Simplicial versus (cubical with connections)

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

Dear categorists,

I would like to comment on Ronnie Brown's message, copied below,
insisting on a parallelism that is not often acknowledged, and may  
'clarify'
- for instance - why simplicial groups somehow behave as
'cubical groups with connections' (see Tonks' paper cited by RB),
rather than as 'ordinary cubical groups'.

     The degeneracies of a simplicial object correspond to the  
connections
     (or higher degeneracies) of a cubical one, introduced by Brown  
and Higgins,
     more than to the ordinary degeneracies.

Formally, this fact can be motivated as follows.

Let us start from the cylinder endofunctor  I(X) = X x [0, 1]  of  
topological spaces.
Its main structure consists of natural transformations of powers of  
I, derived from
(part of) the lattice structure of [0, 1]:

- two faces  1 --> I,   sending x to (x, 0) OR (x, 1),
- a degeneracy  I --> 1,    sending (x, t) to x,
- two connections  I^2 --> I,    sending (x, t, t') to (x, max(t,  
t')) OR (x, min(t, t')).

Then we collapse the higher face of I (for instance), and we get a  
cone functor C, with
a monad structure:

- the lower face of I gives the unit  1 --> C,
- the lower connection gives the multiplication C^2 --> C,
- the other transformations (including the degeneracy of I) induce  
nothing.

Now the cylinder I, with the above structure (which i [myself, not  
the cylinder] call a 'diad'),
operating on any space, gives a cocubical object with connections,
while the monad C gives an augmented cosimplicial object.

[[ Addendum.
If one wants to take on the parallelism to the singular cubical/ 
simplicial set of a space X,
the construction becomes more involved. One should start from:

- the cocubical space I* (with connections) of all standard cubes,  
produced by the cylinder I
    on the singleton space;

- the augmented cosimplicial space Delta* produced by C on the empty  
space 0
    (taking care that C(0), defined as a pushout, is the singleton,  
and C^n(0) is the
    standard simplex of dimension n-1).

Then one applies to these structures the contravariant functor Top(-,  
X) and gets the
singular cubical set of X (with connections) OR the singular  
simplicial set of X (augmented).
]]

With best regards

Marco Grandis


On 12 Sep 2011, at 11:35, Ronnie Brown wrote:

> The reference is included in this review *MR1173825 *of the cubical  
> case.
>
> Tonks, A. P. <http://0-ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/ 
> mathscinet/search/publications.html?pg1=IID&s1=325533>(4-NWAL)  
> <http://0-ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/mathscinet// 
> search/institution.html?code=4_NWAL>
> Cubical groups which are Kan.
> /J. Pure Appl. Algebra/ <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/journaldoc.html? 
> cn=J_Pure_Appl_Algebra> 81 <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html? 
> pg1=ISSI&s1=118323>(1992), <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html? 
> pg1=ISSI&s1=118323>no. 1, <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html? 
> pg1=ISSI&s1=118323> 83–87. <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/mscdoc.html? 
> code=55U10,%2818D35,18G30%29><javascript:openWin('http:// 
> unicat.bangor.ac.uk:4550/resserv', 'AMS:MathSciNet', 'atitle=Cubical 
> %20groups%20which%20are% 
> 20Kan&aufirst=A.&auinit=AP&auinit1=A&auinitm=P&aulast=Tonks&coden=JPAA 
> A2&date=1992&epage=87&genre=article&issn=0022-4049&issue=1&pages=83-87 
> &spage=83&stitle=J.%20Pure%20Appl.%20Algebra&title=Journal%20of% 
> 20Pure%20and%20Applied%20Algebra&volume=81')>
>
> The author shows that group objects in the category of cubical sets  
> with connections [R. Brown and P. J. Higgins, J. Pure Appl. Algebra  
> 21 (1981), no. 3, 233--260; MR0617135 (82m:55015a) <http://0- 
> ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/ 
> publdoc.html?r=1&pg1=CNO&s1=617135&loc=fromrevtext>] satisfy the  
> Kan extension condition. This is a very nice correspondence with  
> the simplicial case [J. C. Moore, in Séminaire Henri Cartan de  
> l'Ecole Normale Supérieure, 1954/1955, Exp. No. 18, Secrétariat  
> Math., Paris, 1955; see MR0087934 (19,438e) <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publdoc.html? 
> r=1&pg1=CNO&s1=87934&loc=fromrevtext>].

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