[Paper Review] Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom

Cohen, Cyril, Coquand, Thierry+2 more|arXiv (Cornell University)|May 31, 2014

Black Holes and Theoretical Physics8 references21 citations

TL;DR

This paper constructs a univalent universe in type-theoretic model categories based on elegant Reedy categories, such as cubical sets and simplicial sets, using two methods: a reduction to Voevodsky’s simplicial construction and a direct generalization of Voevodsky’s proof via minimal fibrations. The key contribution is proving that intensional type theory with the univalence axiom can be strictly interpreted in the homotopy theory of cubical sets, providing a constructive and explicit model for univalent foundations.

ABSTRACT

This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.

Motivation & Objective

• To establish the existence of univalent universes in model categories of presheaves over elegant Reedy categories, such as cubical sets and simplicial sets.
• To provide a constructive and explicit model of intensional type theory with the univalence axiom in the homotopy theory of cubical sets.
• To generalize Voevodsky’s proof of univalence to broader classes of model categories, including those based on Eilenberg-Zilber Reedy categories.
• To resolve coherence issues in interpreting type theory in model categories by leveraging strong properness and fibrancy conditions.

Proposed method

• Reduces the construction of a univalent universe in a presheaf category over an elegant Reedy category to the known case of simplicial sets via a left Quillen equivalence.
• Uses Shulman’s result on univalent universes in simplicial presheaves and applies a homotopy-theoretic lifting argument to transfer the universe to the locally constant model structure.
• Applies the theory of minimal fibrations in presheaves over Eilenberg-Zilber Reedy categories to generalize Voevodsky’s proof of univalence.
• Establishes strong properness of the Grothendieck model structure on presheaves over elegant local test categories, enabling fibrancy and lifting properties essential for univalence.
• Uses the fact that representable presheaves are cofibrant and that fibrations over them satisfy the right lifting property to verify univalence.
• Employs the nerve functor and category of elements to relate weak homotopy equivalences in the model structure to the homotopy theory of ∞-groupoids.

Experimental results

Research questions

• RQ1 Can univalent universes be constructed in model categories of cubical sets, such as cubical sets with or without connections?
• RQ2 Does Voevodsky’s proof of univalence extend to presheaf categories over Eilenberg-Zilber Reedy categories?
• RQ3 Is the Grothendieck model structure on presheaves over an elegant local test category strongly proper, enabling fibrancy and lifting properties?
• RQ4 Can a univalent universe in simplicial presheaves be strictly pulled back to a univalent universe in the underlying presheaf category?
• RQ5 Are there natural generating sets of trivial cofibrations with representable codomains in general elegant local test categories?

Key findings

• A univalent universe exists in the category of simplicial presheaves over an elegant Reedy category, classifying fibrations with κ-small fibers.
• The Grothendieck model structure on presheaves over an elegant local test category is strongly proper, ensuring fibrancy and lifting under right properness.
• The univalent universe constructed in simplicial presheaves induces a univalent fibration in the underlying presheaf category, preserving univalence in the strict sense.
• The proof generalizes Shulman’s result on univalent universes in simplicial presheaves to a broader class of categories, including cubical sets.
• The construction is explicit and constructive, avoiding reliance on higher category theory or abstract ∞-topos theory.
• The method provides a new proof of univalence in simplicial presheaves by verifying fibrancy via minimal model structures and localizing by representable maps.