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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: Ruiyuan Chen
Journal: Trans. Amer. Math. Soc. 372 (2019), 8955-8983
MSC (2010): Primary 03E15; Secondary 03G30, 03C15, 18C10
DOI: https://doi.org/10.1090/tran/7950
Published electronically: September 23, 2019
MathSciNet review: 4029718
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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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Additional Information

Ruiyuan Chen
Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801
Email: ruiyuan@illinois.edu

DOI: https://doi.org/10.1090/tran/7950
Keywords: Strong conceptual completeness, infinitary logic, Borel functors, pretopos.
Received by editor(s): April 4, 2018
Received by editor(s) in revised form: June 16, 2019
Published electronically: September 23, 2019
Additional Notes: The author’s research was partially supported by NSERC PGS D
Article copyright: © Copyright 2019 American Mathematical Society