Projective crossed modules in semi-abelian categories

TL;DR

This paper characterizes projective objects in the category of internal crossed modules within semi-abelian categories, establishing conditions for the existence of free objects and projectives, and exploring implications for derived functors.

Contribution

It provides a characterization of projective objects in internal crossed modules in semi-abelian categories and links the properties of the base variety to the structure of these modules.

Findings

Projective objects are characterized in the category of internal crossed modules.

When the category is a variety of algebras, internal crossed modules form a semi-abelian variety with free objects.

The variety satisfies Condition (P) if and only if the base variety does, affecting the structure of projectives.

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.