From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3581 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Exactness without pullbacks Date: Thu, 18 Jan 2007 22:36:25 -0800 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241019390 9337 80.91.229.2 (29 Apr 2009 15:36:30 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:36:30 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Jan 19 08:58:34 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Jan 2007 08:58:34 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1H7tH8-0006v0-Vc for categories-list@mta.ca; Fri, 19 Jan 2007 08:55:23 -0400 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 74 Original-Lines: 23 Xref: news.gmane.org gmane.science.mathematics.categories:3581 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), I might say that any pullback of a regular epi is regular-epic *if* it exists. (I might instead use a weaker variant, requiring this only in the case that *all* pullbacks of the regular epi in question exist; or else requiring that all pullbacks of *all* regular epis exist, yielding a stronger variant). For a more specific example, the category of smooth manifolds misses many pullbacks but has the property above (at least the weaker form; as I recall, the surjective submersions are precisely those regular epis that have all pullbacks, but I forget if any other regular epis exist; in any case, the pullback of a surjective submersion along any smooth map exists and is also surjective-submersive). --Toby