Three responses to "query re cubical sets"........DMD
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From: Andy Tonks <tonks@manwe.mat.uab.es>
Subject: Re: query re cubical sets
Date: Sun, 23 Jul 100 20:30:42 +0200 (METDST)

> Is a theory of "cubical" sets written somewhere ? For instance, one would
> like to find in this context a theory of products, the analogs of
> Eilenberg-Zilber's theorem, Moore's theorem, etc...

I don't think that the complete theory of cubical homotopy theory has
even been written up, although my impression is that there is no
real reason why not.

The following short note might be of interest:

   Cubical Groups which are Kan, Jour. Pure and Appl. Algebra 47 (1991)

Here the main result is that if a cubical set is in fact a cubical
group then the extension condition ("any box can be filled") is automatic.
(As one has in simplicial homotopy theory).

The only unexpected part is the definition of cubical sets themselves:
it is not enough to consider only the degeneracies corresponding to
the projections (x_1,x_2,x_3,x_4) -> (x_1,x_3,x_4) etc, one must also
consider the so-called "connection" degeneracies [Brown-Higgins]
corresponding to folding or to the minimum function on [0,1]^2:

       (x_1,x_2,x_3,x_4) -> (x_1,min(x_2,x_3),x_4), etc.

   With the connection degeneracies included it can also be shown that
a cubical abelian group can be reconstructed from the "cubical Moore
complex" (the elements with all faces but the "top" one trivial). In
fact one can just as easily give the more general non-abelian version
(cf [Conduché], [Cegarra--Carrasco] simplicially):

   Thm: Each group G_n of a cubical group with connections G has a natural
decomposition as an iterated semidirect product of groups (NG)_k
in the Moore complex for k<=n.

This is not published anywhere - for the details I have a 2-page preprint
from March 1994 I can put on my web page, http://mat.uab.es/tonks.

  For products and the Eilenberg-Zilber theorem there is a "nonresult"
that the classical cubical set product of two intervals, for example,
fails to be contractible. It seems that this problem also disappears
when the connection degeneracies are included. My write-up of this
from summer 1998 is incomplete but I will put it on my web page too
for anyone who is interested.

Yours,

Andy Tonks,
Universitat Autònoma de Barcelona
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Date: Sun, 23 Jul 2000 23:35:16 -0500 (CDT)
From: Brayton Gray <brayton@math.uic.edu>
Subject: Re: query re cubical sets

This is in some of the early papers by Kan. I think parts 1 and 2 of a 4
part series.

Brayton
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Date: Mon, 24 Jul 2000 10:46:54 +0100
From: Ronnie Brown <r.brown@bangor.ac.uk>
Subject: Re: query re cubical sets

There were notes on algebraic topology of Federer from Brown University many
years ago. Massey's book on singular theory uses a cubical approach.

See also

31. (with P.J. HIGGINS), ``On the algebra of cubes'', {\em J. Pure Appl.
Algebra} 21 (1981) 233-260.
32. (with P.J. HIGGINS), ``Colimit theorems for relative homotopy groups'',
{\em J. Pure Appl. Algebra}     22  (1981) 11-41.

Note that within this context and this length and assuming mainly some
geometric facts on cubes we get  as far as a Van Kampen Theorem for relative
homotopy groups which implies the relative Hurewicz Theorem (in the form of
of a result on \pi_n(X \cup CA, X)). The important point is that we need for
this purpose to use an additional structure of `connections' which give
additional degeneracies coming from the monoid structures  max and min on the
unit interval. (the above papers use only max.)

See also:

A. Tonks, Cubical groups which are Kan. Journal of Pure and Applied Algebra
81 (1992) pp.83-87

     The Kan extension condition is shown to be automatic for cubical groups
which have extra connection
     degeneracies; the cubical theory is then parallel to the simplicial.

For a recent reference you might like to look at

AL-AGL, A.A., BROWN, R. & STEINER, R. (00.11) :

Multiple categories: the equivalence of a globular and a cubical approach

Abstract: We show the equivalence of two kinds of strict multiple category,
namely:   the well known globular $\omega$-categories, the cubical
$\omega$-categories with connections.

xxx archive: math.CT/0007009

What has not been worked out, as far as I am aware, is the realisation
theory: it seems likely (possible?) that the geometric realisation of cubical
sets with connection behaves well up to homotopy for the cartesian product.

Hope that helps.

Ronnie Brown