Homology, Homotopy and Applications

Volume 22 (2020)

Number 2

Galois theory and the categorical Peiffer commutator

Pages: 323 – 346

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n2.a20

Authors

Alan S. Cigoli (Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium)

Arnaud Duvieusart (Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium)

Marino Gran (Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium)

Sandra Mantovani (Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy)

Abstract

We show that the Peiffer commutator previously defined by Cigoli, Mantovani and Metere can be used to characterize central extensions of precrossed modules with respect to the subcategory of crossed modules in any semi-abelian category satisfying an additional property. We prove that this commutator also characterizes double central extensions, obtaining then some Hopf formulas for the second and third homology objects of internal precrossed modules.

Keywords

crossed module, Galois theory, Peiffer commutator, central extension, semiabelian category

2010 Mathematics Subject Classification

17B55, 18D35, 18G50, 20J05

Full Text (PDF format)

Copyright © 2020, Alan S. Cigoli, Arnaud Duvieusart, Marino Gran and Sandra Mantovani. Permission to copy for private use granted.

The second author is a Research Fellow of the Fonds de la Recherche Scientifique-FNRS.

Received 22 July 2019

Accepted 17 January 2020

Published 13 May 2020