Subject: categories: Re: question about omega-category

From: Sjoerd Erik CRANS <[email-masked]>
Date: Wed, 20 May 1998 12:46:09 +1000 (EST)

Philippe Gaucher <[email-masked]> wrote:

> Dear categorician (or categorist, I do not know the word in English),
> 
> 
> I posted the following question some days ago in sci.math.research.
> Maybe this list is more appropirated : 
> 
> I would need to understand a proof of the following proposition :  
> 
> There is only one functor up to isomorphism TENS : omega-Cat x
> omega-Cat -> omega-Cat for which C TENS - and - TENS C have right
> adjoint for every omega-category C and which satisfies I^p TENS I^q =
> I^{p+q} where I^p is the omega-category canonically associated to the
> p-cube, (using for example the set of composable sub pasting schemes
> of the pasting scheme associated to the p-cube). 
> 
> I have a paper from Crans ("Pasting schemes for the monoidal biclosed
> structure on omega-Cat") which proves explicitely the proposition. I
> do not understand the construction, which is very technical (*). Is there
> a less complicated way to prove this proposition ? I do not need an
> explicit construction.   
> 
> Any help is welcome.
> 
> pg.
> 
> 
> (*) What is a pasting presentation for example ? I know the definition
> of pasting scheme, realization of pasting schemes, but I do not know the
> one of "pasting presentation". Another question : if f, g are morphisms
> in C and h, k morphisms in D, is there in C TENS D elements corresponding
> to f TENS h and g TENS k, and if the answer is yes, is it true that 
> (f TENS h) o_p (g TENS k) = (f o_p g) TENS (h o_p k) ? I suppose that
> it is false : if it should be true, why the construction of this monoidal
> structure is so complicated ?
> 

A pasting presentation for an omega-category is similar to a presentation
for a group, but taking into account that there are cells of different
dimensions. In particular, generators in dimension n are ``labeled''
n-dimensional pasting schemes with the labeling involving generators
*and relations* in lower dimensions. More details about pasting presentations
can be found in my paper "Pasting presentations for omega-categories", which
is available via my web-page: http://www.mpce.mq.edu.au/~scrans/papers/
or via hypatia: http://hypatia.dcs.qmw.ac.uk/author/C/CransSE/.

The proposition above follows by general categorical methods, using the
adjunction between omega-categories and cubical sets. Apart from my proof
cited above there are earlier proofs by Al-Agl and Steiner [Nerves of
multiple categories, Proc. London Math. Soc. (3) 66 (1993) 92-128] and
by Brown and Higgins [Tensor products and homotopies for omega-groupoids
and crossed complexes] (who only do omega-groupoids but their proof holds
for omega-categories as well).

Yes, there are elements in C TENS D corresponding to f TENS h and g TENS k.
But is not just that (f TENS h) o_p (g TENS k) = (f o_p g) TENS (h o_p k)
is not true: it does not even make sense, because the sources and targets
don't match up.

Sjoerd Crans