Structurable equivalence relations and Lω₁ω interpretations

We show that the category of countable Borel equivalence relations (CBERs) is dually equivalent to the category of countable Lω₁ω theories which admit a one-sorted interpretation of a particular theory we call TLN Tₛep that witnesses embeddability into 2N and the Lusin--Novikov uniformization theorem. This allows problems about Borel combinatorial structures on CBERs to be translated into syntactic definability problems in Lω₁ω, modulo the extra structure provided by TLN Tₛep, 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 ω-colorings and the Slaman--Steel marker lemma. We also generalize this correspondence to locally countable Borel groupoids and theories interpreting TLN, which admit a characterization analogous to that of Hjorth--Kechris for essentially countable isomorphism relations.