From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2362 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: first isomorphism theorem Date: Thu, 19 Jun 2003 00:59:18 -0700 Message-ID: <20030619075917.GL13356@math-cl-n01.ucr.edu> References: <13560194@newdancer.Dartmouth.EDU> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018604 3778 80.91.229.2 (29 Apr 2009 15:23:24 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:24 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Jun 19 08:00:11 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 19 Jun 2003 08:00:11 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19Sx7c-0006U1-00 for categories-list@mta.ca; Thu, 19 Jun 2003 07:58:28 -0300 Content-Disposition: inline In-Reply-To: <13560194@newdancer.Dartmouth.EDU> User-Agent: Mutt/1.4i Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 53 Original-Lines: 38 Xref: news.gmane.org gmane.science.mathematics.categories:2362 Archived-At: Andrei Prokopiw wrote: >Is there a suitable first isomorphism theorem in category theoretic language? >One barrier for me seems to be the correct notion of an image. What seems best >right now would be that given f:X->Y, ker(coker f) = coker ( ker f). Here the >left hand side would represent im f, and the right hand side X/kerf. If this >is the right notion, what are the necessary conditions on the category for it >to hold? I'm not really answering your question, instead addressing the "if". One of the classic examples of the isomorphism theorems is for rings, and the category of rings (with unit) has neither kernels nor cokernels. There are ways around that limitation in this case, such as: * Don't require rings to have units; or * Analyse any specific f in a category of X-modules. But more importantly, I think that much of the point of the theorem is that the image is a purely set-theoretic notion, definable without reference to the algebraic properties of f. In the category of sets, the image is (of course) not realisable as the kernel of the cokernel, but it does exist as the equaliser of the cokernel pair. It seems to me that the essence of the isomorphism theorem is that the forgetful functor from algebras to sets not only preserves limits (which we all know) but also preserves images and coimages (defined as above), even though it does *not* preserve coequalisers or cokernel pairs. The classical form of the theorem is simply a reflection of an inability to speak explicitly about functors' preserving categorical constructions. -- Toby Bartels