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Formalisation of Dependent Type Theory: The Example of CaTT

Author Thibaut Benjamin

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Thibaut Benjamin
  • Université Paris-Saclay, CEA, List, F-91120, Palaiseau, France


I want to thank Samuel Mimram and Eric Finster for their guidance in this project and the anonymous reviewers for their helpful comments.

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Thibaut Benjamin. Formalisation of Dependent Type Theory: The Example of CaTT. In 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 239, pp. 2:1-2:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We present the type theory CaTT, originally introduced by Finster and Mimram to describe globular weak ω-categories, formalise this theory in the language of homotopy type theory and discuss connections with the open problem internalising higher structures. Most of the studies about this type theory assume that it is well-formed and satisfy the usual syntactic properties that dependent type theories enjoy, without being completely clear and thorough about what these properties are exactly. We use our formalisation to list and formally prove all of these meta-properties, thus filling a gap in the foundational aspect. We discuss the aspects of the formalisation inherent to CaTT. We present the formalisation in a way that not only handles the type theory CaTT but also related type theories that share the same structure, and in particular we show that this formalisation provides a proper ground to the study of the theory MCaTT which describes the globular monoidal weak ω-categories. The article is accompanied by a development in the proof assistant Agda to check the formalisation that we present.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • Theory of computation → Categorical semantics
  • Dependent type theory
  • homotopy type theory
  • higher categories
  • formalisation
  • Agda
  • proof assistant


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