[Paper Review] Cubical sets and the topological topos

Bas Spitters|arXiv (Cornell University)|Oct 17, 2016

Homotopy and Cohomology in Algebraic Topology13 references3 citations

TL;DR

This paper establishes that cubical sets classify flat distributive lattices, providing a geometric realization of cubical sets via the topological topos. It constructs a Moore path model of identity types in the topos of cubical sets and proves the existence of a pre-model structure on fibrant cubical sets using internal factorization systems, advancing the foundational semantics of cubical type theory and univalent foundations.

ABSTRACT

Coquand's cubical set model for homotopy type theory provides the basis for a computational interpretation of the univalence axiom and some higher inductive types, as implemented in the cubical proof assistant. This paper contributes to the understanding of this model. We make three contributions: 1. Johnstone's topological topos was created to present the geometric realization of simplicial sets as a geometric morphism between toposes. Johnstone shows that simplicial sets classify strict linear orders with disjoint endpoints and that (classically) the unit interval is such an order. Here we show that it can also be a target for cubical realization by showing that Coquand's cubical sets classify the geometric theory of flat distributive lattices. As a side result, we obtain a simplicial realization of a cubical set. 2. Using the internal `interval' in the topos of cubical sets, we construct a Moore path model of identity types. 3. We construct a premodel structure internally in the cubical type theory and hence on the fibrant objects in cubical sets.

Motivation & Objective

• To clarify the logical and categorical foundations of Coquand’s cubical set model for homotopy type theory.
• To show that the topos of cubical sets classifies flat distributive lattices, extending Johnstone’s topological topos framework.
• To construct a Moore path model of identity types using the internal interval in the topos of cubical sets.
• To establish a pre-model structure on fibrant cubical sets using two internal factorization systems: Gambino-Garner and a display map system.
• To support the internal type theory of cubical sets with a uniform, functorial factorization system suitable for univalent foundations.

Proposed method

• Uses Stone duality between finite distributive lattices and finite posets to show that cubical sets classify flat distributive lattices.
• Applies Lawvere theory duality to show that the cube category is the Kleisli category of the free distributive lattice monad on finite sets.
• Constructs a cocubical object in the topos of cubical sets from a flat distributive lattice, enabling geometric realization.
• Defines Moore paths using the internal interval object in the topos of cubical sets, with judgmental computation rules for the J eliminator.
• Introduces two internal factorization systems: one via the Gambino-Garner construction through the graph of a function, and another via display maps.
• Establishes a pre-model structure by verifying 3-for-2 closure for weak equivalences and orthogonal factorization for cofibrations and fibrations.

Experimental results

Research questions

• RQ1 How does the topos of cubical sets relate to the classifying topos of flat distributive lattices?
• RQ2 Can a Moore path model of identity types be constructed internally in the topos of cubical sets using the internal interval?
• RQ3 Does a pre-model structure exist on the fibrant objects in the topos of cubical sets using internal type-theoretic constructions?
• RQ4 How do the Gambino-Garner and display map factorization systems interact to form a pre-model structure?
• RQ5 What is the role of flatness and filtered colimits in the classification of cubical sets as models of distributive lattice theory?

Key findings

• The topos of cubical sets classifies flat distributive lattices, generalizing Johnstone’s result for simplicial sets.
• A simplicial realization of a cubical set is obtained as a side result via the duality between distributive lattices and finite posets.
• A Moore path model of identity types is constructed in the internal language of the topos of cubical sets, satisfying judgmental computation rules for the J eliminator.
• A pre-model structure exists on the fibrant cubical sets, with cofibrations defined as having the left lifting property with respect to trivial fibrations.
• The weak equivalences satisfy the 3-for-2 property and are closed under retracts, as required for a pre-model structure.
• The two factorization systems—Gambino-Garner and display maps—combine to yield a pre-model structure, with trivial fibrations defined as display maps that are also equivalences.