Geodesic
A tensor product for Gray-categories
Theory and applications of categories, Tome 5 (1999), pp. 12-69.

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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.
Classification : 18D05 (18A05, 18D10, 18D20).
Keywords: 
@article{TAC_1999_5_a1,
     author = {Sjoerd Crans},
     title = {A tensor product for {Gray-categories}},
     journal = {Theory and applications of categories},
     pages = {12--69},
     publisher = {mathdoc},
     volume = {5},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_1999_5_a1/}
}
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Sjoerd Crans. A tensor product for Gray-categories. Theory and applications of categories, Tome 5 (1999), pp. 12-69. http://geodesic.mathdoc.fr/item/TAC_1999_5_a1/