Borel functors, interpretations, and strong conceptual completeness for ℒ_{𝜔₁𝜔}

Chen, Ruiyuan

doi:10.1090/tran/7950

Borel functors, interpretations, and strong conceptual completeness for $\mathcal {L}_{\omega _1\omega }$
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Trans. Amer. Math. Soc. 372 (2019), 8955-8983 Request permission

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 $\text {Mod}(\mathcal {L}’, \mathcal {T}’) \to \text {Mod}(\mathcal {L}, \mathcal {T})$ 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 Montalbán in the $\aleph _0$-categorical case.

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