Groupoid equivalence and the associated iterated crossed product

TL;DR

This paper explores the relationship between groupoid equivalences and iterated crossed products, establishing a natural isomorphism between two different constructions involving groupoid dynamical systems.

Contribution

It introduces a framework connecting groupoid equivalences with iterated crossed products, providing a new isomorphism result in the theory of groupoid dynamical systems.

Findings

The iterated crossed product $(A\rtimes G\ltimes X)\rtimes H$ is isomorphic to $A\rtimes (G\ltimes X\rtimes H)$.

A natural action of $H$ on $A\rtimes G\ltimes X$ is constructed.

The paper generalizes the understanding of crossed products in the context of groupoid equivalences.

Abstract

Given groupoids $G$ and $H$ and a $(G,H)$-equivalence $X$ we may form the transformation groupoid $G\ltimes X\rtimes H$. Given a separable groupoid dynamical system $(A,G\ltimes X\rtimes H,\omega)$ we may restrict $\omega$ to an action of $G\ltimes X$ on $A$ and form the crossed product $A\rtimes G\ltimes X$. We show that there is an action of $H$ on $A\rtimes G\ltimes X$ and that the iterated crossed product $(A\rtimes G\ltimes X)\rtimes H$ is naturally isomorphic to the crossed product $A\rtimes (G\ltimes X\rtimes H)$.