On low dimensional cohomology of crossed modules with nontrivial coefficients

Abstract

For a crossed module ${\bf G}_*$ and abelian crossed module ${\bf A}_*$ equipped with an action of ${\bf G}_*$ we construct a truncated cochain complex $C^*({\bf G}_*, {\bf A}_*)$. We show that the corresponding cohomology groups ${\sf H}^i({\bf G}_*,{\bf A}_*)$, $0\leq i \leq 2$ have several nice properties and can be considered as a "cocycle" description of cotriple cohomology of crossed modules first introduced in \cite{ccg}, and further studied in [9], [15].