The bicategory of groupoid correspondences

Abstract

We define a bicategory with etale, locally compact groupoids as objects and suitable correspondences, that is, spaces with two commuting ac-tions as arrows; the 2-arrows are injective, equivariant continuous maps. We prove that the usual recipe for composition makes this a bicategory, carefully treating also non-Hausdorff groupoids and correspondences. We extend the groupoid C*-algebra construction to a homomorphism from this bicategory to that of C*-algebra correspondences. We describe the C*-algebras of self -similar groups, higher-rank graphs, and discrete Conduche fibrations in our setup.