From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5668 Path: news.gmane.org!not-for-mail From: Aleks Kissinger Newsgroups: gmane.science.mathematics.categories Subject: Re: Are there exactly 11 categories with 3 arrows? Date: Sat, 27 Mar 2010 09:01:33 +0800 Message-ID: References: Reply-To: Aleks Kissinger NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1269720942 1348 80.91.229.12 (27 Mar 2010 20:15:42 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 27 Mar 2010 20:15:42 +0000 (UTC) To: Mark Spezzano Original-X-From: categories@mta.ca Sat Mar 27 21:15:38 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 1NvcPy-0001UR-My for gsmc-categories@m.gmane.org; Sat, 27 Mar 2010 21:15:38 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NvbqY-0005su-3p for categories-list@mta.ca; Sat, 27 Mar 2010 16:39:02 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5668 Archived-At: 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. On Fri, Mar 26, 2010 at 10:07 PM, Mark Spezzano wrote: > Hi, > > This is a beginner's question. I have a textbook (Walters) that asks to show that there are exactly 11 categories with 3 arrows. > > Now, my logic tells me that I need to cover three possibilities: > > a) One object with three arrows. How many are there of these? > > b) Two objects with three arrows. How many are there of these? > > c) Three objects with three arrows. I think that the answer to this is the easiest. The answer is 1 categories because they are all endomorphisms, each object containing just the identity morphism. > > So the other 10 arrows must come from a) and b), but I keep getting different answers like 12 and 13 categories as the total. > > Can someone please explain to me the combinations of categories that need to be covered and why some are missed out during the calculation. > > Any help would be immensely appreciated. > > Thanks, > > Mark Spezzano > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]