Subject: categories: (-1)-categories and (-2)-categories

From: [email-masked]
Date: Mon, 4 Dec 2000 12:41:36 -0800 (PST)

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