Cubical sets and the topological topos

[Bas Spitters]

I have just put a note on my homepage

Cubical sets and the topological topos

http://www.cs.au.dk/~spitters/cubtop.pdf

Comments, questions and suggestions are welcome.

Bas

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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.