Homology, Homotopy and Applications

Volume 8 (2006)

Number 1

Split extensions, semidirect product and holomorph of categorical groups

Pages: 145 – 167

DOI: https://dx.doi.org/10.4310/HHA.2006.v8.n1.a4


Stefano Kasangian (Dipartimento di Matematica, Università di Milano, Italy)

Giuseppe Metere (Dipartimento di Matematica, Università di Milano, Italy)

Enrico M. Vitale (Département de Mathématique Pure et Appliquée, Université catholique de Louvain, Louvain-la-Neuve, Belgium)


Working in the context of categorical groups, we show that the semidirect product provides a biequivalence between actions and points. From this biequivalence, we deduce a two-dimensional classification of split extensions of categorical groups, as well as the universal property of the holomorph of a categorical group. We also discuss the link between the holomorph and inner autoequivalences.


categorical groups, semidirect product, split extensions, holomorph

2010 Mathematics Subject Classification

18D05, 18D10, 18D35, 18G50

Published 1 January 2006