From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8344 Path: news.gmane.org!not-for-mail From: Adam Gal Newsgroups: gmane.science.mathematics.categories Subject: Re: Soundness of commutative diagram proofs Date: Thu, 25 Sep 2014 21:29:22 -0400 Message-ID: References: Reply-To: Adam Gal NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1411755231 27860 80.91.229.3 (26 Sep 2014 18:13:51 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 26 Sep 2014 18:13:51 +0000 (UTC) Cc: categories To: Aleks Kissinger Original-X-From: majordomo@mlist.mta.ca Fri Sep 26 20:13:45 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XXa1Y-0004T4-T0 for gsmc-categories@m.gmane.org; Fri, 26 Sep 2014 20:13:45 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50191) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XXa1E-0004DR-Go; Fri, 26 Sep 2014 15:13:24 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XXa1F-0000uS-3d for categories-list@mlist.mta.ca; Fri, 26 Sep 2014 15:13:25 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8344 Archived-At: Hi Aleks, I don't have a complete answer but maybe this will help: In your example, the problem is that your diagram has several faces on the same plane as another face. I think if you have a diagram where this does not happen you should be fine. E.g in your example it should have been a tetrahedron, and then the outer triangle is just another face. Best, Adam On Thursday, September 25, 2014, Aleks Kissinger wrote: > A common style of proof in CT papers is to draw a huge commutative > diagram, number some subset of the faces, and justify why each of > these faces commute. However, such an argument alone doesn't imply > that the overall diagram commutes. Consider for example a triangle of > arrows with three additional arrows connecting each of the corners to > a fourth object in the centre. It is very easy to find examples where > the three little triangles commute, but not the big outside triangle. > E.g. take the three inward-pointing arrows to be 0 morphisms, then we > can take f,g,h to be arbitrary on the outside. > > So, my question is: > > Is there a simple way of judging soundness for a commutative diagram proof? > > One answer is to determine what constitutes a legal pasting of > diagrams, then only admit those which were obtained inductively by > legal pastings of commuting faces. However, its not immediately > obvious that, given a diagram without such a decomposition into legal > pastings, we can obtain the decomposition efficiently. Has this > problem been studied formally somewhere? > > > Best, > > Aleks > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]