2016/10/17 by Bas Spitters, Spitters, Bas
Mathematics · #03B15 (higher-order logic and type theory) #03B70 (logic in computer science) #55U35 (abstract and axiomatic homotopy theory) #Algebraic Geometry and Number Theory #Category Theory (math.CT) #F.4.1 #FOS: Computer and information sciences #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Logic in Computer Science (cs.LO)
paper · pdf · doi:10.48550/arxiv.1610.05270
openalex publication_date 2016/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.