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

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 |