From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2948 Path: news.gmane.org!not-for-mail From: "Clemens.BERGER" Newsgroups: gmane.science.mathematics.categories Subject: Re: name for a concept Date: Thu, 08 Dec 2005 12:06:15 +0100 Organization: Laboratoire J.-A. =?ISO-8859-1?Q?Dieudonn=E9?= Message-ID: <439813A7.9000909@math.unice.fr> Reply-To: cberger@math.unice.fr NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019001 6469 80.91.229.2 (29 Apr 2009 15:30:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:30:01 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Dec 9 09:34:45 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Dec 2005 09:34:45 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EkiIY-0004DF-4B for categories-list@mta.ca; Fri, 09 Dec 2005 09:28:30 -0400 X-Accept-Language: en-us, en Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 23 Original-Lines: 33 Xref: news.gmane.org gmane.science.mathematics.categories:2948 Archived-At: Let me suggest still another terminology: For this, call a class S of maps in an arbitrary category *(co)stable* iff S is closed under composition and under (co)base change. Then call a commutative square *S-exact * (resp. *S-coexact*) iff the induced map to the pullback (resp. from the pushout) belongs to S. It is then easy to check that S-(co)exact squares compose for any (co)stable class S (which I believe is the minimal condition to impose on any reasonable distinguished class of commutative squares). In an abelian category, the class M of monos (resp. the class E of epis) is not only stable (resp. costable) but also costable (resp. stable). With this terminology, Hilton's exact squares can either be identified with the E-exact squares or with the M-coexact squares, which explains why it is a self-dual concept, cf. the first message of Michael Barr and the last message of Marco Grandis. In homotopy theory, there is the important concept of a *homotopy pullback* which is the ``homotopy invariant'' substitute for an ordinary pullback. For those who are familiar with Quillen model categories, it is very useful in practice that if a Quillen model category is *right proper* (i.e. its class of fibrations is stable), then a commutative square with two parallel fibrations is a homotopy pullback *if and only if* the square is exact with respect to the class of trivial fibrations (those fibrations which are also weak equivalences). There is of course a dual statement for homotopy pushouts in a left proper Quillen model category. With best regards, Clemens Berger.