Re: Gray tensor product

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

There are quite a few paper on this particularly by Sjoerd  Crans. Part
of the problem is that a definition of tensor product will always be in
terms of a structure generated by c \tens d subject to certain
relations, so it will not necessarily be easy to get at the elements of
the tensor product. Hence the exponential law which you quote shows that
it is more explicit to give the elements of Ps(D,E), and these are in
each dimension families of functions satisfying certain conditions.

It is actually easier, I feel,  to do this cubically, as is done
explicitly for the omega-case in Section 10 of
Al-Agl, F.~A., Brown, R. and Steiner, R.
{Multiple categories: the equivalence of a globular and a  cubical
approach}.
{Adv. Math.} \textbf{170}~(1) (2002) 71--118.
which gives an explicit description of n-fold left homotopy in that
cubical context, and so in principle a translation to the globular case.

In the omega-groupoid case it is possible to say more: see the book
project, and references there,  advertised on
http://pages.bangor.ac.uk/~mas010/nonab-a-t.html


Ronnie Brown




On 27/05/2011 00:57, Mike Stay wrote:
> Has anyone "unpacked" the meaning of the Gray tensor product of strict
> 2-categories?  I'm looking for something like "the Gray product C
> tensor D is the 2-category whose
> - objects are pairs (c,d)
> - morphisms are ...
> - 2-morphisms are ..."
>
> My higher-category-fu isn't strong enough yet to grok the implicit definition
>      2Cat(C tensor D, E) ~= 2Cat(C, Ps(D,E)),
> where Ps(D,E) is the 2-category of 2-functors D->E, pseudonatural
> transformations, and modifications.


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