Coherence for Monoidal Groupoids in HoTT

Author Stefano Piceghello

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Stefano Piceghello
  • Department of Informatics and Department of Mathematics, University of Bergen, Norway


The author wishes to thank Kristian S. Alfsvåg, Marc Bezem, Floris van Doorn, Bjœrn Ian Dundas, Peter Dybjer, Favonia and Håkon R. Gylterud for providing valuable insight on the topic and the anonymous reviewers for constructive comments, and acknowledges the support of the Centre for Advanced Study (CAS) in Oslo, Norway, which funded and hosted the research project Homotopy Type Theory and Univalent Foundations during the 2018/19 academic year.

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Stefano Piceghello. Coherence for Monoidal Groupoids in HoTT. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We present a proof of coherence for monoidal groupoids in homotopy type theory. An important role in the formulation and in the proof of coherence is played by groupoids with a free monoidal structure; these can be represented by 1-truncated higher inductive types, with constructors freely generating their defining objects, natural isomorphisms and commutative diagrams. All results included in this paper have been formalised in the proof assistant Coq.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • Theory of computation → Constructive mathematics
  • homotopy type theory
  • coherence
  • monoidal categories
  • groupoids
  • higher inductive types
  • formalisation
  • Coq


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