[Paper Review] An Equivalent Presentation of the Bezem-Coquand-Huber Category of Cubical Sets

Andrew M. Pitts|arXiv (Cornell University)|Jan 30, 2014

Homotopy and Cohomology in Algebraic Topology3 references6 citations

TL;DR

This paper establishes an equivalence between the category of cubical sets—used in homotopy type theory—and a category of nominal sets equipped with a 01-substitution operation. By rephrasing cubical sets in terms of nominal sets with substitution structure, the author provides a more elegant and computationally tractable framework for constructing univalent models of constructive type theory.

ABSTRACT

Staton has shown that there is an equivalence between the category of presheaves on (the opposite of) finite sets and partial bijections and the category of nominal restriction sets: see [2, Exercise 9.7]. The aim here is to see that this extends to an equivalence between the category of cubical sets introduced in [1] and a category of nominal sets equipped with a "01-substitution" operation. It seems to me that presenting the topos in question equivalently as 01-substitution sets rather than cubical sets will make it easier (and more elegant) to carry out the constructions and calculations needed to build the intended univalent model of intentional constructive type theory.

Motivation & Objective

• To provide an equivalent categorical presentation of the Bezem-Coquand-Huber category of cubical sets.
• To simplify constructions in univalent type theory by replacing cubical sets with a nominal set-based structure featuring 01-substitution operations.
• To generalize the equivalence between presheaves on finite sets with partial bijections and nominal restriction sets to the cubical setting.
• To facilitate the development of univalent models by leveraging the algebraic and nominal-theoretic properties of 01-substitution sets.
• To reformulate Kan-fibrancy and filling operations in terms of name abstraction and substitution, enhancing clarity and computability.

Proposed method

• Introduces the category $\mathbf{01Sub}$ of nominal sets equipped with a 01-substitution operation satisfying freshness, commutativity, and equivariance axioms.
• Defines the 01-substitution operation via morphisms $s: X \times \mathbb{A} \times 2 \to X$ with properties ensuring compatibility with name abstraction and permutation actions.
• Establishes a categorical equivalence between $[\mathbf{C}^{\mathrm{op}}, \mathbf{Set}]$ (cubical sets) and $\mathbf{01Sub}$ using the restriction along the full subcategory $\mathbf{I} \subset \mathbf{C}$ of injective functions.
• Uses name abstraction $[\mathbb{A}]X$ to lift the 01-substitution operation to higher-dimensional cubes, defining $\Box_n X$ as $n$-fold name abstractions with compatible substitution.
• Reformulates uniform-Kan fibrations using split morphisms $i_n: \sqcup_n X \to \Box_{n+1}X$ and $j_n: \sqcap_n X \to \Box_{n+1}X$, ensuring filling operations are uniformly definable.
• Characterizes fibrations via equivariant and substitution-commuting filling operations $\uparrow u$ and $\downarrow u$ for 1-open and 0-open boxes, respectively.

Experimental results

Research questions

• RQ1 Can the category of cubical sets be equivalently presented using nominal sets with 01-substitution operations?
• RQ2 How does the 01-substitution structure simplify the construction of univalent models in type theory?
• RQ3 What is the role of name abstraction and degeneracy in modeling higher-dimensional cubes within the nominal framework?
• RQ4 Can uniform-Kan fibrations be characterized via split morphisms in the $\mathbf{01Sub}$ category?
• RQ5 How do equivariance and substitution compatibility ensure coherence in filling operations for fibrations?

Key findings

• The category of cubical sets is categorically equivalent to the category $\mathbf{01Sub}$ of nominal sets with 01-substitution operations.
• The 01-substitution operation lifts to name abstractions, enabling a recursive construction of $n$-dimensional cubes via $\Box_n X$.
• Uniform-Kan fibrations are characterized by the splitness of morphisms $p_n: \Box_{n+1}X \to \sqcup_n X$ and $q_n: \Box_{n+1}X \to \sqcap_n X$, ensuring uniform filling.
• Filling operations $\uparrow u$ and $\downarrow u$ are equivariant and commute with substitutions for names fresh with respect to the box and its base.
• The equivalence allows replacing cubical sets with $\mathbf{01Sub}$, leading to more elegant and computationally transparent constructions in univalent type theory.
• The nominal-theoretic formulation via name abstraction and substitution provides a cleaner algebraic foundation than the original cubical set model.