From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5671 Path: news.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Re: Are there exactly 11 categories with 3 arrows? Date: Sat, 27 Mar 2010 11:15:20 +1030 Message-ID: References: Reply-To: David Roberts NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1269721063 1705 80.91.229.12 (27 Mar 2010 20:17:43 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 27 Mar 2010 20:17:43 +0000 (UTC) To: Mark Spezzano Original-X-From: categories@mta.ca Sat Mar 27 21:17:39 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 1NvcRr-0002QW-W4 for gsmc-categories@m.gmane.org; Sat, 27 Mar 2010 21:17:36 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Nvbpq-0005rg-KY for categories-list@mta.ca; Sat, 27 Mar 2010 16:38:18 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5671 Archived-At: Hi Mark, Let us phrase the question as: Given three sets, {a},{b,c},{d,e,f}, how many categories are there with one of these sets as its set of objects and exactly three arrows? (any other set of objects is disallowed, as you pointed out) This is to remove questions of 'up to equivalence' because technically, there are an infinite number of categories with three arrows, and taking categories up to equivalence will, I think, give too few for your liking (for example, in my list of four categories with two objects, we would only get two categories). Here's a start: there are two categories b -> c, c -> b (omitting identity arrows) with no non-trivial isomorphisms, and two categories Z/2 \sqcup {*} where * is one of b or c, and the other is the identity element of Z/2. Given your unique category with three elements this brings us to five categories. Then there is Z/3 (with identity element a), giving us six. What other categories with three arrows have you identified? They will, I imagine, have two objects. David On 27 March 2010 00:37, 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/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]