Protomodularity, descent, and semidirect products. (English) Zbl 0890.18003

Bourn, D.; Janelidze, G.

Theory Appl. Categ. 4, 37-46 (1998).

The notion of protomodular category was introduced by the first author of this paper in 1991, as a generalization of the modular categories of A. Carboni and a subclass of the Mal’cev categories studied by Carboni, J. Lambek and M. C. Pedicchio, amongst others. He showed that they form a convenient setting for doing nonabelian homological algebra. The present paper builds on this observation in two ways. First, the authors use descent theory to give a number of new characterizations of protomodular categories, in terms of the ‘short five lemma’; and secondly, by considering the cases when the pullback functor associated with a morphism in a protomodular category is monadic, they develop a categorical notion of semidirect product, which specializes to the usual one in the protomodular category of groups.

Reviewer: P.T.Johnstone (Cambridge)

MSC:

18B99	Special categories
18C10	Theories (e.g., algebraic theories), structure, and semantics
18G50	Nonabelian homological algebra (category-theoretic aspects)
20J05	Homological methods in group theory

Keywords:

protomodular category; modular categories; Mal’cev categories; nonabelian homological algebra; descent theory; short five lemma; monadic pullback functor; semidirect product