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Documents authored by Angiuli, Carlo

Document
DOI: 10.4230/LIPIcs.LICS.2026.58
Fat Cell Structures and Generalized Algebraic Theories

Authors: Xu Huang and Carlo Angiuli

Published in: LIPIcs, Volume 380, 41st Annual Symposium on Logic in Computer Science (LICS 2026)

Abstract
We give a new syntax-independent account of finitely-presented generalized algebraic theories (GATs) as finite cell complexes in the category of categories with families (CwFs), in which GATs are constructed by successive pushouts along the CwF morphisms generically postulating a sort, an operation, or an equation. Inspired by the fat small object argument of Makkai, Rosický, and Vokřínek, we introduce fat GAT presentations, thereby allowing infinite presentations with non-linear dependency structure. Then, motivated by wanting our GATs to self-describe, we extend presentations to admit infinitary arities, including infinitely deep dependency chains. Finally, we verify that these generalized GATs satisfy expected semantic properties including Frey’s Gabriel–Ulmer duality.
Cite as

Xu Huang and Carlo Angiuli. Fat Cell Structures and Generalized Algebraic Theories. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 58:1-58:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{huang_et_al:LIPIcs.LICS.2026.58,
  author =	{Huang, Xu and Angiuli, Carlo},
  title =	{{Fat Cell Structures and Generalized Algebraic Theories}},
  booktitle =	{41st Annual Symposium on Logic in Computer Science (LICS 2026)},
  pages =	{58:1--58:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-434-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{380},
  editor =	{Faggian, Claudia and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.LICS.2026.58},
  URN =		{urn:nbn:de:0030-drops-268451},
  doi =		{10.4230/LIPIcs.LICS.2026.58},
  annote =	{Keywords: Generalized algebraic theories, quotient inductive-inductive types, categories with families, cell complexes, logical frameworks, Gabriel-Ulmer duality}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2019.31
Cubical Syntax for Reflection-Free Extensional Equality

Authors: Jonathan Sterling, Carlo Angiuli, and Daniel Gratzer

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

Abstract
We contribute XTT, a cubical reconstruction of Observational Type Theory [Altenkirch et al., 2007] which extends Martin-Löf’s intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of identity proofs principle (UIP): any two elements of the same equality type are judgmentally equal. Moreover, we conjecture that the typing relation can be decided in a practical way. In this paper, we establish an algebraic canonicity theorem using a novel extension of the logical families or categorical gluing argument inspired by Coquand and Shulman [Coquand, 2018; Shulman, 2015]: every closed element of boolean type is derivably equal to either true or false.
Cite as

Jonathan Sterling, Carlo Angiuli, and Daniel Gratzer. Cubical Syntax for Reflection-Free Extensional Equality. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 31:1-31:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{sterling_et_al:LIPIcs.FSCD.2019.31,
  author =	{Sterling, Jonathan and Angiuli, Carlo and Gratzer, Daniel},
  title =	{{Cubical Syntax for Reflection-Free Extensional Equality}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{31:1--31:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.31},
  URN =		{urn:nbn:de:0030-drops-105387},
  doi =		{10.4230/LIPIcs.FSCD.2019.31},
  annote =	{Keywords: Dependent type theory, extensional equality, cubical type theory, categorical gluing, canonicity}
}
Document
DOI: 10.4230/LIPIcs.CSL.2018.6
Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities

Authors: Carlo Angiuli, Kuen-Bang Hou (Favonia), and Robert Harper

Published in: LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

Abstract
We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The fibrant fragment validates Voevodsky's univalence axiom and includes a circle type, while the non-fibrant fragment includes exact (strict) equality types satisfying equality reflection. Our type theory is defined by a semantics in cubical partial equivalence relations, and is the first two-level type theory to satisfy the canonicity property: all closed terms of boolean type evaluate to either true or false.
Cite as

Carlo Angiuli, Kuen-Bang Hou (Favonia), and Robert Harper. Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{angiuli_et_al:LIPIcs.CSL.2018.6,
  author =	{Angiuli, Carlo and Hou (Favonia), Kuen-Bang and Harper, Robert},
  title =	{{Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities}},
  booktitle =	{27th EACSL Annual Conference on Computer Science Logic (CSL 2018)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-088-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{119},
  editor =	{Ghica, Dan R. and Jung, Achim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.6},
  URN =		{urn:nbn:de:0030-drops-96734},
  doi =		{10.4230/LIPIcs.CSL.2018.6},
  annote =	{Keywords: Homotopy Type Theory, Two-Level Type Theory, Computational Type Theory, Cubical Sets}
}
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