Dear Mike, the objects, as you say, are pairs (c,d) consisting of an object of C and an object of D morphisms are freely generated by those of the form (f:c->c',1_d:d->d) and (1_c:c->c,g:d->d') subject to the relations (f',1)(f,1)=(f'f,1) and (1,g')(1,g)=(1,g'g). (this is very similar to the description of the elements of the tensor product of two abelian groups). 2-cells are best described by the fact that there is a 2-functor from the Gray tensor C@D to the product CxD which is the identity on objects and on the generating 1-cells, and is locally fully faithful (fully faithful on 2-cells). Thus this 2-functor C@D->CxD is in fact a biequivalence. On 27/05/2011, at 9:57 AM, 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. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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