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Documents authored by Wullaert, Kobe

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DOI: 10.4230/LIPIcs.TYPES.2025.8
The Rezk Completion for Elementary Topoi

Authors: Kobe Wullaert and Niels van der Weide

Published in: LIPIcs, Volume 384, 31st International Conference on Types for Proofs and Programs (TYPES 2025)

Abstract
The development of category theory in univalent foundations and the formalization thereof is an active field of research. Categories in that setting are often assumed to be univalent which means that identities and isomorphisms of objects coincide. One consequence hereof is that equivalences and identities coincide for univalent categories and that structure on univalent categories transfers along equivalences. However, constructions such as the Kleisli category, the Karoubi envelope, and the tripos-to-topos construction, do not necessarily give univalent categories. To deal with that problem, one uses the Rezk completion, which completes a category into a univalent one. However, to use the Rezk completion when considering categories with structure, one also needs to show that the Rezk completion inherits the structure from the original category. In this work, we present a modular framework for lifting the Rezk completion from categories to categories with structure. We demonstrate the modularity of our framework by lifting the Rezk completion from categories to elementary topoi in manageable steps.
Cite as

Kobe Wullaert and Niels van der Weide. The Rezk Completion for Elementary Topoi. In 31st International Conference on Types for Proofs and Programs (TYPES 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 384, pp. 8:1-8:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{wullaert_et_al:LIPIcs.TYPES.2025.8,
  author =	{Wullaert, Kobe and van der Weide, Niels},
  title =	{{The Rezk Completion for Elementary Topoi}},
  booktitle =	{31st International Conference on Types for Proofs and Programs (TYPES 2025)},
  pages =	{8:1--8:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-441-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{384},
  editor =	{Nordvall Forsberg, Fredrik and McKinna, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2025.8},
  URN =		{urn:nbn:de:0030-drops-270260},
  doi =		{10.4230/LIPIcs.TYPES.2025.8},
  annote =	{Keywords: univalent foundations, univalent categories, Rezk completions, UniMath, formalization, elementary topoi}
}
Document
DOI: 10.4230/LIPIcs.ITP.2025.33
Scott’s Representation Theorem and the Univalent Karoubi Envelope

Authors: Arnoud van der Leer, Kobe Wullaert, and Benedikt Ahrens

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

Abstract
Lambek and Scott constructed a correspondence between simply-typed lambda calculi and Cartesian closed categories. Scott’s Representation Theorem is a cousin to this result for untyped lambda calculi. It states that every untyped lambda calculus arises from a reflexive object in some category. We present a formalization of Scott’s Representation Theorem in univalent foundations, in the (Rocq-)UniMath library. Specifically, we implement two proofs of that theorem, one by Scott and one by Hyland. We also explain the role of the Karoubi envelope - a categorical construction - in the proofs and the impact the chosen foundation has on this construction. Finally, we report on some automation we have implemented for the reduction of λ-terms.
Cite as

Arnoud van der Leer, Kobe Wullaert, and Benedikt Ahrens. Scott’s Representation Theorem and the Univalent Karoubi Envelope. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanderleer_et_al:LIPIcs.ITP.2025.33,
  author =	{van der Leer, Arnoud and Wullaert, Kobe and Ahrens, Benedikt},
  title =	{{Scott’s Representation Theorem and the Univalent Karoubi Envelope}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{33:1--33:20},
  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.2025.33},
  URN =		{urn:nbn:de:0030-drops-246318},
  doi =		{10.4230/LIPIcs.ITP.2025.33},
  annote =	{Keywords: Lambda calculi, algebraic theories, categorical semantics, Karoubi envelope, formalization, Rocq-UniMath, univalent foundations}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2024.25
Substitution for Non-Wellfounded Syntax with Binders Through Monoidal Categories

Authors: Ralph Matthes, Kobe Wullaert, and Benedikt Ahrens

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract
We describe a generic construction of non-wellfounded syntax involving variable binding and its monadic substitution operation. Our construction of the syntax and its substitution takes place in category theory, notably by using monoidal categories and strong functors between them. A language is specified by a multi-sorted binding signature, say Σ. First, we provide sufficient criteria for Σ to generate a language of possibly infinite terms, through ω-continuity. Second, we construct a monadic substitution operation for the language generated by Σ. A cornerstone in this construction is a mild generalization of the notion of heterogeneous substitution systems developed by Matthes and Uustalu; such a system encapsulates the necessary corecursion scheme for implementing substitution. The results are formalized in the Coq proof assistant, through the UniMath library of univalent mathematics.
Cite as

Ralph Matthes, Kobe Wullaert, and Benedikt Ahrens. Substitution for Non-Wellfounded Syntax with Binders Through Monoidal Categories. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{matthes_et_al:LIPIcs.FSCD.2024.25,
  author =	{Matthes, Ralph and Wullaert, Kobe and Ahrens, Benedikt},
  title =	{{Substitution for Non-Wellfounded Syntax with Binders Through Monoidal Categories}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{25:1--25:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.25},
  URN =		{urn:nbn:de:0030-drops-203540},
  doi =		{10.4230/LIPIcs.FSCD.2024.25},
  annote =	{Keywords: Non-wellfounded syntax, Substitution, Monoidal categories, Actegories, Tensorial strength, Proof assistant Coq, UniMath library}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2022.15
Univalent Monoidal Categories

Authors: Kobe Wullaert, Ralph Matthes, and Benedikt Ahrens

Published in: LIPIcs, Volume 269, 28th International Conference on Types for Proofs and Programs (TYPES 2022)

Abstract
Univalent categories constitute a well-behaved and useful notion of category in univalent foundations. The notion of univalence has subsequently been generalized to bicategories and other structures in (higher) category theory. Here, we zoom in on monoidal categories and study them in a univalent setting. Specifically, we show that the bicategory of univalent monoidal categories is univalent. Furthermore, we construct a Rezk completion for monoidal categories: we show how any monoidal category is weakly equivalent to a univalent monoidal category, universally. We have fully formalized these results in UniMath, a library of univalent mathematics in the Coq proof assistant.
Cite as

Kobe Wullaert, Ralph Matthes, and Benedikt Ahrens. Univalent Monoidal Categories. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 15:1-15:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{wullaert_et_al:LIPIcs.TYPES.2022.15,
  author =	{Wullaert, Kobe and Matthes, Ralph and Ahrens, Benedikt},
  title =	{{Univalent Monoidal Categories}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{15:1--15:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-285-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{269},
  editor =	{Kesner, Delia and P\'{e}drot, Pierre-Marie},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2022.15},
  URN =		{urn:nbn:de:0030-drops-184580},
  doi =		{10.4230/LIPIcs.TYPES.2022.15},
  annote =	{Keywords: Univalence, Monoidal categories, Rezk completion, Displayed (bi)categories, Proof assistant Coq, UniMath library}
}
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