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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.

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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}
}

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Fat Cell Structures and Generalized Algebraic Theories

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