Subject: categories: question about omega-category

From: [email-masked] (Philippe Gaucher)
Date: Wed, 13 May 1998 15:35:53 +0200

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 ?