Representing Polish groupoids via metric structures

Abstract

We prove that every open $\sigma$-locally Polish groupoid $G$ is Borel equivalent to the groupoid of models on the Urysohn sphere $\mathbb{U}$ of an $\mathcal{L}_{\omega_1\omega}$-sentence in continuous logic. In particular, the orbit equivalence relations of such groupoids are up to Borel bireducibility precisely those of Polish group actions, answering a question of Lupini. Analogously, every non-Archimedean (i.e., every unit morphism has a neighborhood basis of open subgroupoids) open quasi-Polish groupoid is Borel equivalent to the groupoid of models on $\mathbb{N}$ of an $\mathcal{L}_{\omega_1\omega}$-sentence in discrete logic. The proof in fact gives a topological representation of $G$ as the groupoid of isomorphisms between a "continuously varying" family of structures over the space of objects of $G$, constructed via a topological Yoneda-like lemma of Moerdijk for localic groupoids and its metric analog. Other ingredients in our proof include the Lopez-Escobar theorem for continuous logic, a uniformization result for full Borel functors between open quasi-Polish groupoids, a uniform Borel version of Kat\v{e}tov's construction of $\mathbb{U}$, groupoid versions of the Pettis and Birkhoff--Kakutani theorems, and a development of the theory of non-Hausdorff topometric spaces and their quotients.