#
A tensor product for Gray-categories

##
Sjoerd Crans

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.

Keywords:

1991 MSC: 18D05 (18A05, 18D10, 18D20).

*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

http://www.tac.mta.ca/tac/volumes/1999/n2/n2.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1999/n2/n2.dvi

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