Projective crossed modules in semi-abelian categories
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917190322618368 |
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| author | Culot, Maxime |
| author_facet | Culot, Maxime |
| contents | We characterize projective objects in the category of internal crossed modules within any semi-abelian category. When this category forms a variety of algebras, the internal crossed modules again constitute a semi-abelian variety, ensuring the existence of free objects, and thus of enough projectives. We show that such a variety is not necessarily Schreier, but satisfies the so-called Condition (P) -- meaning the class of projectives is closed under protosplit subobjects -- if and only if the base variety satisfies this condition. As a consequence, the non-additive left chain-derived functors of the connected components functor are well defined in this context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19165 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projective crossed modules in semi-abelian categories Culot, Maxime Category Theory 18E13, 18G05, 18G10, 18G45 We characterize projective objects in the category of internal crossed modules within any semi-abelian category. When this category forms a variety of algebras, the internal crossed modules again constitute a semi-abelian variety, ensuring the existence of free objects, and thus of enough projectives. We show that such a variety is not necessarily Schreier, but satisfies the so-called Condition (P) -- meaning the class of projectives is closed under protosplit subobjects -- if and only if the base variety satisfies this condition. As a consequence, the non-additive left chain-derived functors of the connected components functor are well defined in this context. |
| title | Projective crossed modules in semi-abelian categories |
| topic | Category Theory 18E13, 18G05, 18G10, 18G45 |
| url | https://arxiv.org/abs/2502.19165 |