Structurable equivalence relations and ${\mathcal{L}}_{\ddot{I}{‰}_1\ddot{I}‰}$ interpretations

Abstract

We show that the category of countable Borel equivalence relations (CBERs) is dually equivalent to the category of countable ${\mathcal{L}}_{\ddot{I}{‰}_1\ddot{I}‰}$ theories which admit a one-sorted interpretation of a particular theory we call ${\mathcal{T}}_{\mathsf{LN}}\bigsqcup {\mathcal{T}}_{\mathsf{sep}}$ 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 ${\mathcal{L}}_{\ddot{I}{‰}_1\ddot{I}‰}$, modulo the extra structure provided by ${\mathcal{T}}_{\mathsf{LN}}\bigsqcup {\mathcal{T}}_{\mathsf{sep}}$, 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 $\ddot{I}‰$-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.

55 pages