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DOI: 10.4230/LIPIcs.CSL.2026.40

The Groupoid-Syntax of Type Theory Is a Set

Authors: Thorsten Altenkirch, Ambrus Kaposi, and Szumi Xie

Published in: LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026)

Abstract

Categories with families (CwFs) have been used to define the semantics of type theory in type theory. In the setting of Homotopy Type Theory (HoTT), one of the limitations of the traditional notion of CwFs is the requirement to set-truncate types, which excludes models based on univalent categories, such as the standard set model. To address this limitation, we introduce the concept of a Groupoid Category with Families (GCwF). This framework truncates types at the groupoid level and incorporates coherence equations, providing a natural extension of the CwF framework when starting from a 1-category. We demonstrate that the initial GCwF for a type theory with a base family of sets and Π-types (groupoid-syntax) is set-truncated. Consequently, this allows us to utilize the conventional intrinsic syntax of type theory while enabling interpretations in semantically richer and more natural models. All constructions in this paper were formalised in Cubical Agda.

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Thorsten Altenkirch, Ambrus Kaposi, and Szumi Xie. The Groupoid-Syntax of Type Theory Is a Set. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 40:1-40:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{altenkirch_et_al:LIPIcs.CSL.2026.40,
  author =	{Altenkirch, Thorsten and Kaposi, Ambrus and Xie, Szumi},
  title =	{{The Groupoid-Syntax of Type Theory Is a Set}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{40:1--40:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-411-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{363},
  editor =	{Guerrini, Stefano and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.40},
  URN =		{urn:nbn:de:0030-drops-254650},
  doi =		{10.4230/LIPIcs.CSL.2026.40},
  annote =	{Keywords: Categorical models of type theory, category with families, groupoids, coherence, homotopy type theory}
}

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DOI: 10.4230/LIPIcs.FSCD.2024.10

Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics

Authors: Ambrus Kaposi and Szumi Xie

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Programming languages can be defined from the concrete to the abstract by abstract syntax trees, well-scoped syntax, well-typed (intrinsic) syntax, algebraic syntax (well-typed syntax quotiented by conversion). Another aspect is the representation of binding structure for which nominal approaches, De Bruijn indices/levels and higher order abstract syntax (HOAS) are available. In HOAS, binders are given by the function space of an internal language of presheaves. In this paper, we show how to combine the algebraic approach with the HOAS approach: following Uemura, we define languages as second-order generalised algebraic theories (SOGATs). Through a series of examples we show that non-substructural languages can be naturally defined as SOGATs. We give a formal definition of SOGAT signatures (using the syntax of a particular SOGAT) and define two translations from SOGAT signatures to GAT signatures (signatures for quotient inductive-inductive types), based on parallel and single substitutions, respectively.

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Ambrus Kaposi and Szumi Xie. Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kaposi_et_al:LIPIcs.FSCD.2024.10,
  author =	{Kaposi, Ambrus and Xie, Szumi},
  title =	{{Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{10:1--10:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.10},
  URN =		{urn:nbn:de:0030-drops-203396},
  doi =		{10.4230/LIPIcs.FSCD.2024.10},
  annote =	{Keywords: Type theory, universal algebra, inductive types, quotient inductive types, higher-order abstract syntax, logical framework}
}

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The Groupoid-Syntax of Type Theory Is a Set

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Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics

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