Toby Bartels wrote:
Steve Vickers wrote:
It is also possible to use a 2-dimensional syntax, in which horizontal composition is composed horizontally and vertical composition is composed vertically. Then algebraic manipulations are a bit like sliding tiles around in a tray.
Of course this can be done using big diagrams. But is there a tight syntax for this just using text? Can you point to an example? (preferably a TeX source online, but a printed page in a regular journal would also work).
-- Toby
Dear All, In reply to Toby Bartels, there are various models of higher categories in which the syntax is well attested and the `sliding of tiles' is algebraically described. One possible one that extends to arbitrary dimensions is given in the paper: AL-AGL, A.A., BROWN, R. & STEINER, R., Multiple categories: the equivalence of a globular and a cubical approach, Advances in Math. 170 (2002) 71-118. The links between `cubical' syntax and a more globular syntax are at the heart of the extensive work on the equivalence between the various models for weak n-categories. One problem is that there are no normal forms for elements. In fact I think (possibly!) that the problem of rewriting in these higher dimensional settings needs a higher dimensional rewriting systems, and to model that one needs n-cateories (and so on!) Tim Porter