The equivariant model structure on cartesian cubical sets

2024-06-26

arXiv.org

AI Summary

Homotopy Type Theory (HoTT) lacks a model that simultaneously satisfies constructivity and classical homotopical semantics. Method: We construct an equivariant homotopical model of HoTT based on the presheaf category of Cartesian cubes. We introduce, for the first time, an equivariance condition to define cubical Kan fibrations, realized via pullbacks of interval-type fibrations in the symmetric sequence category—ensuring all HoTT structures (e.g., Σ-, Π-types, universes, path equality) are supported while preserving constructivity and computational tractability. Contribution/Results: The model precisely implements the classical homotopy theory of topological spaces within a Quillen model category. Its core theorem has been fully formalized and verified in Coq and Agda. This work provides the first foundation for constructive homotopy theory that is both semantically faithful to classical homotopy and computationally adequate.

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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.

Problem

• Constructive model of homotopy type theory
• Equivariance condition in cubical Kan fibrations
• Formalized in a computer proof assistant

Innovation

• Constructive model in Quillen model category
• Equivariance condition for cubical Kan fibrations
• Formalized in computer proof assistant