Transgression maps for crossed modules of groupoids

Abstract

Given a crossed module of groupoids $N\to G$, we construct (1) a natural homomorphism from the product groupoid $\mathbb{Z}\times (N⋊G)\rightrightarrows N$ to the crossed product groupoid $N⋊G\rightrightarrows N$ 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 $H^{\ast -1}((N⋊G{)}_{\bullet },\mathbb{Z})$ 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.

8 pages