A functorial approach to $0$-abelian categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911183805612032 |
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| author | Gulisz, Vitor |
| author_facet | Gulisz, Vitor |
| contents | We use functorial methods to define and study $0$-abelian categories, which we propose to be the case $n = 0$ of Jasso's $n$-abelian categories. In particular, we define a bifunctor for $0$-abelian categories with enough injectives or projectives, which is analogous to the extension bifunctor for an abelian category. We prove a few results concerning this bifunctor, including $0$-abelian versions of the long exact sequence involving the extension bifunctors, of a conjecture due to Auslander on the direct summands of the extension functors, and of the Hilton-Rees theorem. These results are then applied to the study of the stable categories of a $0$-abelian category, and a similar discussion is carried out for the stable categories of an abelian category. Moreover, by specializing our results to modules over rings, we show that $0$-abelian categories with additive generators are in correspondence with semi-hereditary rings. We present applications to these rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24810 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A functorial approach to $0$-abelian categories Gulisz, Vitor Category Theory Representation Theory 18A25 (Primary) 18E99, 18E05, 16E99 (Secondary) We use functorial methods to define and study $0$-abelian categories, which we propose to be the case $n = 0$ of Jasso's $n$-abelian categories. In particular, we define a bifunctor for $0$-abelian categories with enough injectives or projectives, which is analogous to the extension bifunctor for an abelian category. We prove a few results concerning this bifunctor, including $0$-abelian versions of the long exact sequence involving the extension bifunctors, of a conjecture due to Auslander on the direct summands of the extension functors, and of the Hilton-Rees theorem. These results are then applied to the study of the stable categories of a $0$-abelian category, and a similar discussion is carried out for the stable categories of an abelian category. Moreover, by specializing our results to modules over rings, we show that $0$-abelian categories with additive generators are in correspondence with semi-hereditary rings. We present applications to these rings. |
| title | A functorial approach to $0$-abelian categories |
| topic | Category Theory Representation Theory 18A25 (Primary) 18E99, 18E05, 16E99 (Secondary) |
| url | https://arxiv.org/abs/2509.24810 |