Existence of groupoid models for diagrams of groupoid correspondences

TL;DR

This paper proves that diagrams of étale groupoid correspondences can be modeled by locally compact étale groupoids under certain conditions, using a relative Stone-ech compactification as a key tool.

Contribution

It establishes the existence of groupoid models for diagrams of étale groupoid correspondences and characterizes their properties under local compactness and properness.

Findings

Every diagram has a groupoid model.

The groupoid model is locally compact and étale if the diagram is locally compact and proper.

The relative Stone-ech compactification is used as a key tool.

Abstract

This article continues the study of diagrams in the bicategory of \'etale groupoid correspondences. We prove that any such diagram has a groupoid model and that the groupoid model is a locally compact \'etale groupoid if the diagram is locally compact and proper. A key tool for this is the relative Stone-\v{C}ech compactification for spaces over a locally compact Hausdorff space.