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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]