From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5672 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Are there exactly 11 categories with 3 arrows? Date: Sat, 27 Mar 2010 16:40:10 -0700 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1269786495 550 80.91.229.12 (28 Mar 2010 14:28:15 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 28 Mar 2010 14:28:15 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Sun Mar 28 16:28:10 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1NvtTF-0005bl-Er for gsmc-categories@m.gmane.org; Sun, 28 Mar 2010 16:28:09 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Nvslp-0002Q5-H9 for categories-list@mta.ca; Sun, 28 Mar 2010 10:43:17 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5672 Archived-At: On 3/26/2010 6:01 PM, Aleks Kissinger wrote: > Probably a good exercise to work out the details yourself, but here's > a hint. The 1-object case is exactly the same as counting monoids. With the exception of one category, so is the general case. It suffices to identify the odd category out and then put the remainder in bijection with the monoids of order at most 3 (see Sloane A058129). For positive integer n, 3 is the least value of n for which the categories with n arrows cannot be put in bijection with the monoids of order at most n. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]