From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3000 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: Normal quotients of categories Date: Wed, 18 Jan 2006 11:26:00 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v733) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019036 6838 80.91.229.2 (29 Apr 2009 15:30:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:30:36 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Thu Jan 19 13:25:31 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 19 Jan 2006 13:25:31 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EzdQy-0004eQ-Nz for categories-list@mta.ca; Thu, 19 Jan 2006 13:18:52 -0400 X-Mailer: Apple Mail (2.733) Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 26 Original-Lines: 33 Xref: news.gmane.org gmane.science.mathematics.categories:3000 Archived-At: On 18 Jan 2006, at 09:36, V. Schmitt wrote: > Marco I am not very awake this morning but i think that this > construction of formally inverted some arrow is well known > for long (cf for instance Borceux's handbooks on localizations). > Am i wrong? > Cheers, > Vincent > > Categories of fractions are indeed very well-known, but satisfy a different universal property: to make *invertible* the assigned arrows (instead of making them *identities*). But you can view categories of fractions at the light of what I was saying. Take in Cat the (closed) ideal of functors which send every map to an isomorphism, or equivalently of those functors which factor through a groupoid. With respect to this ideal, the kernel of a functor f: X -> Y is the (wide and replete) subcategory of maps which f turns into isomorphisms, while the cokernel is the category of fractions of Y which inverts all arrows reached by f. Best regards Marco G. PS. And - thinking of Jean Pradine's message - yes, of course, quotient of groupoids are important, but have special features of their own; as he is pointing out.