Re: Simplicial versus (cubical with connections)

[Marco Grandis <grandis@dima.unige.it>]

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

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