From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6134 Path: news.gmane.org!not-for-mail From: Colin McLarty Newsgroups: gmane.science.mathematics.categories Subject: Re: Grothendieck: more complete URL Date: Fri, 10 Sep 2010 08:26:41 -0400 Message-ID: References: Reply-To: Colin McLarty NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1284166490 12907 80.91.229.12 (11 Sep 2010 00:54:50 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 11 Sep 2010 00:54:50 +0000 (UTC) To: Categories list , toby+categories@ugcs.caltech.edu Original-X-From: majordomo@mlist.mta.ca Sat Sep 11 02:54:48 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OuEMg-000881-A7 for gsmc-categories@m.gmane.org; Sat, 11 Sep 2010 02:54:46 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:55187) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OuELZ-0000ya-Oa; Fri, 10 Sep 2010 21:53:37 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OuELX-0004Wr-1t for categories-list@mlist.mta.ca; Fri, 10 Sep 2010 21:53:35 -0300 In-Reply-To: <20100910011740.GA27758@ugcs.caltech.edu> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6134 Archived-At: Certainly it depends on how the axioms are phrased. And I only said I would not be surprised! When I wrote that, I had the idea that in Tohoku he phrased AB5 for all filtered colimits of objects, but I see it is in fact phrased for colimits of subobjects of the given one -- which is pretty obvious now that I see it. I suppose the issue for constructivizing the proof then comes from AB3 on arbitrary sums together AB4-5 on their properties. It would depend sensitively on how "arbitrary sums" are defined. colin 2010/9/9 Toby Bartels : > Colin McLarty wrote: > >>I would not be surprised if Grothendieck's Tohoku proof that all AB5 >>categories have enough injectives is straightforwardly constructive -- >>with the unhelpful limitation that from a constructive viewpoint >>nearly no categories satisfy the AB5 axioms. =A0Especially the last >>axiom, 5, requires completeness in a strong sense which is tailored to >>make each step of the classical Baer proof work. > > Does this depend on how the axiom is phrased? > An elementary version says that an Ab5 category > is an cocomplete abelian category such that, > given any object X, subobject A of X, > and directed system (B_i)_i of subobjects of X, > A \cap \sum_i B_i =3D \sum_i (A \cap B_i). > The obvious proof that modules form an Ab5 category > works in the internal language of any topos with NNO. > > It is less clear to me whether it follows from this > that filtered colimits preserve exact sequences, > which is the more abstract form of axiom 5. > > > --Toby > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]