Graph morphisms as groupoid actors

TL;DR

This paper introduces a conceptual framework for understanding graph morphisms and related *-homomorphisms via proper actors from graph groupoids to étale groupoids, encompassing self-similar structures.

Contribution

It provides a new description of graph morphisms and *-homomorphisms using bisections and actors from groupoid models, extending to self-similar groups and graphs.

Findings

Proper actors characterized by bisections

Unified understanding of graph morphisms and *-homomorphisms

Applicable to self-similar groupoid structures

Abstract

We describe proper actors from the underlying groupoid of a graph C*-algebra to another \'etale groupoid in terms of bisections. This allows to understand graph morphisms and the *-homomorphisms that they induce more conceptually. More generally, we describe actors from the groupoid model of a groupoid correspondence to any \'etale groupoid. This also covers the groupoids associated to self-similar groups and self-similar graphs, among others.