Equivalence of definitions of AF groupoid

TL;DR

This paper proves the equivalence of two different definitions of AF groupoids, showing they coincide only under specific conditions related to local homeomorphisms being covering maps.

Contribution

It establishes the equivalence of two prominent definitions of AF groupoids and clarifies the conditions under which they coincide.

Findings

The two definitions of AF groupoids are equivalent if and only if the local homeomorphism is a covering map.

The definitions differ in general, but coincide under specific topological conditions.

The paper provides a rigorous comparison of the foundational concepts in AF groupoid theory.

Abstract

We prove the equivalence of two definitions of AF groupoid in the literature: one by Renault and the other by Farsi, Kumjian, Pask and Sims. In both definitions, an AF groupoid is an increasing union of more basic groupoids, called elementary groupoids. Surprisingly, the two definitions of elementary groupoid are not equivalent; they coincide if and only if the local homeomorphism that characterises them is a covering map.