and paving the way for a bi-monoidal tensor product structure.

Findings

Grothendieck categories form a bicategory of fractions of linear sites

This framework links colimit preserving functors to continuous morphisms of sites

Potential for defining a bi-monoidal tensor product on Grothendieck categories

Abstract

We prove that the 2-category Grt of Grothendieck abelian categories with colimit preserving functors and natural transformations is a bicategory of fractions in the sense of Pronk of the 2-category Site of linear sites with continuous morphisms of sites and natural transformations. This result can potentially be used to make the tensor product of Grothendieck categories from earlier work by Lowen, Shoikhet and the author into a bi-monoidal structure on Grt.