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

Simon Boulier,Nicolas Tabareau

doi:10.1017/s0960129520000213

Abstract

AbstractModel 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.

Highlights

Homotopy type theory (HoTT) can be seen both as a language to formalize mathematics and as a language to do synthetic homotopy theory
Synthetic homotopy theory consists in proving homotopy properties in a syntactic language, here a type theory, which can be interpreted in several models of homotopy theory
A type is interpreted by a Kan simplicial set1 and univalent equality is interpreted by paths in those topological spaces

Summary

Introduction

Homotopy type theory (HoTT) can be seen both as a language to formalize mathematics and as a language to do synthetic homotopy theory. The first model of HoTT has been proposed by Voevodsky using a standard notion in homotopy theory – simplicial sets This model has later been reworked and polished by Kapulkin and Lumsdaine (2012). As Martin-Löf type theory can be interpreted in any presheaf category (and even any topos), they identified nine axioms which are valid through the interpretation in the CCHM model and which are enough to carry several constructions of the model (basically, all but the universe of fibrant types) and to define cubical type theory internally. Model categories are of great importance to compare those different settings in which to formalize homotopy theory, using the notion of Quillen equivalences It has already been shown in HoTT that there is a pre-model structure on the universe of fibrant types. We give a direct link to the corresponding file in the repository

Interval Type Theory

Models of ITT

Degenerate fibrancy

Model Category

Pushout

12. Conclusion and Future Work