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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