A tensor product for Gray-categories

Theory Appl. Categ. 5 (1999), 12-69

Abstract:

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)-CATegory.

``The author acknowledges the support of the Australian Research Council
``© Sjoerd E. Crans 1999. Permission to copy for private use granted.''

ps-file of the revised preprint version | published version | back to the papers page

fichier ps de la version préprint révisée | version publiée | retour à la page papiers

Sjoerd E. Crans
School of Mathematics, Physics, Computing & Electronics
Macquarie University
NSW
Australia