A "working mathematician's" definition of semi-abelian categories

TL;DR

This paper provides a simplified, axiomatic characterization of semi-abelian categories, connecting them to fundamental categorical notions and demonstrating their stability under diagram categories like simplicial objects.

Contribution

It offers a 'working mathematician's' axiomatic framework for semi-abelian categories and shows their preservation in categories of diagrams, expanding their applicability.

Findings

Semi-abelian categories characterized by four axioms

Categories of diagrams in semi-abelian categories are semi-abelian

Simplicial and mma-objects categories are semi-abelian

Abstract

Semi-abelian and finitely cocomplete homological categories are characterized in terms of four resp. three simple axioms, in terms of the basic categorical notions introduced in the first few chapters of MacLane's classical book. As an immediate application we show that categories of diagrams in semi-abelian and similar categories are of the same type; in particular, the category of simplicial or \Gamma-objects in a semi-abelian category is semi-abelian.