The equivariant model structure on cartesian cubical sets
Journal article, 2026
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.
Author
Steve Awodey
Carnegie Mellon University (CMU)
Evan Cavallo
Chalmers, Computer Science and Engineering (Chalmers), Computing Science
University of Gothenburg
Thierry Coquand
University of Gothenburg
Chalmers, Computer Science and Engineering (Chalmers), Computing Science
Emily Riehl
Johns Hopkins University
Christian Sattler
University of Gothenburg
Chalmers, Computer Science and Engineering (Chalmers), Computing Science
Advances in Mathematics
0001-8708 (ISSN) 1090-2082 (eISSN)
Vol. 495 110965Subject Categories (SSIF 2025)
Algebra and Logic
DOI
10.1016/j.aim.2026.110965