The bicategory of groupoid correspondences

TL;DR

This paper constructs a bicategory of étale, locally compact groupoids with correspondences as morphisms, extending the groupoid C*-algebra construction to a broader categorical framework including non-Hausdorff cases.

Contribution

It introduces a bicategory framework for groupoid correspondences and extends the C*-algebra construction to this setting, covering complex examples like self-similar groups and higher-rank graphs.

Findings

Established a bicategory structure for groupoids and correspondences

Extended C*-algebra functor to this bicategory

Applied framework to various complex groupoid examples

Abstract

We define a bicategory with \'etale, locally compact groupoids as objects and suitable correspondences, that is, spaces with two commuting actions 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 Conduch\'e fibrations in our setup.