Groupoid models for diagrams of groupoid correspondences

Abstract

A diagram of groupoid correspondences is a homomorphism to the bicategory of étale groupoid correspondences. We study examples of such diagrams, including complexes of groups and self-similar higher-rank graphs. We encode the diagram in a single groupoid, which we call its groupoid model. The groupoid model is defined so that there is a natural bijection between its actions on a space and suitably defined actions of the diagram. We describe the groupoid model in several cases, including a complex of groups or a self-similar group. We show that the groupoid model is a bilimit in the bicategory of groupoid correspondences.

Existence of groupoid models for diagrams of groupoid correspondences

“…The main result in this article answers an important, but technical question in the previous article [5]. 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].…”

“…By the results in [5], the groupoid model exists if and only if the category of actions of the diagram on spaces defined in [5] has a terminal object, and then it is unique up to isomorphism. To show that such a terminal diagram action exists, we prove that the category of actions is cocomplete and has a coseparating set of objects; this criterion is also used to prove the Special Adjoint Functor Theorem.…”

“…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.…”

“…In favourable cases, the C * -algebra associated to such a diagram is a groupoid C * -algebra of a certain étale groupoid built from the diagram. A candidate for this groupoid is proposed in [5], where it is called the groupoid model of the diagram.…”

“…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.…”