Re: bilax_monoidal_functors?=

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

Michael Batanin wrote in part:

>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?

No, but it seems to me that you are doing something very much like this.

The concept of n-category makes sense for n as low as -2,
so it would be nice to renumber this so that we start at n = 0.
However, if we do so, then we need a word other than "-category";
if "category" = "3-category", then this violates "foo" = "1-foo".

Similarly, the concept of k-braided MC makes sense for k = -1,
so it would be nice to renumber this so that we start at k = 0.
However, if we do so, then we need a word other than "-braided MC";
if "braided MC" = "2-braided MC", then this violates "foo" = "1-foo".

So either we stick with Andre's numbering, inelegant as may be,
or we change Andre's "-braided MC" to John's "-tuply MC".
But you say, no, we do not need "foo" = "1-foo",
simply renumber so that "braided MC" = "2-braided MC".
That is like saying, renumber so that "category" = "3-category".
While it is a more elegant numbering, it is likely to be confusing.

I will say no more about it.  I will be happy to read your papers,
as long as you explain your terminology up front, as we all should.
I may grumble to myself at your violation of "foo" = "1-foo",
but I will nevertheless understand since you have explained.
(But if you later post to the categories list about it,
  then I may be confused if you don't recall the numbering.)


--Toby


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