  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.
