Semidirect products of internal groupoids
A categorical notion of semidirect product has been introduced by \textit{D. Bourn} and \textit{G. Janelidze} [``Protomodularity, descent, and semidirect products'', Theory Appl. Categ. 4, 37--46 (1998; Zbl 0890.18003)] in terms of split pullback diagrams. A notion of semidirect product for groupoids has been given in \textit{R. Brown} [``Groupoids as coefficients'', Proc. Lond. Math. Soc., III. Ser. 25, 413--426 (1972; Zbl 0245.20045)]. A generalization of this construction is known as Grothendieck construction for groupoids [\textit{A. Grothendieck}, Séminaire de Géometrie Algébrique 1, Exposé VI: Catégories fibrées et descente. 1960--1961, Lect. Notes Math. 224 (1971; Zbl 0234.14002)]. In this paper the necessary conditions for the existence of semidirect products are given and then these are studied for categorical groupoids. The groupoid action, split extensions and semidirect product notions for categorical groupoids are defined and the necessary conditions for the existence of semidirect products in categorical groupoids are determined.
scientific article; zbMATH DE number 5697067