Transgression maps for crossed modules of groupoids

Abstract

Given a crossed module of groupoids [Formula: see text], we construct (1) a natural homomorphism from the product groupoid [Formula: see text] to the crossed product groupoid [Formula: see text] and (2) a transgression map from the singular cohomology [Formula: see text] of the nerve of the groupoid [Formula: see text] to the singular cohomology [Formula: see text] of the nerve of the crossed product groupoid [Formula: see text]. The latter turns out to be identical to the transgression map obtained by Tu–Xu in their study of equivariant [Formula: see text]-theory.