Re: bilax_monoidal_functors?=

[Michael Batanin <mbatanin@ics.mq.edu.au>]

>> 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.

Well, I agree with Andre's argument but it does not convince me to use
Andre's terminology nor John's terminology (see my objections below).

The shift of numbers in Andre's terminmology is annoying when you try to
prove stabilisation hypothesis using higher braided operads. I hope to
talk about this proof in Genoa in a couple of months but it follows
readily from another atabilization theorem for n-braided operads. It is
   where I was more or less forced to call braided operads 2-braided
operads despite violation of ("foo" = "1-foo").

Another argument in favor of this terminology is that it provides a
uniform terminology in higher dimensions which agrees with E_n-algebra
point of view developed by Lurie and also his proof of stabilization
hypothesis (see Urs's message).

I agree that it creates some clash in low dimensions but I think it is
not a big deal since classical terminology does not have numbers (nobody
calls a monoidal category 0-braided or symmeteic monoidal category
2-braided monoidal). The low dimensional cases are important but they
are not always good models for higher dimension. As an example, -2 and
-1 categories as Baez and Dolan pointed out can be understood as one
pointed set and two pointed set correspondingly. Should we shift the
numbers and call category a 3-category?


> 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.


> But I will not lose sleep if Andre uses "k-braided" as a synonym for
> "(k+1)-tuply monoidal".

I am glad to join John. I am also grateful to everybody participating
in this discussion. Terminology is a very important issue but I do not
think it is a crime to use a different one if the clarity of exposition
dictates it and if one acknowledges the existence of an alternative.  I
think I will continue to use my own  terminology but I am going to give
more explanation in the introduction   for those who like a different
one.

with best regards,
Michael.


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