From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7007 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: when does preservation of monos imply left exactness? Date: Thu, 27 Oct 2011 08:52:10 +1100 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1319717110 7798 80.91.229.12 (27 Oct 2011 12:05:10 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 27 Oct 2011 12:05:10 +0000 (UTC) Cc: Categories list To: Dmitry Roytenberg Original-X-From: majordomo@mlist.mta.ca Thu Oct 27 14:05:06 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 1RJOhl-0000J8-Mh for gsmc-categories@m.gmane.org; Thu, 27 Oct 2011 14:05:05 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:33071) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RJOdx-0004GD-L4; Thu, 27 Oct 2011 09:01:09 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RJOdw-0002Mk-1Z for categories-list@mlist.mta.ca; Thu, 27 Oct 2011 09:01:08 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7007 Archived-At: Dear Dmitry, There is a result along these lines in Mike Barr's paper "On categories with effective unions", Springer LNM 1348. His Theorem 4.1 says: Suppose F: C --> D is a functor such that C has finite limits, cokernel pairs and effective unions, and F preserves finite products, regular monomorphisms and cokernel pairs. Then F preserves finite limits. In the statement of this result, a category is said to have effective unions if the union of two regular subobjects always exists and is calculated as the pushout over the intersection. Best, Richard On 26 October 2011 22:59, Dmitry Roytenberg wrote: > 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/ ]