Structurable equivalence relations and $\mathcal{L}_{\omega_1\omega}$ interpretations

TL;DR

This paper establishes a duality between countable Borel equivalence relations and certain $rac{ ext{L}}{ ext{omega}_1 ext{omega}}$ theories, translating combinatorial problems into logical definability issues.

Contribution

It introduces a categorical duality linking Borel equivalence relations with $rac{ ext{L}}{ ext{omega}_1 ext{omega}}$ theories, formalizing folklore intuitions in Borel combinatorics.

Findings

Category of CBERs is dually equivalent to certain $rac{ ext{L}}{ ext{omega}_1 ext{omega}}$ theories.

Interpretability relations among standard Borel combinatorial structures are characterized.

Generalization to Borel groupoids and theories interpreting $rac{ ext{L}}{ ext{omega}_1 ext{omega}}$ theories.

Abstract

We show that the category of countable Borel equivalence relations (CBERs) is dually equivalent to the category of countable $\mathcal{L}_{\omega_1\omega}$ theories which admit a one-sorted interpretation of a particular theory we call $\mathcal{T}_\mathsf{LN} \sqcup \mathcal{T}_\mathsf{sep}$ that witnesses embeddability into $2^\mathbb{N}$ and the Lusin--Novikov uniformization theorem. This allows problems about Borel combinatorial structures on CBERs to be translated into syntactic definability problems in $\mathcal{L}_{\omega_1\omega}$, modulo the extra structure provided by $\mathcal{T}_\mathsf{LN} \sqcup \mathcal{T}_\mathsf{sep}$, thereby formalizing a folklore intuition in locally countable Borel combinatorics. We illustrate this with a catalogue of the precise interpretability relations between several standard classes of structures commonly used in Borel combinatorics, such as Feldman--Moore $\omega$-colorings and the Slaman--Steel marker lemma. We also generalize this correspondence to locally countable Borel groupoids and theories interpreting $\mathcal{T}_\mathsf{LN}$, which admit a characterization analogous to that of Hjorth--Kechris for essentially countable isomorphism relations.