Re: bilax_monoidal_functors?

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

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


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