Michael Batanin wrote:
If we follow the principle "foo = 1 foo" and want to agree with historical low dimensional terminology we should call categories 2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess.
Sorry, but I don't think that you understand what we (Mike and I) mean when we say that "foo" should equal "1-foo". In all of these examples, the word "1-foo" (or "1-tuply foo") means the same as the historic low-dimensional term "foo": * "n-category", with the usual meaning; * "n-set", as you suggested above; * "k-tuply monoidal", as used by John Baez; * "k-braided monoidal", as used by Andre Joyal; * "n-stack", with the usual meaning; * "n-sheaf", as Mike Shulman suggested. In only these examples, the word "1-foo" does ~not~ mean the same as "foo": * "n-connected space", with the usual meaning; * "n-category", with the new meaning that you suggested earlier; * "k-braided monoidal", as you used it here: http://permalink.gmane.org/gmane.science.mathematics.categories/5764/. I like some of the terms in the first list more than others. I find some of them sensibly numbered and some of them not (which is part, but not all, of what goes into my liking them). But I find all of them usable and I instantly understand them. I object to the terms in the second list as inherently confusing, even when I find them sensibly numbered. Of the terms on that list, only "n-connected space" has actually been sanctioned by history. (But see http://ncatlab.org/nlab/show/k-simply+connected+n-category for an alternative approach.) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]