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 $({\mathcal{L}}^{\prime },{\mathcal{T}}^{\prime })$ (in possibly different languages $\mathcal{L},{\mathcal{L}}^{\prime }$), every Borel functor $\mathsf{Mod}({\mathcal{L}}^{\prime },{\mathcal{T}}^{\prime })\to \mathsf{Mod}(\mathcal{L},\mathcal{T})$ between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some ${\mathcal{L}}_{{\omega }_1\omega }^{\prime }$-interpretation of $\mathcal{T}$ in ${\mathcal{T}}^{\prime }$. This generalizes a recent result of Harrison-Trainor, Miller, and Montalbán in the case where $\mathcal{T},{\mathcal{T}}^{\prime }$ each have a single countable model up to isomorphism.

arXiv comment49 pages; minor revisions