Morphisms of 2-groupoids and low-dimensional cohomology of crossed modules

Abstract

Given a morphism P : G → H of 2-groupoids, we construct a 6-term 2-exact sequence of cat-groups and pointed groupoids. We use this sequence to obtain an analogue for cat-groups (and, in particular, for crossed modules) of the fundamental exact sequence of non-abelian group cohomology. The link with simplicial topology is also explained. Introduction The aim of this paper is to obtain a basic result in low-dimensional cohomology of crossed modules. Homology and cohomology of crossed modules have been studied extensively, and a satisfactory theory has been developed (see [7] and the references therein, [14, 19, 20, 26]). The existing literature on this subject considers crossed modules and their morphisms as a category. Our point of view is that crossed modules are in a natural way the objects of a 2-category, and therefore they should be studied in a 2-dimensional context. This different point of view leads to 1991 Mathematics Subject Classification. Primary 18G50; Secondary 18B40, 18D05, 18D35, 20L05. Third author supported by FNRS grant 1.5.116.01. c ©0000 American Mathematical Society