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.
The trouble is that n-monoidal categories already exist. They were introduced my Balteanu, Fioderowicz, Shwantzl and Vogt. This is why I also see n-tuply monoidal as confusing. I do not say that they sound identical but certainly very close to each other.
This is a strong point. Obviously n-tuply monoidal category should mean category with n "compatible" monoidal structures; but there many possible meanings of "compatible". One choice leads to a single monoidal structure with an (n-1)-braiding; but a different choice leads to the notion of BFSV. In fact, I think that even the BFSV notion is too strict---it forces all the units to be the same, where I think one should allow them to be different (in general). That is, I think it would be reasonable to use "doubly monoidal category" to mean (pseudo)monoid internal to LAX (rather than STRONG, or even NORMAL). Cheers, Jeff. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]