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- W2598498983 abstract "The categorical notion of semidirect product was introduced by Bourn and Janelidze in (1). A category C with split pullbacks is said to be a category with semidirect products if, for every morphism p:E! B in C, the pullback functor p : Pt(B)! Pt(E) has a left adjoint and is monadic. In this note we consider the case where the category C is pointed, has coequalizers of reexive pairs and binary coproducts. Forming the category of internal actions (as in (2)) we have by denition that C has semidirect products if the category Pt(C) of points is equivalent to the category Act(C) of internal actions. It is well known that: (a) a variety of universal algebras has semidirect products if and only if it is protomodular; (b) every semiabelian category has semidirect products; (c) not every homological category has semidirect products. This talk is divided in two parts. In the rst part we analyze the monadicity of p and give some necessary and sucient conditions for a category C to have semidirect products. In the second part we introduce the notion of strict action and show that C has semidirect products if and only if it is protomodular and every internal action is strict." @default.
- W2598498983 created "2017-04-07" @default.
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- W2598498983 date "2010-01-01" @default.
- W2598498983 modified "2023-09-23" @default.
- W2598498983 title "On categories with semidirect products" @default.
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