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DOI: 10.4230/LIPIcs.MFCS.2026.44

Functorial Semantics for First-Order Theories

Authors: Filippo Bonchi, Alessandro Di Giorgio, Roberto Di Virgilio, and Paweł Sobociński

Published in: LIPIcs, Volume 386, 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)

Abstract

Building on the recent axiomatisation of first-order bicategories, we develop a functorial semantics approach to the model theory of first-order logic. First-order theories 𝕋 are captured by free first-order bicategories ℱ_𝕋 and models of𝕋 are structure-preserving functors from ℱ_𝕋 to a first-order bicategory 𝐂. Elementary morphisms of models arise as lax natural transformations between such functors, and the classical Tarski-Vaught test and downward Löwenheim-Skolem theorem admit direct diagrammatic proofs. Our results instantiate classically when 𝐂 = Rel and hold uniformly for models valued in Rel(𝐃) over an arbitrary Boolean geometric category 𝐃 in which regular epis split.

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Filippo Bonchi, Alessandro Di Giorgio, Roberto Di Virgilio, and Paweł Sobociński. Functorial Semantics for First-Order Theories. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bonchi_et_al:LIPIcs.MFCS.2026.44,
  author =	{Bonchi, Filippo and Di Giorgio, Alessandro and Di Virgilio, Roberto and Soboci\'{n}ski, Pawe{\l}},
  title =	{{Functorial Semantics for First-Order Theories}},
  booktitle =	{51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
  pages =	{44:1--44:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-442-0},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{386},
  editor =	{Kouck\'{y}, Michal and Petrișan, Daniela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.44},
  URN =		{urn:nbn:de:0030-drops-274255},
  doi =		{10.4230/LIPIcs.MFCS.2026.44},
  annote =	{Keywords: First-order logic, Model theory, Functorial semantics, String diagrams}
}

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Functorial Semantics for First-Order Theories

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