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arXiv · 2502.19165

Projective crossed modules in semi-abelian categories

Maxime Culot
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Abstract

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.

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Maxime Culot. 2025-02-26. Projective crossed modules in semi-abelian categories. https://doi.org/10.1007/s10485-025-09828-w

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