Univalent Monoidal Categories

Authors Kobe Wullaert , Ralph Matthes , Benedikt Ahrens

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Author Details

Kobe Wullaert
  • Delft University of Technology, The Netherlands
Ralph Matthes
  • IRIT, Université de Toulouse, CNRS, Toulouse INP, UT3, France
Benedikt Ahrens
  • Delft University of Technology, The Netherlands
  • University of Birmingham, UK


We gratefully acknowledge the work by the Coq development team in providing the Coq proof assistant and surrounding infrastructure, as well as their support in keeping UniMath compatible with Coq. Furthermore, we thank Niels van der Weide for helpful discussions on the subject matter and for reviewing the formalization. We also thank Vikraman Choudhury for a question regarding the connection of the monoidal Rezk completion to the Day convolution product. Furthermore, we thank the anonymous referees for their helpful feedback; besides triggering further thoughts on the technical aspects, it helped us to improve the presentation.

Cite AsGet BibTex

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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • Theory of computation → Logic and verification
  • Univalence
  • Monoidal categories
  • Rezk completion
  • Displayed (bi)categories
  • Proof assistant Coq
  • UniMath library


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