Eduardo wrote:
Andre points out:
"To call a monoidal category 1-braided is kind of confusing because there is no commutation structure on a general monoidal category. A monoidal category is 0-braided."
Being an outsider, with no previous neither usage or opinion on this terminology beyond just monoidal and/or tensor category, this seems to me definitive, and more than enough to settle the question.
I'm glad that's enough to convince you that Michael Batanin's terminology "monoidal = 1-braided" is inferior to Andre's "monoidal = 0-braided". But I think "braided = doubly monoidal" is even better. After all, a monoidal category has one tensor product; a braided monoidal category has two compatible tensor products, and a symmetric monoidal category has three. But I will not lose sleep if Andre uses "k-braided" as a synonym for "(k+1)-tuply monoidal". I don't see it causing any confusion. I just think it will create more +1's in various formulas. E.g.: the classifying space of a k-braided n-category is a (k+1)-fold loop space. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]