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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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


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
  • MR Author ID: 1012788
  • Email:
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 8955-8983
  • MSC (2010): Primary 03E15; Secondary 03G30, 03C15, 18C10
  • DOI:
  • MathSciNet review: 4029718