The low-dimensional homology of crossed modules
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.
scientific article; zbMATH DE number 1479891