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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