From: baez@newmath.UCR.EDU
To: categories@mta.ca
Subject: (-1)-categories and (-2)-categories
Date: Mon, 4 Dec 2000 12:41:36 -0800 (PST) [thread overview]
Message-ID: <200012042041.eB4KfaM14920@math-cl-n06.ucr.edu> (raw)
Robert Dawson wrote:
Surely we should start with the set of (-1)-categories?
Actually we should start at least a little bit before that, with
(-2)-categories. Let me explain....
Once upon a time I showed what I called the "periodic table" to
Chris Isham, a physicist who works on quantum gravity. It starts
like this:
k-tuply monoidal n-categories
n = 0 n = 1 n = 2
k = 0 sets categories 2-categories
k = 1 monoids monoidal monoidal
categories 2-categories
k = 2 commutative braided braided
monoids monoidal monoidal
categories 2-categories
k = 3 " " symmetric sylleptic
monoidal monoidal
categories 2-categories
k = 4 " " " " symmetric
monoidal
2-categories
k = 5 " " " " " "
and it extends infinitely in both directions.
The basic idea is that a "k-tuply monoidal n-category" is a weak
(n+k)-category with only one j-morphism for j < k. There's a
lot of evidence from homotopy theory and elsewhere that each
column of this table must "stabilize" when k reaches n + 2. Of
course, this observation needs to be made more precise before
it can become a theorem, or even a conjecture, so James Dolan
and I called it the "stabilization hypothesis". Carlos Simpson
found one way to make it precise and prove it:
On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani's
weak n-categories, math.CT/9810058.
but everyone who has a definition should take a whack at it!
Anyway, when I showed this pattern to Chris Isham, I was very
proud of it, so I was annoyed when he instantly found fault with
it. He said: what about the (-2)-categories, (-1)-categories,
and monoidal (-1)-categories? You've drawn this big triangle,
but it's missing the upper left-hand corner!
I told him I'd have to think about that.
After that, I kept trying to guess what (-2)-categories,
(-1)-categories and monoidal (-1)-categories should be.
Clearly a monoidal (-1)-category should be a set with
just one element. But what about the other two?
Later, when explaining the concepts of "property", "structure" and
"stuff" to Toby Bartels, James Dolan figured out what (-1)-categories
are. Toby then helped him figure out what (-2)-categories were,
too. Actually, I should be a bit careful here: they really figured
out what (-1)-groupoids and (-2)-groupoids are. However, I believe
that these coincide with (-1)-categories and (-2)-categories.
I have a lot to do today, and this article is already getting too
long, so I'll stop here and leave these as a puzzle for all of you.
It's sort of fun!
I should however mention this: after James and I came to understand this
stuff, someone pointed out an error in our definition of n-categories,
and we were very perturbed until we realized it could be fixed by
changing just one number in our existing definition - which would have
been obvious from the start if we'd understood about (-1)-categories.
John Baez
next reply other threads:[~2000-12-04 20:41 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
2000-12-04 20:41 baez [this message]
2000-12-14 21:07 baez
2000-12-15 19:08 baez
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