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Model structure on the universe of all types in interval type theory

Published online by Cambridge University Press:  14 October 2020

Simon Boulier  and
Nicolas Tabareau Open the ORCID record for Nicolas Tabareau [Opens in a new window]
Simon Boulier
Affiliation:
Inria, France
Nicolas Tabareau*
Affiliation:
Inria, France
*
*Corresponding author. Email: nicolas.tabareau@inria.fr
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Abstract

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Model categories constitute the major context for doing homotopy theory. More recently, homotopy type theory (HoTT) has been introduced as a context for doing syntactic homotopy theory. In this paper, we show that a slight generalization of HoTT, called interval type theory (⫿TT), allows to define a model structure on the universe of all types, which, through the model interpretation, corresponds to defining a model structure on the category of cubical sets. This work generalizes previous works of Gambino, Garner, and Lumsdaine from the universe of fibrant types to the universe of all types. Our definition of ⫿TT comes from the work of Orton and Pitts to define a syntactic approximation of the internal language of the category of cubical sets. In this paper, we extend the work of Orton and Pitts by introducing the notion of degenerate fibrancy, which allows to define a fibrant replacement, at the heart of the model structure on the universe of all types. All our definitions and propositions have been formalized using the Coq proof assistant.

Keywords

Homotopy type theorymodel structurefibrant typesinterval type theory
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 4 , April 2021 , pp. 392 - 423
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Footnotes

†

This work has been supported by the CoqHoTT ERC Grant 637339.

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