From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1737 Path: news.gmane.org!not-for-mail From: baez@newmath.UCR.EDU Newsgroups: gmane.science.mathematics.categories Subject: (-1)-categories and (-2)-categories Date: Mon, 4 Dec 2000 12:41:36 -0800 (PST) Message-ID: <200012042041.eB4KfaM14920@math-cl-n06.ucr.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018056 32669 80.91.229.2 (29 Apr 2009 15:14:16 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:16 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Dec 4 20:04:16 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB4NaI921280 for categories-list; Mon, 4 Dec 2000 19:36:18 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: ELM [version 2.5 PL2] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 10 Original-Lines: 86 Xref: news.gmane.org gmane.science.mathematics.categories:1737 Archived-At: 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