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DOI: 10.4230/LIPIcs.TYPES.2024.6

Data Types with Symmetries via Action Containers

Authors: Philipp Joram and Niccolò Veltri

Published in: LIPIcs, Volume 336, 30th International Conference on Types for Proofs and Programs (TYPES 2024)

Abstract

We study two kinds of containers for data types with symmetries in homotopy type theory, and clarify their relationship by introducing the intermediate notion of action containers. Quotient containers are set-valued containers with groups of permissible permutations of positions, interpreted as (possibly non-finitary) analytic functors on the category of sets. Symmetric containers encode symmetries in a groupoid of shapes, and are interpreted accordingly as polynomial functors on the 2-category of groupoids. Action containers are endowed with groups that act on their positions, with morphisms preserving the actions. We show that, as a category, action containers are equivalent to the free coproduct completion of a category of group actions. We derive that they model non-inductive single-variable strictly positive types in the sense of Abbott et al.: The category of action containers is closed under arbitrary (co)products and exponentiation with constants. We equip this category with the structure of a locally groupoidal 2-category, and prove that it locally embeds into the 2-category of symmetric containers. This follows from the embedding of a 2-category of groups into the 2-category of groupoids, extending the delooping construction.

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Philipp Joram and Niccolò Veltri. Data Types with Symmetries via Action Containers. In 30th International Conference on Types for Proofs and Programs (TYPES 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 336, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{joram_et_al:LIPIcs.TYPES.2024.6,
  author =	{Joram, Philipp and Veltri, Niccol\`{o}},
  title =	{{Data Types with Symmetries via Action Containers}},
  booktitle =	{30th International Conference on Types for Proofs and Programs (TYPES 2024)},
  pages =	{6:1--6:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-376-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{336},
  editor =	{M{\o}gelberg, Rasmus Ejlers and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2024.6},
  URN =		{urn:nbn:de:0030-drops-233681},
  doi =		{10.4230/LIPIcs.TYPES.2024.6},
  annote =	{Keywords: Containers, Homotopy Type Theory, Agda, 2-categories}
}

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DOI: 10.4230/LIPIcs.ITP.2023.20

Constructive Final Semantics of Finite Bags

Authors: Philipp Joram and Niccolò Veltri

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)

Abstract

Finitely-branching and unlabelled dynamical systems are typically modelled as coalgebras for the finite powerset functor. If states are reachable in multiple ways, coalgebras for the finite bag functor provide a more faithful representation. The final coalgebra of this functor is employed as a denotational domain for the evaluation of such systems. Elements of the final coalgebra are non-wellfounded trees with finite unordered branching, representing the evolution of systems starting from a given initial state. This paper is dedicated to the construction of the final coalgebra of the finite bag functor in homotopy type theory (HoTT). We first compare various equivalent definitions of finite bags employing higher inductive types, both as sets and as groupoids (in the sense of HoTT). We then analyze a few well-known, classical set-theoretic constructions of final coalgebras in our constructive setting. We show that, in the case of set-based definitions of finite bags, some constructions are intrinsically classical, in the sense that they are equivalent to some weak form of excluded middle. Nevertheless, a type satisfying the universal property of the final coalgebra can be constructed in HoTT employing the groupoid-based definition of finite bags. We conclude by discussing generalizations of our constructions to the wider class of analytic functors.

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Philipp Joram and Niccolò Veltri. Constructive Final Semantics of Finite Bags. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{joram_et_al:LIPIcs.ITP.2023.20,
  author =	{Joram, Philipp and Veltri, Niccol\`{o}},
  title =	{{Constructive Final Semantics of Finite Bags}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{20:1--20:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.20},
  URN =		{urn:nbn:de:0030-drops-183954},
  doi =		{10.4230/LIPIcs.ITP.2023.20},
  annote =	{Keywords: finite bags, final coalgebra, homotopy type theory, Cubical Agda}
}

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Data Types with Symmetries via Action Containers

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Constructive Final Semantics of Finite Bags

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