On a Model Invariance Problem in Homotopy Type Theory

  • Anthony BordgEmail author
Open Access


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.


Univalent Foundations Homotopy Type Theory Univalence Axiom Type-theoretic fibration category Quillen model category Injective model structure Groupoid Groupoid model Universe Model invariance problem 



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© The Author(s) 2019

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Authors and Affiliations

  1. 1.Department of Computer Science and TechnologyUniversity of CambridgeCambridgeUK

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