10 Search Results for "Hou (Favonia), Kuen-Bang"

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.ITP.2025.20
Formalizing Colimits in 𝒞at

Authors: Mario Carneiro and Emily Riehl

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
Certain results involving "higher structures" are not currently accessible to computer formalization because the prerequisite ∞-category theory has not been formalized. To support future work on formalizing ∞-category theory in Lean’s mathematics library, we formalize some fundamental constructions involving the 1-category of categories. Specifically, we construct the left adjoint to the nerve embedding of categories into simplicial sets, defining the homotopy category functor. We prove further that this adjunction is reflective, which allows us to conclude that 𝒞at has colimits. To our knowledge this is the first formalized proof that the nerve functor is a fully faithful right adjoint and that the category of categories is cocomplete.
Cite as

Mario Carneiro and Emily Riehl. Formalizing Colimits in 𝒞at. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{carneiro_et_al:LIPIcs.ITP.2025.20,
  author =	{Carneiro, Mario and Riehl, Emily},
  title =	{{Formalizing Colimits in 𝒞at}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{20:1--20:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.20},
  URN =		{urn:nbn:de:0030-drops-246186},
  doi =		{10.4230/LIPIcs.ITP.2025.20},
  annote =	{Keywords: category theory, infinity-category theory, nerve, simplicial set, colimit}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2024.9
Complexity of Cubical Cofibration Logics I: coNP-Complete Examples

Authors: Robert Rose and Daniel R. Licata

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

Abstract
We provide a comprehensive classification of the cofibration entailment problem, COFENT, for the cofibration logics of various cubical type theories in use today. The problem COFENT arose from the need of cubical proof assistants to automate reasoning about cubical complexes included in an n-dimensional hypercube. Intuitively, it asks: given logical descriptions of two such complexes, is one a subcomplex of the other? We show that the common variants of COFENT are coNP-complete.
Cite as

Robert Rose and Daniel R. Licata. Complexity of Cubical Cofibration Logics I: coNP-Complete Examples. In 30th International Conference on Types for Proofs and Programs (TYPES 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 336, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{rose_et_al:LIPIcs.TYPES.2024.9,
  author =	{Rose, Robert and Licata, Daniel R.},
  title =	{{Complexity of Cubical Cofibration Logics I: coNP-Complete Examples}},
  booktitle =	{30th International Conference on Types for Proofs and Programs (TYPES 2024)},
  pages =	{9:1--9:21},
  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.2024.9},
  URN =		{urn:nbn:de:0030-drops-233711},
  doi =		{10.4230/LIPIcs.TYPES.2024.9},
  annote =	{Keywords: cubical sets, internal language, intuitionistic logic, dependent type theory, homotopy type theory, decision procedures}
}
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.CSL.2020.13
Internal Parametricity for Cubical Type Theory

Authors: Evan Cavallo and Robert Harper

Published in: LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

Abstract
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, and we give an account of the identity extension lemma for internal parametricity.
Cite as

Evan Cavallo and Robert Harper. Internal Parametricity for Cubical Type Theory. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cavallo_et_al:LIPIcs.CSL.2020.13,
  author =	{Cavallo, Evan and Harper, Robert},
  title =	{{Internal Parametricity for Cubical Type Theory}},
  booktitle =	{28th EACSL Annual Conference on Computer Science Logic (CSL 2020)},
  pages =	{13:1--13:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-132-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{152},
  editor =	{Fern\'{a}ndez, Maribel and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.13},
  URN =		{urn:nbn:de:0030-drops-116564},
  doi =		{10.4230/LIPIcs.CSL.2020.13},
  annote =	{Keywords: parametricity, cubical type theory, higher inductive types}
}
Document
DOI: 10.4230/LIPIcs.CSL.2020.14
Unifying Cubical Models of Univalent Type Theory

Authors: Evan Cavallo, Anders Mörtberg, and Andrew W Swan

Published in: LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

Abstract
We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure.
Cite as

Evan Cavallo, Anders Mörtberg, and Andrew W Swan. Unifying Cubical Models of Univalent Type Theory. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cavallo_et_al:LIPIcs.CSL.2020.14,
  author =	{Cavallo, Evan and M\"{o}rtberg, Anders and Swan, Andrew W},
  title =	{{Unifying Cubical Models of Univalent Type Theory}},
  booktitle =	{28th EACSL Annual Conference on Computer Science Logic (CSL 2020)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-132-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{152},
  editor =	{Fern\'{a}ndez, Maribel and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.14},
  URN =		{urn:nbn:de:0030-drops-116578},
  doi =		{10.4230/LIPIcs.CSL.2020.14},
  annote =	{Keywords: Cubical Set Models, Cubical Type Theory, Homotopy Type Theory, Univalent Foundations}
}
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.TYPES.2016.11
Covering Spaces in Homotopy Type Theory

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

Published in: LIPIcs, Volume 97, 22nd International Conference on Types for Proofs and Programs (TYPES 2016)

Abstract
Broadly speaking, algebraic topology consists of associating algebraic structures to topological spaces that give information about their structure. An elementary, but fundamental, example is provided by the theory of covering spaces, which associate groups to covering spaces in such a way that the universal cover corresponds to the fundamental group of the space. One natural question to ask is whether these connections can be stated in homotopy type theory, a new area linking type theory to homotopy theory. In this paper, we give an affirmative answer with a surprisingly concise definition of covering spaces in type theory; we are able to prove various expected properties about the newly defined covering spaces, including the connections with fundamental groups. An additional merit is that our work has been fully mechanized in the proof assistant Agda.
Cite as

Kuen-Bang Hou (Favonia) and Robert Harper. Covering Spaces in Homotopy Type Theory. In 22nd International Conference on Types for Proofs and Programs (TYPES 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 97, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hou(favonia)_et_al:LIPIcs.TYPES.2016.11,
  author =	{Hou (Favonia), Kuen-Bang and Harper, Robert},
  title =	{{Covering Spaces in Homotopy Type Theory}},
  booktitle =	{22nd International Conference on Types for Proofs and Programs (TYPES 2016)},
  pages =	{11:1--11:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-065-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{97},
  editor =	{Ghilezan, Silvia and Geuvers, Herman and Ivetic, Jelena},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2016.11},
  URN =		{urn:nbn:de:0030-drops-98512},
  doi =		{10.4230/LIPIcs.TYPES.2016.11},
  annote =	{Keywords: homotopy type theory, covering space, fundamental group, mechanized reasoning}
}
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}
}
Document
DOI: 10.4230/LIPIcs.CSL.2016.22
The Seifert-van Kampen Theorem in Homotopy Type Theory

Authors: Kuen-Bang Hou (Favonia) and Michael Shulman

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

Abstract
Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic theorems, yielding "synthetic homotopy theory". Here we consider the Seifert-van Kampen theorem, which characterizes the loop structure of spaces obtained by gluing. This is useful in homotopy theory because many spaces are constructed by gluing, and the loop structure helps distinguish distinct spaces. The synthetic proof showcases many new characteristics of synthetic homotopy theory, such as the "encode-decode" method, enforced homotopy-invariance, and lack of underlying sets.
Cite as

Kuen-Bang Hou (Favonia) and Michael Shulman. The Seifert-van Kampen Theorem in Homotopy Type Theory. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{hou(favonia)_et_al:LIPIcs.CSL.2016.22,
  author =	{Hou (Favonia), Kuen-Bang and Shulman, Michael},
  title =	{{The Seifert-van Kampen Theorem in Homotopy Type Theory}},
  booktitle =	{25th EACSL Annual Conference on Computer Science Logic (CSL 2016)},
  pages =	{22:1--22:16},
  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.2016.22},
  URN =		{urn:nbn:de:0030-drops-65626},
  doi =		{10.4230/LIPIcs.CSL.2016.22},
  annote =	{Keywords: homotopy type theory, fundamental group, homotopy pushout, mechanized reasoning}
}
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