Re: bilax_monoidal_functors?=

[John Baez <john.c.baez@gmail.com>]

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


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