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.


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
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