Categorical non abelian cohomology, and the Schreier theory of groupoids

Abstract

By regarding the classical non abelian cohomology of groups from a 2-dimensional categorical viewpoint, we are led to a non abelian cohomology of groupoids which continues to satisfy classification, interpretation and representation theorems generalizing the classical ones. This categorical approach is based on the fact that if groups are regarded as categories, then, on the one hand, crossed modules are 2-groupoids and, cocycles are lax 2-functors and the cocycle conditions are precisely the coherence axioms for lax 2-functors, and, on the other hand group extensions are fibrations of categories. Furthermore, $n$-simplices in the nerve of a 2-category are lax 2-functors.