On a model invariance problem in Homotopy Type Theory

TL;DR

This paper constructs a new model of Martin-Löf type theory using functor categories and injective model structures, demonstrating a counterexample to the model invariance problem in Homotopy Type Theory.

Contribution

It introduces a novel model of type theory with a univalent universe in a functor category, showing that Quillen equivalent categories can host different models of type theory.

Findings

Constructed a type-theoretic fibration category with a univalent universe.

Provided a counterexample to the model invariance problem.

Showed that different model categories can host non-invariant models of type theory.

Abstract

In this article the author endows the functor category [B(Z2),Gpd] with the structure of a type-theoretic fibration category with a univalent universe using the so-called injective model structure. It gives us a new model of Martin-L\"of type theory with dependent sums, dependent products, identity types and a univalent universe. This model, together with the model (developed by the author in an other work) in the same underlying category together with the very same universe that turned out to be provably not univalent with respect to projective fibrations, provides 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.