On satellites in semi-abelian categories: Homology without projectives

TL;DR

This paper develops a new approach to homology in semi-abelian categories that does not rely on projective objects, using Janelidze's theory of satellites to establish universal properties and relate to existing homology notions.

Contribution

It introduces a projective-free definition of homology in semi-abelian categories using satellite theory, extending the applicability of homological methods.

Findings

Established a projective-independent homology framework

Proved higher Hopf formulae in this new context

Provided examples illustrating the theory

Abstract

Working in a semi-abelian context, we use Janelidze's theory of generalised satellites to study universal properties of the Everaert long exact homology sequence. This results in a new definition of homology which does not depend on the existence of projective objects. We explore the relations with other notions of homology, and thus prove a version of the higher Hopf formulae. We also work out some examples.