Re: Simplicial versus (cubical with connections)

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

The result of Maltsiniotis referred to by Jonathan is very welcome.  But 
I wonder if there is still a problem with cubical sets with connection:

the geometric realisation of a simplicial group is, in a convenient 
category, a topological group, because of the homeomorphism

f: |K \times Y| \to |K| \times |Y| .

However in the case of cubical sets with connections this map f is a 
homotopy equivalence but it seems is not a homeomorphism (?).  As 
Grothendieck wrote: `homotopically speaking' that is not a problem!

For homotopies and higher homotopies cubes are nice and easy because of 
the basic formula

I^m \times I^n = I^{m+n}.

This leads to monoidal closed structures on strict cubical higher 
categories and groupoids.

For a basic discussion of other issues such as algebraic inverses to 
subdivision and commutative cubes  I refer to my 2009  Liverpool seminar 
on`What is and what should be `Higher dimensional group theory'?'

http://pages.bangor.ac.uk/~mas010/pdffiles/liverpool-beamer-handout.pdf

Ronnie




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