Borel functors, interpretations, and strong conceptual completeness for $\mathcal L_{\omega_1\omega}$

Abstract

We prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic $\mathcal L_{\omega_1\omega}$: every countable $\mathcal L_{\omega_1\omega}$-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories $(\mathcal L, \mathcal T)$ and $(\mathcal L', \mathcal T')$ (in possibly different languages $\mathcal L, \mathcal L'$), every Borel functor $\mathsf{Mod}(\mathcal L', \mathcal T') \to \mathsf{Mod}(\mathcal L, \mathcal T)$ between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some $\mathcal L'_{\omega_1\omega}$-interpretation of $\mathcal T$ in $\mathcal T'$. This generalizes a recent result of Harrison-Trainor, Miller, and Montalb\'an in the case where $\mathcal T, \mathcal T'$ each have a single countable model up to isomorphism.