9 Search Results for "Capriotti, Paolo"

Document
DOI: 10.4230/LIPIcs.LICS.2026.59
Generalized Decidability via Brouwer Trees

Authors: Tom de Jong, Nicolai Kraus, Aref Mohammadzadeh, and Fredrik Nordvall Forsberg

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

Abstract
In the setting of constructive mathematics, we suggest and study a framework for decidability of properties, which allows for finer distinctions than just "decidable, semidecidable, or undecidable". We work in homotopy type theory and use Brouwer tree ordinals to specify the level of decidability of a property. In this framework, we express the property that a proposition is α-decidable, for an ordinal α, and show that it generalizes decidability and semidecidability. Further generalizing known results, we show that α-decidable propositions are closed under binary conjunction, and discuss for which α they are closed under binary disjunction. We prove that if each P(i) is semidecidable, then the countable meet ∀ i ∈ ℕ. P(i) is ω²-decidable, and similar results for countable joins and iterated quantifiers. We also discuss the relationship with countable choice. All our results are formalized in Cubical Agda.
Cite as

Tom de Jong, Nicolai Kraus, Aref Mohammadzadeh, and Fredrik Nordvall Forsberg. Generalized Decidability via Brouwer Trees. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 59:1-59:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dejong_et_al:LIPIcs.LICS.2026.59,
  author =	{de Jong, Tom and Kraus, Nicolai and Mohammadzadeh, Aref and Nordvall Forsberg, Fredrik},
  title =	{{Generalized Decidability via Brouwer Trees}},
  booktitle =	{41st Annual Symposium on Logic in Computer Science (LICS 2026)},
  pages =	{59:1--59:27},
  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.2026.59},
  URN =		{urn:nbn:de:0030-drops-268466},
  doi =		{10.4230/LIPIcs.LICS.2026.59},
  annote =	{Keywords: Decidability in constructive mathematics, homotopy type theory, ordinals, Brouwer trees, countable choice}
}
Document
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.CSL.2026.20
On Left Adjoints Preserving Colimits in HoTT

Authors: Perry Hart

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

Abstract
We examine how the standard proof that left adjoints preserve colimits behaves in the setting of wild categories, a natural setting for synthetic homotopy theory inside homotopy type theory. We prove that the proof may fail for adjunctions between wild categories. Our core contribution, however, is a sufficient condition on the left adjoint for the proof to go through. The condition, called 2-coherence, expresses that the naturality structure of the hom-isomorphism commutes with composition of morphisms. We present two useful examples of this condition in action. First, we use it, along with a new version of a known trick for homogeneous types, to show that the suspension functor preserves graph-indexed colimits. Second, we show that every modality, viewed as a functor on coslices of a type universe, is 2-coherent as a left adjoint to the forgetful functor from the subcategory of modal types, thereby proving this subcategory is cocomplete. We have formalized our main results in Agda.
Cite as

Perry Hart. On Left Adjoints Preserving Colimits in HoTT. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{hart:LIPIcs.CSL.2026.20,
  author =	{Hart, Perry},
  title =	{{On Left Adjoints Preserving Colimits in HoTT}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{20:1--20:17},
  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.2026.20},
  URN =		{urn:nbn:de:0030-drops-254442},
  doi =		{10.4230/LIPIcs.CSL.2026.20},
  annote =	{Keywords: wild categories, colimits, adjunctions, homotopy type theory, category theory, synthetic homotopy theory, higher inductive types, modalities}
}
Document
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.46
Coslice Colimits in Homotopy Type Theory

Authors: Perry Hart and Kuen-Bang Hou (Favonia)

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
We contribute to the theory of (homotopy) colimits inside homotopy type theory. The heart of our work characterizes the connection between colimits in coslices of a universe, called coslice colimits, and colimits in the universe (i.e., ordinary colimits). To derive this characterization, we find an explicit construction of colimits in coslices that is tailored to reveal the connection. We use the construction to derive properties of colimits. Notably, we prove that the forgetful functor from a coslice creates colimits over trees. We also use the construction to examine how colimits interact with orthogonal factorization systems and with cohomology theories. As a consequence of their interaction with orthogonal factorization systems, all pointed colimits (special kinds of coslice colimits) preserve n-connectedness, which implies that higher groups are closed under colimits on directed graphs. We have formalized our main construction of the coslice colimit functor in Agda.
Cite as

Perry Hart and Kuen-Bang Hou (Favonia). Coslice Colimits in Homotopy Type Theory. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hart_et_al:LIPIcs.CSL.2025.46,
  author =	{Hart, Perry and Hou (Favonia), Kuen-Bang},
  title =	{{Coslice Colimits in Homotopy Type Theory}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{46:1--46:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.46},
  URN =		{urn:nbn:de:0030-drops-228039},
  doi =		{10.4230/LIPIcs.CSL.2025.46},
  annote =	{Keywords: colimits, homotopy type theory, category theory, higher inductive types, synthetic homotopy theory}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2019.5
From Cubes to Twisted Cubes via Graph Morphisms in Type Theory

Authors: Gun Pinyo and Nicolai Kraus

Published in: LIPIcs, Volume 175, 25th International Conference on Types for Proofs and Programs (TYPES 2019)

Abstract
Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of higher groupoids. Bezem, Coquand, and Huber [Bezem et al., 2014] have presented a constructive model of univalence using a specific cube category, which we call the BCH cube category. The higher categories encoded with the BCH cube category have the property that all morphisms are invertible, mirroring the fact that equality is symmetric. This might not always be desirable: the field of directed type theory considers a notion of equality that is not necessarily invertible. This motivates us to suggest a category of twisted cubes which avoids built-in invertibility. Our strategy is to first develop several alternative (but equivalent) presentations of the BCH cube category using morphisms between suitably defined graphs. Starting from there, a minor modification allows us to define our category of twisted cubes. We prove several first results about this category, and our work suggests that twisted cubes combine properties of cubes with properties of globes and simplices (tetrahedra).
Cite as

Gun Pinyo and Nicolai Kraus. From Cubes to Twisted Cubes via Graph Morphisms in Type Theory. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{pinyo_et_al:LIPIcs.TYPES.2019.5,
  author =	{Pinyo, Gun and Kraus, Nicolai},
  title =	{{From Cubes to Twisted Cubes via Graph Morphisms in Type Theory}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-158-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{175},
  editor =	{Bezem, Marc and Mahboubi, Assia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.5},
  URN =		{urn:nbn:de:0030-drops-130694},
  doi =		{10.4230/LIPIcs.TYPES.2019.5},
  annote =	{Keywords: homotopy type theory, cubical sets, directed equality, graph morphisms}
}
Document
DOI: 10.4230/LIPIcs.CSL.2016.21
Extending Homotopy Type Theory with Strict Equality

Authors: Thorsten Altenkirch, Paolo Capriotti, and Nicolai Kraus

Published in: LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

Abstract
In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2-level theory which features both strict and weak equality. This can essentially be represented as two type theories: an "outer" one, containing a strict equality type former, and an "inner" one, which is some version of HoTT. Our type theory is inspired by Voevodsky's suggestion of a homotopy type system (HTS) which currently refers to a range of ideas. A core insight of our proposal is that we do not need any form of equality reflection in order to achieve what HTS was suggested for. Instead, having unique identity proofs in the outer type theory is sufficient, and it also has the meta-theoretical advantage of not breaking decidability of type checking. The inner theory can be an easily justifiable extensions of HoTT, allowing the construction of "infinite structures" which are considered impossible in plain HoTT. Alternatively, we can set the inner theory to be exactly the current standard formulation of HoTT, in which case our system can be thought of as a type-theoretic framework for working with "schematic" definitions in HoTT. As demonstrations, we define semi-simplicial types and formalise constructions of Reedy fibrant diagrams.
Cite as

Thorsten Altenkirch, Paolo Capriotti, and Nicolai Kraus. Extending Homotopy Type Theory with Strict Equality. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{altenkirch_et_al:LIPIcs.CSL.2016.21,
  author =	{Altenkirch, Thorsten and Capriotti, Paolo and Kraus, Nicolai},
  title =	{{Extending Homotopy Type Theory with Strict Equality}},
  booktitle =	{25th EACSL Annual Conference on Computer Science Logic (CSL 2016)},
  pages =	{21:1--21:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-022-4},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{62},
  editor =	{Talbot, Jean-Marc and Regnier, Laurent},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.21},
  URN =		{urn:nbn:de:0030-drops-65612},
  doi =		{10.4230/LIPIcs.CSL.2016.21},
  annote =	{Keywords: homotopy type theory, coherences, strict equality, homotopy type system}
}
Document
DOI: 10.4230/LIPIcs.CSL.2015.359
Functions out of Higher Truncations

Authors: Paolo Capriotti, Nicolai Kraus, and Andrea Vezzosi

Published in: LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

Abstract
In homotopy type theory, the truncation operator ||-||n (for a number n greater or equal to -1) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ||A||n -> B if B is not an n-type. This makes it desirable to derive more powerful elimination theorems. We show a first general result: If B is an (n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B that are constant on all (n+1)-st loop spaces. We give one "elementary" proof and one proof that uses a higher inductive type, both of which require some effort. As a sample application of our result, we show that we can construct "set-based" representations of 1-types, as long as they have "braided" loop spaces. The main result with one of its proofs and the application have been formalised in Agda.
Cite as

Paolo Capriotti, Nicolai Kraus, and Andrea Vezzosi. Functions out of Higher Truncations. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 359-373, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{capriotti_et_al:LIPIcs.CSL.2015.359,
  author =	{Capriotti, Paolo and Kraus, Nicolai and Vezzosi, Andrea},
  title =	{{Functions out of Higher Truncations}},
  booktitle =	{24th EACSL Annual Conference on Computer Science Logic (CSL 2015)},
  pages =	{359--373},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-90-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{41},
  editor =	{Kreutzer, Stephan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.359},
  URN =		{urn:nbn:de:0030-drops-54257},
  doi =		{10.4230/LIPIcs.CSL.2015.359},
  annote =	{Keywords: homotopy type theory, truncation elimination, constancy on loop spaces}
}
Document
DOI: 10.4230/LIPIcs.TLCA.2015.17
Non-Wellfounded Trees in Homotopy Type Theory

Authors: Benedikt Ahrens, Paolo Capriotti, and Régis Spadotti

Published in: LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

Abstract
We prove a conjecture about the constructibility of conductive types - in the principled form of indexed M-types - in Homotopy Type Theory. The conjecture says that in the presence of inductive types, coinductive types are derivable. Indeed, in this work, we construct coinductive types in a subsystem of Homotopy Type Theory; this subsystem is given by Intensional Martin-Löf type theory with natural numbers and Voevodsky's Univalence Axiom. Our results are mechanized in the computer proof assistant Agda.
Cite as

Benedikt Ahrens, Paolo Capriotti, and Régis Spadotti. Non-Wellfounded Trees in Homotopy Type Theory. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 17-30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{ahrens_et_al:LIPIcs.TLCA.2015.17,
  author =	{Ahrens, Benedikt and Capriotti, Paolo and Spadotti, R\'{e}gis},
  title =	{{Non-Wellfounded Trees in Homotopy Type Theory}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{17--30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-87-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{38},
  editor =	{Altenkirch, Thorsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.17},
  URN =		{urn:nbn:de:0030-drops-51522},
  doi =		{10.4230/LIPIcs.TLCA.2015.17},
  annote =	{Keywords: Homotopy Type Theory, coinductive types, computer theorem proving, Agda}
}
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