Structurable equivalence relations and ${\mathcal{L}}_{{\omega }_1\omega }$ interpretations

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 ${\mathcal{T}}_{\mathsf{L}\mathsf{N}}\bigsqcup {\mathcal{T}}_{\mathsf{s}\mathsf{e}\mathsf{p}}$ 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 ${\mathcal{T}}_{\mathsf{L}\mathsf{N}}\bigsqcup {\mathcal{T}}_{\mathsf{s}\mathsf{e}\mathsf{p}}$, 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 ${\mathcal{T}}_{\mathsf{L}\mathsf{N}}$, which admit a characterization analogous to that of Hjorth–Kechris for essentially countable isomorphism relations.