Internal Universes in Models of Homotopy Type Theory

Authors Daniel R. Licata , Ian Orton , Andrew M. Pitts , Bas Spitters

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Daniel R. Licata
  • Wesleyan University Dept. Mathematics & Computer Science, Middletown, USA
Ian Orton
  • University of Cambridge Dept. Computer Science & Technology, Cambridge, UK
Andrew M. Pitts
  • University of Cambridge Dept. Computer Science & Technology, Cambridge, UK
Bas Spitters
  • Aarhus University Dept. Computer Science, Aarhus, DK

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Daniel R. Licata, Ian Orton, Andrew M. Pitts, and Bas Spitters. Internal Universes in Models of Homotopy Type Theory. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-Mörtberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • cubical sets
  • dependent type theory
  • homotopy type theory
  • internal language
  • modalities
  • univalent foundations
  • universes


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