From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7008 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: when does preservation of monos imply left exactness? Date: Thu, 27 Oct 2011 12:32:43 +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="UTF-8"; reply-type=original Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1319717149 8105 80.91.229.12 (27 Oct 2011 12:05:49 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 27 Oct 2011 12:05:49 +0000 (UTC) To: "Dmitry Roytenberg" , "Categories list" Original-X-From: majordomo@mlist.mta.ca Thu Oct 27 14:05:44 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RJOiO-0000dC-43 for gsmc-categories@m.gmane.org; Thu, 27 Oct 2011 14:05:44 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:39227) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RJOgp-0004aa-4O; Thu, 27 Oct 2011 09:04:07 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RJOgn-0002Ov-BQ for categories-list@mlist.mta.ca; Thu, 27 Oct 2011 09:04:05 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7008 Archived-At: All right, somebody should answer... Dear Dmitry, First, may I suggest that looking at category theory as merely a 'generalized abelian category theory' is not really a good idea? Specifically, in the abelian context 'right exact + preserves monos => left exact' is simple but extremely important, while for general categories it is indeed wrong, and adding conditions to make this work seems to be a strange thing to do! Now, more specifically, concerning Barr exact: Let 2 be the ordered set {0,1} considered as a category, let S be the category of sets, and let F : S --> 2 be the left adjoint of the canonical embedding 2 --> S (hence, for a set X, F(X) = 0 if and only if X is empty). Then: (a) 2 and S are Barr exact. (b) F preserves all colimits since it is a left adjoint. (c) F preserves products (moreover, the fact that it preserves all products is one of the standard forms of the Axiom of Choice). (d) F preserves monomorphisms since every morphism in 2 is a monomorphism. (e) F does not preserve all equalizers since two parallel morphisms between non-empty sets might have the empty equalizer. Best regards George Janelidze -------------------------------------------------- From: "Dmitry Roytenberg" Sent: Wednesday, October 26, 2011 1:59 PM To: "Categories list" Subject: categories: Re: when does preservation of monos imply left exactness? > I'll try again... > > On Mon, Oct 24, 2011 at 11:23 AM, Dmitry Roytenberg > wrote: >> Dear category theorists, >> >> It is well known that any functor that preserves finite limits >> preserves monomorphisms, and that for an additive right-exact functor >> between abelian categories, the converse is also true. Is it known how >> far this extends to the non-additive setting? In other words, what >> exactness properties of two categories and a functor between them >> would suffice to conclude that the functor preserves finite limits if >> and only if it preserves monos? For instance, is it enough to assume >> that the categories be Barr-exact and that the functor preserve all >> colimits, finite products and monos to conclude that it also preserves >> equalizers? >> >> Any references would be extremely helpful. >> >> Thanks, >> Dmitry >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]