Higher exact dg-categories
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910107598585856 |
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| author | Mochizuki, Nao Nakaoka, Hiroyuki |
| author_facet | Mochizuki, Nao Nakaoka, Hiroyuki |
| contents | We introduce the notion of an $n$-exact dg-category. This notion provides a higher analogue of Chen's exact dg-category, in the sense that the case where $n$ equals 1 recovers exact dg-categories.
We prove that, under a suitable vanishing condition on the cohomologies of $\mathrm{Hom}$-complexes of an $n$-exact dg-category $\mathscr{A}$, its homotopy category admits a natural $n$-exangulated structure. Thus $n$-exact dg-categories provide dg-enhancements of $n$-exangulated categories. At the same time, our framework can be regarded as a dg-categorical generalization of $n$-exangulated categories applicable even without the vanishing condition.
In the latter part of the article, we show that an $n$-cluster tilting subcategory of an exact dg-category naturally carries the structure of an $n$-exact dg-category. This result indicates that $n$-exact dg-structures provide an intrinsic dg-categorical axiomatization of $n$-cluster tilting subcategories, highlighting the advantages of studying dg-generalizations of $n$-exangulated categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05493 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Higher exact dg-categories Mochizuki, Nao Nakaoka, Hiroyuki Category Theory Representation Theory We introduce the notion of an $n$-exact dg-category. This notion provides a higher analogue of Chen's exact dg-category, in the sense that the case where $n$ equals 1 recovers exact dg-categories. We prove that, under a suitable vanishing condition on the cohomologies of $\mathrm{Hom}$-complexes of an $n$-exact dg-category $\mathscr{A}$, its homotopy category admits a natural $n$-exangulated structure. Thus $n$-exact dg-categories provide dg-enhancements of $n$-exangulated categories. At the same time, our framework can be regarded as a dg-categorical generalization of $n$-exangulated categories applicable even without the vanishing condition. In the latter part of the article, we show that an $n$-cluster tilting subcategory of an exact dg-category naturally carries the structure of an $n$-exact dg-category. This result indicates that $n$-exact dg-structures provide an intrinsic dg-categorical axiomatization of $n$-cluster tilting subcategories, highlighting the advantages of studying dg-generalizations of $n$-exangulated categories. |
| title | Higher exact dg-categories |
| topic | Category Theory Representation Theory |
| url | https://arxiv.org/abs/2604.05493 |