From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6842 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Goursat's lemma Date: Sat, 27 Aug 2011 02:46:04 +0200 Message-ID: References: Reply-To: "George Janelidze" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1"; reply-type=response Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1314455384 5129 80.91.229.12 (27 Aug 2011 14:29:44 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 27 Aug 2011 14:29:44 +0000 (UTC) To: "Michael Barr" ,"Categories list" Original-X-From: majordomo@mlist.mta.ca Sat Aug 27 16:29:40 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QxJtD-0007OQ-P7 for gsmc-categories@m.gmane.org; Sat, 27 Aug 2011 16:29:40 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58920) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1QxJry-0004tk-Um; Sat, 27 Aug 2011 11:28:22 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QxJry-0002Jn-8n for categories-list@mlist.mta.ca; Sat, 27 Aug 2011 11:28:22 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6842 Archived-At: Let us define a short exact sequence 0 --> U --> V --> W --> 0 by saying that U --> V is the kernel of V --> W ("left exactness") and V --> W is the cokernel of U --> V ("right exactness"). If so, your formulation becomes is trivial in full generality: The kernel of a morphism X --> Y (in any pointed category with zero object) is the pullback of X --> Y and 0 --> Y. Therefore: (a) D' is the pullback of G --> C and 0 --> C; (b) the kernel of D --> G --> C is the pullback of D --> G --> C and 0 --> C; (c) as follows from (a) and (b), the kernel of D --> G --> C is the pullback of D --> G and D' --> C; (d) similarly, the kernel of D' --> G --> C' is the same pullback. Since this is true in every category with zero object (you don't need any normality argument), the dual statement is also always true (and everything with pushouts and cokernels instead of pullbacks and kernels is equally clear). Moreover (obviously), for the coincidence of the kernels we need only the "left exactness", and dually, for the coincidence of the cokernels we need only the "right exactness" of our original sequences. However, what Jim's says is not true in every pointed Barr exact category with cokernels: it will be true if and only if every regular epimorphism is normal (that is, is the cokernel of something, or, equivalently, the cokernel of its kernel). George -------------------------------------------------- From: "Michael Barr" Sent: Friday, August 26, 2011 1:03 AM To: "Categories list" Subject: categories: Goursat's lemma > This is something that Lambek used for diagram chasing in his book on > Rings and Modules. I have discovered a very simple proof of a slight > generalization. > > The lemma states (as least Jim stated it this way) that if G is a > submodule of A x B and C = image of G --> A, C' the image of G --> B, D > the kernel of G --> B and D' the kernel of G --> A, then C/D is isomorphic > to C'/D'. > > Forget that G is a subgroup of A x B and just suppose you have two exact > sequences 0 --> D' --> G --> C --> 0 and 0 --> D --> G --> C' --> 0, then > the composites D --> G --> C and D' --> G --> C' have the same cokernel. > The dual claim is that they have the same kernel. But thinking of D and > D' are submodules of G, then the kernels are quite obviously D \cap D' and > the original conclusion follows by duality. > > However, the same thing is true for groups (both D and D', being kernels, > are normal in G and hence in C, C'). I see no such simple duality > argument there. > > Michael > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]