Module-theoretic approach to dualizable Grothendieck categories
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909211018919936 |
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| author | Kanda, Ryo |
| author_facet | Kanda, Ryo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16468 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Module-theoretic approach to dualizable Grothendieck categories Kanda, Ryo Category Theory Rings and Algebras Representation Theory 18E10 (Primary), 16D90, 18C35, 18E20 (Secondary) 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. |
| title | Module-theoretic approach to dualizable Grothendieck categories |
| topic | Category Theory Rings and Algebras Representation Theory 18E10 (Primary), 16D90, 18C35, 18E20 (Secondary) |
| url | https://arxiv.org/abs/2405.16468 |