The low-dimensional homology of crossed modules
type: Item
instance of: scholarly article
title: The low-dimensional homology of crossed modules
zbMATH Open document ID: 0972.18004
author: N. D. Gilbert
published in: Homology, Homotopy and Applications
publication date: 2000-07-23T00:00:00Z
full work available at URL: https://eudml.org/doc/222930
full work available at URL: volume2-1.htm
review text: According to an observation of Verdier, the category of crossed modules is equivalent to that of group objects in the category of groupoids. In the paper under review, a notion of 1- and 2-dimensional homology of a crossed module is studied which arises from this equivalence of categories. It is shown that, for a perfect crossed module, the second homology occurs as the kernel of a suitably defined universal central extension for crossed modules given in \textit{K. Norrie}'s thesis [University of London, 1987]. The second homology crossed module defined by \textit{M. Ladra} and \textit{A. R. Grandjeán} [J. Pure Appl. Algebra 95, No. 1, 41-55 (1994; Zbl 0807.18008)] by means of a Hopf formula is in general different from that in the paper under review but the two notions coincide for an aspherical perfect crossed module.
reviewed by: J. Huebschmann
Mathematics Subject Classification ID: 18D35
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 18G50
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 20L05
zbMATH DE Number: 1479891
zbMATH Keywords: groupoids
zbMATH Keywords: crossed module
zbMATH Keywords: perfect crossed module
zbMATH Keywords: low-dimensional homology
zbMATH Keywords: Hopf formula
MaRDI profile type: Publication
OpenAlex ID: W2034383483
DOI: 10.4310/HHA.2000.V2.N1.A4
description: scientific article; zbMATH DE number 1479891