Given a crossed module of groupoids $N\rightarrow G$, we construct (1) a natural homomorphism from the product groupoid $\mathbb{ Z}\times(N\rtimes G)\rightrightarrows N$ to the crossed product groupoid $N\rtimes G\rightrightarrows N$ and (2) a transgression map from the singular cohomology $H^\ast(G_\bullet,\mathbb{ Z})$ of the nerve of the groupoid $G$ to the singular cohomology $H^{ \ast-1}\big((N\rtimes G)_\bullet,\mathbb{ Z}\big)$ of the nerve of the crossed product groupoid $N\rtimes G$. The latter turns out to be identical to the transgression map obtained by Tu--Xu in their study of equivariant $K$-theory.
Categories: math.AT hep-th