In this paper I extend Gray's tensor product of 2-categories to a new tensor product of Gray-categories. I give a description in terms of generators and relations, one of the relations being an ``interchange'' relation, and a description similar to Gray's description of his tensor product of 2-categories. I show that this tensor product of Gray-categories satisfies a universal property with respect to quasi-functors of two variables, which are defined in terms of lax-natural transformations between Gray-categories. The main result is that this tensor product is part of a monoidal structure on Gray-Cat, the proof requiring interchange in an essential way. However, this does not give a monoidal {(bi)closed} structure, precisely because of interchange. And although I define composition of lax-natural transformations, this composite need not be a lax-natural transformation again, making Gray-Cat only a partial Gray-Cat$_\otimes$-CATegory.
Theory and Applications of Categories, Vol. 5, 1999, No. 2, pp 12-69 http://www.tac.mta.ca/tac/volumes/1999/n2/n2.dvi http://www.tac.mta.ca/tac/volumes/1999/n2/n2.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1999/n2/n2.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1999/n2/n2.psTAC Home