Derivations of categorical groups

Abstract

In this paper we introduce and study the categorical group of derivations, Der(G, A), from a categorical group G into a braided categorical group (A, c) equipped with a given coherent left action of G. Categorical groups provide a 2-dimensional vision of groups and so this object is a sort of 0-cohomology at a higher level for categorical groups.We show that the functor Der(-, A) is corepresentable by the semidirect product A G and that Der(G, -) preserves homotopy kernels.Well-known cohomology groups, and exact sequences relating these groups, in several different contexts are then obtained as examples of this general theory.