Projective crossed modules in semi-abelian categories
Category Theory
2026-01-09 v4
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.
Cite
@article{arxiv.2502.19165,
title = {Projective crossed modules in semi-abelian categories},
author = {Maxime Culot},
journal= {arXiv preprint arXiv:2502.19165},
year = {2026}
}