Structurable equivalence relations and L䃑 interpretations

We show that the category of countable Borel equivalence relations (CBERs) is dually equivalent to the category of countable L䃑 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 L䃑, 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 -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.