Module-theoretic approach to dualizable Grothendieck categories
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.16468
- arXiv:
- arXiv:2405.16468
- Bibcode:
- 2024arXiv240516468K
- Keywords:
-
- Mathematics - Category Theory;
- Mathematics - Rings and Algebras;
- Mathematics - Representation Theory;
- 18E10 (Primary);
- 16D90;
- 18C35;
- 18E20 (Secondary)
- E-Print:
- 8 pages