The low-dimensional homology of crossed modules

This paper is prompted by work of Grandjeán and Ladra on homology crossed modules. We define the homology of a crossed module in dimensions one and two via the equivalence of categories with group objects in groupoids. We show that for any perfect crossed module, its second homology crossed module occurs as the kernel of its universal central extension as defined by Norrie. Our first homology is the same as that defined by Grandjeán and Ladra, but the second homology crossed modules are in general different. However, they coincide for aspherical perfect crossed modules. Our methods can in principle be applied to define a homology crossed module in any dimension.

Keywords

homology, crossed module, groupoid

2010 Mathematics Subject Classification

18D35, 18G50, 20J05, 20Lxx

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Published 1 January 2000