Transgression maps for crossed modules of groupoids

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