Relative Semi-abelian Categories

We introduce a relative semi-abelian category as a pair (C,E), where C is a pointed category with finite limits, and E is a class of normal epimorphisms in C satisfying certain conditions, stronger than those defining a relative homological category [2]. In the absolute case, i.e. when E is the class of all regular epimorphisms in C, the pair (C,E) is relative semi-abelian if and only if C is semi-abelian in the sense of [1]. Another extreme case is the “trivial” one, where E is the class of isomorphisms in C, and then (C,E) is always relative semi-abelian. Some results of [1] on the equivalence of the so-called old-style and new-style axioms are extended to the relative case.