Univalent Enriched Categories and the Enriched Rezk Completion

Author Niels van der Weide



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Niels van der Weide
  • Institute for Computing and Information Sciences, Radboud University, Nijmegen, The Netherlands

Acknowledgements

The author thanks the anonymous reviewers of HoTT/UF and FSCD for their useful comments, Nima Rasekh for useful discussions, and Dan Frumin and Kobe Wullaert for proof reading earlier versions of this paper. The author also thanks the Coq developers for providing the Coq proof assistant and their continuous support to keep https://github.com/UniMath/UniMath compatible with Coq.

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Niels van der Weide. Univalent Enriched Categories and the Enriched Rezk Completion. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.FSCD.2024.4

Abstract

Enriched categories are categories whose sets of morphisms are enriched with extra structure. Such categories play a prominent role in the study of higher categories, homotopy theory, and the semantics of programming languages. In this paper, we study univalent enriched categories. We prove that all essentially surjective and fully faithful functors between univalent enriched categories are equivalences, and we show that every enriched category admits a Rezk completion. Finally, we use the Rezk completion for enriched categories to construct univalent enriched Kleisli categories.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Theory of computation → Type theory
Keywords
  • enriched categories
  • univalent categories
  • homotopy type theory
  • univalent foundations
  • Rezk completion

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