[Paper Review] Uniform Fibrations and the Frobenius Condition

Nicola Gambino, Christian Sattler|arXiv (Cornell University)|Oct 2, 2015

Homotopy and Cohomology in Algebraic Topology22 references3 citations

TL;DR

This paper introduces uniform fibrations in categories with a functorial cylinder, proving they form the right class of a weak factorization system and satisfy the Frobenius condition in presheaf categories like simplicial and cubical sets. This ensures pushforwards along uniform fibrations preserve fibrancy, offering a constructive foundation for Voevodsky’s univalent foundations and extending cubical type theory models.

ABSTRACT

We introduce and study the notion of a uniform fibration in categories with a functorial cylinder. In particular, we show that in a wide class of presheaf categories, including simplicial sets and cubical sets with connections, uniform fibrations are the right class of a natural weak factorization system and satisfy the Frobenius condition. This implies that pushforward along a uniform fibration preserves uniform fibrations. When instantiated in simplicial sets, this result gives a constructive counterpart of one of the key facts underpinning Voevodsky's simplicial model of univalent foundations, while in cubical sets it extends some of the existing work on cubical models of type theory by Coquand and others.

Motivation & Objective

• To define and study uniform fibrations in categories equipped with a functorial cylinder.
• To establish that uniform fibrations form the right class of a weak factorization system in simplicial and cubical sets with connections.
• To prove that uniform fibrations satisfy the Frobenius condition, ensuring stability under pushforward.
• To provide a constructive alternative to key facts in Voevodsky’s simplicial model of univalent foundations.
• To extend existing results in cubical type theory by generalizing the behavior of fibrations under pushforward.

Proposed method

• The authors define uniform fibrations using a lifting property relative to a functorial cylinder structure.
• They establish that in presheaf categories with connections—such as simplicial and cubical sets—uniform fibrations form the right class of a weak factorization system.
• The Frobenius condition is proven by analyzing the interaction between pushforwards and lifting properties.
• The construction relies on the functoriality of the cylinder to ensure uniform behavior across morphisms.
• The method applies to both simplicial sets and cubical sets with connections, leveraging their categorical structure.
• The framework is applied to recover and generalize results in type theory and homotopy foundations.

Experimental results

Research questions

• RQ1 How can fibrations be uniformly defined in categories with a functorial cylinder to ensure stability under pushforward?
• RQ2 What conditions ensure that the class of fibrations forms a weak factorization system in presheaf categories?
• RQ3 Does the Frobenius condition hold for fibrations in simplicial and cubical sets with connections?
• RQ4 Can uniform fibrations provide a constructive foundation for Voevodsky’s univalent foundations in simplicial sets?
• RQ5 How do uniform fibrations extend or generalize existing results in cubical type theory models?

Key findings

• Uniform fibrations form the right class of a weak factorization system in simplicial sets and cubical sets with connections.
• The Frobenius condition holds for uniform fibrations, ensuring that pushforwards preserve fibrancy.
• The construction provides a constructive counterpart to a key fact in Voevodsky’s simplicial model of univalent foundations.
• The framework extends existing work on cubical models of type theory by Coquand and others.
• The results apply uniformly across a wide class of presheaf categories with functorial cylinders.
• The method ensures that fibrations remain well-behaved under categorical operations like pushforward.