Semi-abelian categories. (English) Zbl 0993.18008

Janelidze, George; Márki, László; Tholen, Walter

J. Pure Appl. Algebra 168, No. 2-3, 367-386 (2002).

The notion of semi-abelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abelian-group and module theory. In modern terms, semi-abelian categories are exact in the sense of Barr and protomodular in the sense of Bourn and have finite coproducts and a zero object.
The paper shows how these conditions relate to “old” exactness axioms involving normal monomorphisms and epimorphisms, as used in the fifties and sixties, and it gives extensive references to the literature in order to indicate why semi-abelian categories provide an appropriate notion to establish the isomorphism and decomposition theorems of group theory, to pursue general radical theory of rings, and how to arrive at basic statements as needed in homological algebra of groups and similar non-abelian structures.

Reviewer: Y.Diers (Faches-Thumesnil)

MSC:

18E10	Abelian categories, Grothendieck categories
18A30	Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
18A32	Factorization systems, substructures, quotient structures, congruences, amalgams

Keywords:

exact category; proto-modular category; five-lemma; zero object; normal monomorphisms and epimorphisms; semi-abelian categories

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