Componentwise Polish groupoids and equivalence relations

TL;DR

This paper explores the structure of Borel equivalence relations and groupoids with componentwise Polish topologies, showing they are Borel equivalent to action groupoids of Polish groups, and extends classical tools to this setting.

Contribution

It introduces the concept of componentwise Polish topologies on Borel groupoids and proves their equivalence to global Polish groupoid actions, extending classical results.

Findings

Every such groupoid is Borel equivalent to a Polish group action groupoid.

The results extend to Borel groupoids with quasi-Polish topologies.

Standard tools like Vaught transforms and Effros's theorem are generalized.

Abstract

We study Borel equivalence relations equipped with a uniformly Borel family of Polish topologies on each equivalence class, and more generally, standard Borel groupoids equipped with such a family of topologies on each connected component. Such "componentwise Polish topologies" capture precisely the topological information determined by the Borel structure of a Polish group action, by the Becker--Kechris theorem. We prove that conversely, every abstract such Borel componentwise Polish groupoid obeying suitable axioms admits a Borel equivalence of groupoids to a global open Polish groupoid. Together with known results, this implies that every such groupoid is Borel equivalent to an action groupoid of a Polish group action; in particular, the induced equivalence relations are Borel bireducible. Our results are also valid for Borel groupoids with componentwise quasi-Polish topologies; and under stronger uniformity assumptions, we show that such groupoids in fact themselves admit global quasi-Polish topologies. As a byproduct, we also generalize several standard tools for Polish groups and their actions to the setting of componentwise quasi-Polish groupoids, including Vaught transforms, Effros's theorem on orbits, and the open mapping theorem.