On a Model Invariance Problem in Homotopy Type Theory

Anthony Bordg

doi:10.1007/s10485-019-09558-w

Abstract

In this article, the author endows the functor category [mathbf {B}(mathbb {Z}_2),mathbf {Gpd}] with the structure of a type-theoretic fibration category with a univalent universe, using the so-called injective model structure. This gives a new model of Martin-Löf type theory with dependent sums, dependent products, identity types and a univalent universe. This model, together with the model (developed by the author in another work) in the same underlying category and with the same universe, which turns out to be provably not univalent with respect to projective fibrations, provide an example of two Quillen equivalent model categories that host different models of type theory. Thus, we provide a counterexample to the model invariance problem formulated by Michael Shulman.

Highlights

This article is a contribution to the ongoing effort to find models of the Univalent Foundations [3] introduced by Vladimir Voevodsky
Shulman [6–8] with his notion of typetheoretic fibration categories prompted the development of models of the Univalence Axiom in functor categories
[6] Shulman endowed the category [D, sSet], where D is any elegant Reedy category and sSet is the category of simplicial sets, with the structure of a type-theoretic fibration category with a univalent universe, again by using the Reedy

Summary

Introduction

This article is a contribution to the ongoing effort to find models of the Univalent Foundations [3] introduced by Vladimir Voevodsky. In [6] Shulman endowed the category [D, sSet], where D is any elegant Reedy category and sSet is the category of simplicial sets, with the structure of a type-theoretic fibration category with a univalent universe, again by using the Reedy Before Shulman’s work in [8] the author had worked out in his PhD thesis [1, Chapter 5] the details of a type-theoretic fibration category with a univalent universe using the injective model structure on [B(Z2), Gpd], overcoming the technical challenge of the presence of a non-trivial automorphism in the index category at least in the simple but important case of the target category Gpd with its univalent universe of sets (discrete groupoids). The details depend on the chosen type theory.” In the present work, we consider a type theory with a unit type, dependent sums, dependent products, intensional identity types and a universe type, and by an interpretation of that type theory we mean precisely Shulman’s notion of a type-theoretic fibration category with a universe which is intended to give an interpretation of such a type theory

The Injective Type-Theoretic Fibration Structure on GpdZ2

The Injective Model Structure on GpdZ2 Made Explicit

Conclusion