The equivariant model structure on cartesian cubical sets
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.
Cite
@article{arxiv.2406.18497,
title = {The equivariant model structure on cartesian cubical sets},
author = {Steve Awodey and Evan Cavallo and Thierry Coquand and Emily Riehl and Christian Sattler},
journal= {arXiv preprint arXiv:2406.18497},
year = {2026}
}
Comments
v2: Corrected a mistake in the treatment of notions of fibred structure. Some numbering has changed in sections 2.1 and 2.3. v3: Final journal version