Andre Joyal wrote in part:
We should use a similar terminology for spaces and maps. E-n space <--> E-n map Also for (higher) categories and functors. monoidal category <---> monoidal functor braided monoidal category <----> braided monoidal functor 2-braided monoidal category <--> 2-braided monoidal functor 3-braided monoidal category <--> 3-braided monoidal functor ...... symmetric monoidal category <--> symmetric monoidal functor
I agree, one should say "symmetric monoidal functor"; if nothing else, that indicates that the source and target are symmetric (not merely braided) monoidal categories. I only put "BMF" in my table to show a particular pattern. Depending on how you write down the definitions, that a braided monoidal functor between symmetric monoidal categories is the same thing as a symmetric monoidal functor between them is either an utter triviality or a deep and interesting theorem; but in either case, we need the words to state it. (I do agree with John about preferring "k-tuply monoidal", but I'll let him make that argument.)
A (n+1)-braided monoidal n-category is symmetric by the stabilisation hypothesis. I believe that a (n+1)-braided monoidal functor between (n+1)-braided monoidal n-categories is symmetric.
I think that you mean to say (which is even stronger) that an n-braided monoidal functor between SM n-categories is symmetric. More generally, a k-braided monoidal l-transfor between SM n-categories is symmetric as long as k + l is greater than or equal to n. (A 0-transfor is a functor, a 1-transfor is a natural transformation, etc. This numbering is due to Sjoerd Crans; feel free to argue that it's off.) More generally yet, a k-braided monoidal l-transfor between m-braided monoidal n-categories is m-braided, as long as k + l >= n, regardless of the value of m (although we need m >= k for the antecedent to make sense).
Is this part of the official stabilisation hypothesis?
I don't know what's official, but I'll claim the conjecture above as mine if nobody else has written it down yet. (^_^) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]