From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6133 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: Grothendieck: more complete URL Date: Thu, 9 Sep 2010 18:17:40 -0700 Message-ID: References: Reply-To: Toby Bartels NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: dough.gmane.org 1284166471 12854 80.91.229.12 (11 Sep 2010 00:54:31 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 11 Sep 2010 00:54:31 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Sat Sep 11 02:54:29 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 1OuEMP-00083A-9s for gsmc-categories@m.gmane.org; Sat, 11 Sep 2010 02:54:29 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:55182) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OuEKw-0000wA-Sb; Fri, 10 Sep 2010 21:52:58 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OuEKt-0004W2-Lu for categories-list@mlist.mta.ca; Fri, 10 Sep 2010 21:52:55 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6133 Archived-At: 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. Especially 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 = \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/ ]