The equivariant model structure on cartesian cubical sets

TL;DR

This paper constructs a homotopy type theory model using presheaves over the cartesian cube category, introducing an equivariance condition in cubical Kan fibrations, with formalized proofs.

Contribution

It introduces an equivariance condition in cubical Kan fibrations within a presheaf model, advancing the formalization of homotopy type theory.

Findings

Developed a constructive model of homotopy type theory in a Quillen model category.

Formalized the main technical results in a computer proof assistant.

Established a well-behaved model based on presheaves over the cartesian cube category.

Abstract

We develop a constructive model of homotopy type theory in a Quillen model category that classically presents the usual homotopy theory of spaces. Our model is based on presheaves over the cartesian cube category, a well-behaved Eilenberg-Zilber category. The key innovation is an additional equivariance condition in the specification of the cubical Kan fibrations, which can be described as the pullback of an interval-based class of uniform fibrations in the category of symmetric sequences of cubical sets. The main technical results in the development of our model have been formalized in a computer proof assistant.