André Joyal wrote: I am using the following terminology for
higher braided monoidal (higher) categories:
Monoidal< braided < 2-braided <.......<symmetric
A (n+1)-braided n-category is symmetric according to your stabilisation hypothesis.
Is this a good terminology?
I use "k-tuply monoidal" to mean what you'd call "(k-1)-braided". This seems preferable to me, not because it sounds nicer - it doesn't - but because it starts counting at a somewhat more natural place. I believe that counting monoidal structures is more natural than counting braidings. For example, a doubly monoidal n-category, one with two compatible monoidal structures, is a braided monoidal n-category. I believe this is a theorem proved by you and Ross when n = 1. This way of thinking clarifies the relation between braided monoidal categories and double loop spaces. Various numbers become more complicated when one counts braidings rather than monoidal structures: An n-tuply monoidal k-category is (conjecturally) a special sort of (n+k)-category... while an n-braided category is a special sort of (n+k+1)-category. Similarly: n-dimensional surfaces in (n+k)-dimensional space are n-morphisms in a k-tuply monoidal n-category... but they are n-morphisms in an (k-1)-braided n-category. And so on. On the other hand, if it's braidings that you really want to count, rather than monoidal structures, your terminology is perfect. By the way: I don't remember anyone on this mailing list ever asking if their own terminology is good. I only remember them complaining about other people's terminology. I applaud your departure from this unpleasant tradition! Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]