The bicategory of groupoid correspondences

Abstract

We define a bicategory with étale, 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 Conduché fibrations in our setup.

Existence of groupoid models for diagrams of groupoid correspondences

“…Groupoid correspondences may be composed, and this gives rise to a bicategory Gr (see [1]). We only need this structure to talk about bicategory homomorphisms into Gr.…”

“…These examples of C * -algebras are defined by some combinatorial or dynamical data. This data is interpreted in [1,5] as a diagram in a certain bicategory, whose objects are étale groupoids and whose arrows are called groupoid correspondences. A groupoid correspondence is a space with commuting actions of the two groupoids, subject to some conditions.…”

“…Therefore, we assume that the reader has already seen [5] and we do not attempt to make this article self-contained. In Section 2, we only recall the most crucial results from [1,5]. In Section 3, we prove that any diagram has a groupoid model -not necessarily Hausdorff or locally compact.…”

Groupoid models for diagrams of groupoid correspondences

“…Similarly, a self-similarity of a group may be interpreted in this way, namely, as a generalised endomorphism of a group. The right setting for such "endomorphisms" is the bicategory Gr of étale groupoid correspondences introduced in [4]. A correspondence between two groupoids is a space with commuting actions of the two groupoids, subject to some conditions.…”

“…A groupoid correspondence X from an étale groupoid G to itself gives rise to a C * -correspondence C * (X ) from C * (G) to itself, which then has a Cuntz-Pimsner algebra O C * (X ) . It is shown in [4] that this construction gives many important classes of C * -algebras. Katsura's topological graph C * -algebras arise when G is a locally compact Hausdorff space, viewed as an étale groupoid.…”