Pushouts of extensions of groupoids by bundles of abelian groups

TL;DR

This paper investigates extensions of groupoids by abelian group bundles, introduces a pushout construction for these extensions, and relates the resulting structures to $C^*$-algebras, providing new insights and examples.

Contribution

It introduces a pushout construction for extensions of groupoids by abelian groups and relates these to $C^*$-algebra isomorphisms, advancing the understanding of such extensions.

Findings

The full $C^*$-algebra of the constructed twist is isomorphic to that of the extension.

The isomorphism between full $C^*$-algebras descends to reduced algebras.

Examples and applications illustrating the theory are provided.

Abstract

We analyse extensions $\Sigma$ of groupoids $G$ by bundles $A$ of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid $G$ by a given bundle $A$. There is a natural action of $\Sigma$ on the dual of $A$, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of $A$ with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of $G$ on the dual of $A$. We prove that the full $C^*$-algebra of this twist is isomorphic to the full $C^*$-algebra of $\Sigma$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.