RE : bilax monoidal functors

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

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/ ]