Module-theoretic approach to dualizable Grothendieck categories

TL;DR

This paper characterizes dualizable Grothendieck categories using module-theoretic methods, showing they satisfy certain conditions and providing a complete answer to a conjecture on their dualizability.

Contribution

It proves that dualizable Grothendieck categories with duals are characterized by Grothendieck's conditions Ab6 and Ab4*, resolving a modified conjecture.

Findings

Dualizable Grothendieck categories satisfy Ab6 and Ab4*

The class of dualizable linear cocomplete categories equals those satisfying Ab6 and Ab4*

Provides a complete answer to a conjecture on dualizability

Abstract

We prove that every dualizable Grothendieck category whose dual is again a Grothendieck category satisfies Grothendieck's conditions Ab6 and Ab4*, by taking a module-theoretic approach based on the Gabriel-Popescu embedding. Combining this with a result by Stefanich, we conclude that the class of dualizable linear cocomplete categories is precisely the class of linear Grothendieck category satisfying Ab6 and Ab4*. This provides a complete answer to a modified conjecture on the dualizability, originally posed by Brandenburg, Chirvasitu, and Johnson-Freyd.