Higher-dimensional generalization of abelian categories via DG-categories
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909677628948480 |
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| author | Mochizuki, Nao |
| author_facet | Mochizuki, Nao |
| contents | In this paper, we introduce abelian $n$-truncated DG-categories as an $n$-dimensional analogue of abelian categories in the setting of DG-categories. When $n=1$, this recovers ordinary abelian categories, and when $n=\infty$, it corresponds to stable DG-categories. This notion serves as a DG-categorical analogue of abelian $(n,1)$-categories in the context of $(n,1)$-categories. We show that the homotopy categories of abelian $n$-truncated DG-categories acquire the structure of extriangulated and pretriangulated categories. Furthermore, we develop a general theory of abelian $n$-truncated DG-categories, including the analogues of the existence epi-mono factorizations of morphisms, as in classical abelian categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06955 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher-dimensional generalization of abelian categories via DG-categories Mochizuki, Nao Category Theory Rings and Algebras Representation Theory 18G35, 18N20, 18G90 In this paper, we introduce abelian $n$-truncated DG-categories as an $n$-dimensional analogue of abelian categories in the setting of DG-categories. When $n=1$, this recovers ordinary abelian categories, and when $n=\infty$, it corresponds to stable DG-categories. This notion serves as a DG-categorical analogue of abelian $(n,1)$-categories in the context of $(n,1)$-categories. We show that the homotopy categories of abelian $n$-truncated DG-categories acquire the structure of extriangulated and pretriangulated categories. Furthermore, we develop a general theory of abelian $n$-truncated DG-categories, including the analogues of the existence epi-mono factorizations of morphisms, as in classical abelian categories. |
| title | Higher-dimensional generalization of abelian categories via DG-categories |
| topic | Category Theory Rings and Algebras Representation Theory 18G35, 18N20, 18G90 |
| url | https://arxiv.org/abs/2501.06955 |