This is about two points of a recent message of Dmitry Roytenberg, forwarded by Urs Schreiber.
I have not been able to find an abstract description of any of the cubical sites, in the spirit of the simplex category being the category of non-empty finite ordinals
I do not know of any such description. But there is a nice abstract description of the site of cubical sets with connections, parallel to a well-known characterisation of the simplicial site: - the free strict monoidal category with an assigned dioid. See [GM], Thm. 5.2. (There are analogous results for the other cubical sites.) A `dioid' is a set with two monoid operations, where the unit of each operation is absorbant for the other. Typically, an abstract interval has such a structure, and a cylinder functor has the structure of a 'diad'. See [Gr]. (I was also using the terms 'cubical monoid' and 'cubical monad', for an obvious analogy; later I abandoned them because they could be misleading - obviously again). Every lattice is an idempotent dioid, but idempotency is - apparently - of no interest in homotopy. This leads us to the second point: smooth homotopy.
In any case, using these connections in a differential-geometric context is problematic, not (just) because of a clash with established terminology, but because the max and min maps are only piecewise smooth.
For smooth homotopy one should use a different (non-idempotent) dioid, still commutative and involutive: NOT the standard interval with min, max, linked by the involution t' = 1 - t, BUT the standard interval with multiplication and *, linked by the same involution: x*y = (x'.y')' = x + y - xy. See [Gr]. [Gr] M. Grandis, Cubical monads and their symmetries, in: Proc. of the Eleventh Intern. Conf. on Topology, Trieste 1993, Rend. Ist. Mat. Univ. Trieste 25 (1993), 223-262. http://www.dmi.units.it/~rimut/volumi/25/index.html [GM] M. Grandis - L. Mauri, Cubical sets and their site, Theory Appl. Categ. 11 (2003), No. 8, 185-211. Best regards Marco Grandis [For admin and other information see: http://www.mta.ca/~cat-dist/ ]