  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.
