Groupoid models for diagrams of groupoid correspondences

TL;DR

This paper introduces a unified groupoid model for diagrams of groupoid correspondences, including complexes of groups and self-similar higher-rank graphs, establishing a natural bijection with their actions.

Contribution

It defines a novel groupoid model that encodes diagrams of groupoid correspondences and proves it as a bilimit in the bicategory, linking actions on spaces.

Findings

The groupoid model provides a natural bijection with diagram actions.

The model applies to complexes of groups and self-similar higher-rank graphs.

It is characterized as a bilimit in the bicategory of groupoid correspondences.

Abstract

A diagram of groupoid correspondences is a homomorphism to the bicategory of \'etale 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.