Cubical sets and the topological topos
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.
 Publication:

arXiv eprints
 Pub Date:
 October 2016
 arXiv:
 arXiv:1610.05270
 Bibcode:
 2016arXiv161005270S
 Keywords:

 Computer Science  Logic in Computer Science;
 Mathematics  Category Theory;
 Mathematics  Logic;
 03B70 (logic in computer science);
 03B15 (higherorder logic and type theory);
 55U35 (abstract and axiomatic homotopy theory);
 F.4.1