Philippe Gaucher <gaucher@irma.u-strasbg.fr> 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