Re: Simplicial versus (cubical with connections)

[Urs Schreiber <urs.schreiber@googlemail.com>]

Dear category theorists,

a while back we had a discussion here about model structures on
cubical sets. My colleague Dmitry Roytenberg sent the following
message to the list, which, for some reason, did not seem to have gone
through.

Since it might be of interest to some members of the list I would like
to repost it hereby.


---------- Forwarded message ----------
From: Dmitry Roytenberg <starrgazerr@gmail.com>
Date: Thu, Sep 22, 2011 at 12:18 PM
Subject: Re: categories: Re: Simplicial versus (cubical with connections)
To: categories <categories@mta.ca>


Dear colleagues,

Thanks to everyone who has replied, either privately or on the list.
Having read up on the subject a bit I've discovered that quite a lot
is known by now about the homotopy theory of cubical sets, thanks
mainly to the work of Cisinski and Jardine (Jardine's "Categorical
homotopy theory" contains a good exposition of Cisinski's methods,
especially useful for non-French speakers). The spatial model
structure mentioned by Urs has inclusions as cofibrations and cubical
Kan fibrations as fibrations; it is proper, combinatorial and monoidal
with respect to the tensor product described by Ronnie, with weak
equivalences stable under filtered colimits. There is a monoidal left
Quillen equivalence from cubical to simplcial sets mapping the
n-dimensional cube to the n-th power of the 1-simplex (the simplicial
realization). This is enough to conclude that cubical sets form an
excellent model category, in view of Lemma A.3.2.20 and Remark
A.3.2.21 in HTT. It then follows from (HTT, Theorem A.3.2.24) that a
cubical category is fibrant iff all its function complexes are Kan.

As for presheaves on other cubical sites (i.e. with more than just the
classical faces and degeneracies), Isaacson in his thesis

http://www.ma.utexas.edu/users/isaacson/PDFs/diss.pdf

constructs a cubical site containing Brown-Higgins connections (there
called conjunctions and disjunctions, as in logic) as well as
symmetries of hypercubes, closely related but different from Grandis
and Mauri's site !K. Isaacson constructs a _symmetric_ monoidal models
tructure on the resulting category of symmetric cubical sets, and
equips it with a monoidla left Quillen equivalence from the ordinary
cubical sets. However, this model category is not excellent, as  not
all monomorphisms are cofibrations.

As far as I can tell, it is not known if there is an excellent model
structure on cubical sets with connections (but without symmetries).
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.

Finally, to my astonishment 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. Clearly the
cubes are to be viewed as finite power sets, but which structure on
the power sets is preserved by the morphisms in each case?

Best,

Dmitry

On Fri, Sep 16, 2011 at 3:24 PM, Fernando Muro <fmuro@us.es> wrote:
>
> For a published reference see:
>
> MR2591923 (2010k:18022)
> Maltsiniotis, Georges(F-PARIS7-IMJ)
> La catégorie cubique avec connexions est une catégorie test stricte. (French. English summary) [The category of cubes with connections is a strict test category]
> Homology, Homotopy Appl. 11 (2009), no. 2, 309–326.
>
> Best,
>
> Fernando
>

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