From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3584 Path: news.gmane.org!not-for-mail From: Eduardo Dubuc Newsgroups: gmane.science.mathematics.categories Subject: Re: Exactness without pullbacks Date: Fri, 19 Jan 2007 15:35:22 -0300 (ART) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019392 9352 80.91.229.2 (29 Apr 2009 15:36:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:36:32 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Jan 19 19:42:59 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Jan 2007 19:42:59 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1H83Gk-0002Cx-9K for categories-list@mta.ca; Fri, 19 Jan 2007 19:35:38 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 77 Original-Lines: 38 Xref: news.gmane.org gmane.science.mathematics.categories:3584 Archived-At: > > Has anybody considered (and are there any references with standard results) > categories that do *not* have *all* pullbacks > but nevertheless have some nice exactness properties? > > For example, instead of saying that regular epis are stable under pullback > (so that the pullback of a regular epi along any map is also regular-epic), grothendieck notion of strict epi (SGA4) is equivalent to the notion of regular epi in the presence of the kernel-pair, but it makes sense in the absence of pull-backs. you can say that a strict epi is "stable under pullbacks" also in the absence of pullbacks: Z_i -------> X | | |f_i |f \/ h \/ Z --------> Y a strict epi f is universal if given any h there exists a strict epi family f_i as indicated in the diagram. this exactness property is as good as stability under pullbacks see the links http://arXiv.org/abs/math/0611701 http://arXiv.org/abs/math/0612727 i am afraid thought that you have different examples in mind. eduardo j. dubuc