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A Model of Type Theory in Cubical Sets

Authors Marc Bezem, Thierry Coquand, Simon Huber

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Marc Bezem
Thierry Coquand
Simon Huber

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Marc Bezem, Thierry Coquand, and Simon Huber. A Model of Type Theory in Cubical Sets. In 19th International Conference on Types for Proofs and Programs (TYPES 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 26, pp. 107-128, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014) https://doi.org/10.4230/LIPIcs.TYPES.2013.107

Abstract

We present a model of type theory with dependent product, sum, and
identity, in cubical sets. We describe a universe and explain how
to transform an equivalence between two types into an equality.  We
also explain how to model propositional truncation and the circle.
While not expressed internally in type theory, the model is expressed in a constructive metalogic. Thus it is a step towards a computational interpretation of Voevodsky's Univalence Axiom.

Subject Classification

Keywords
  • Models of dependent type theory
  • cubical sets
  • Univalent Foundations

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