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

TL;DR

This paper establishes a strong form of conceptual completeness for the infinitary logic _{\u03c9_1} by showing that theories can be reconstructed from their models' groupoids, linking Borel functors to interpretations.

Contribution

It proves a strong conceptual completeness theorem for _{\u03c9_1}, connecting Borel functors between model groupoids to interpretations, generalizing previous results to theories with multiple models.

Findings

Every countable _{\u03c9_1} theory can be recovered from its models' Borel groupoid.

Borel functors between model groupoids correspond to interpretations between theories.

The result generalizes recent work to theories with multiple models.

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.