The equivariant model structure on cartesian cubical sets
This work provides a new constructive model for homotopy type theory, which is foundational for the field of type theory and its applications in computer-assisted proofs.
The authors developed a constructive model of homotopy type theory in a Quillen model category that classically presents the usual homotopy theory of spaces, using presheaves over the cartesian cube category with an additional equivariance condition. The main technical results have been formalized in a computer proof assistant.
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.