Re: bilax_monoidal_functors?=

[Michael Shulman <shulman@uchicago.edu>]

I think it is the least confusing for everyone if when "foo"s start
being decorated with numbers, a "1-foo" is the same thing as what an
unadorned "foo" used to be.  So I definitely have to agree that an
ordinary braided monoidal category should be called "1-braided" if the
naming scheme is going to go by decorating "braided" with numbers.

On the other hand, occasionally it seems to happen that after "foo"s
have been studied for a while, someone introduces a categorified "foo"
and calls it a "bar," and then later someone else comes along and
categorifies again but now starts introducing numbers with "2-bar,"
"3-bar," and so on.  So what really should have been called a "2-foo"
is called a "bar," what really should have been called a "3-foo" is
called a "2-bar," and so on with the numbers all off by one.  As John
points out, the use of "braided = 1-braided" and then "2-braided,"
etc. could be viewed this way, with "monoidal" as the basic "foo" that
we should have started numbering at.

(One other example of this that comes to mind is the original use of
"stack" to mean essentially "2-sheaf," leading to "2-stack" for
something that is really a 3-categorical object, and so on.
Fortunately this particular trend seems to be reversing somewhat.)

However, in the case at hand, it seems to me that there is also an
advantage to the term "braided" over "doubly monoidal."  To give a
category a braided monoidal structure may be *equivalent* to giving it
two interchanging monoidal structures, but that's only true because in
the latter case, the interchange law forces the two monoidal
structures to be essentially the same.  In practice, I find that I
very rarely think about a braided monoidal category as if it were
equipped with two monoidal structures; rather I think of it as having
one monoidal structure together with an extra structure called a
"braiding."  So there are arguments on both sides of this issue, and
as John says probably neither usage will create any confusion.

Mike

On Mon, May 10, 2010 at 1:16 PM, John Baez <john.c.baez@gmail.com> wrote:
> 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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