From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2931 Path: news.gmane.org!not-for-mail From: "Ronald Brown" Newsgroups: gmane.science.mathematics.categories Subject: Re: Name for a concept Date: Fri, 2 Dec 2005 11:19:22 -0000 Message-ID: <003801c5f732$57cff880$06fb4c51@brown1> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018990 6406 80.91.229.2 (29 Apr 2009 15:29:50 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:29:50 +0000 (UTC) To: "Categories list" Original-X-From: rrosebru@mta.ca Fri Dec 2 14:02:54 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Dec 2005 14:02:54 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EiFBf-0006Ey-Ve for categories-list@mta.ca; Fri, 02 Dec 2005 13:59:12 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 6 Original-Lines: 58 Xref: news.gmane.org gmane.science.mathematics.categories:2931 Archived-At: This kind of condition occurs in topology as a fibrant square - but where all the maps are fibrations as is the map A ---> B x_D C . This can be generalised to cubes. See a paper by R. Steiner on `Resolutions of spaces by n-cubes of fibrations', J. London Math. Soc.(2), 34, 169-176, 1986 used to build a complete (strict) algebraic model of homotopy n-types which allows some computations. This raises the spectre in algebra of Resolutions of A by free crossed n-cubes of A. to give a more `nonabelian' homological algebra. Of course crossed n-cubes of A should be equivalent to n-fold groupoids in A. This would presumably bring in higher versions of nonabelian tensor products in A; a bibliography of such a tensor, mainly for n=2, has 90 items. This probably does not help to answer Mike's question on the name! Ronnie www.bangor.ac.uk/r.brown/nonabtens.html ----- Original Message ----- From: "Michael Barr" To: "Categories list" Sent: Thursday, December 01, 2005 1:48 AM Subject: categories: Name for a concept > Is there a standard name for a square > A ----> B > | | > | | > | | > v v > C ----> D > in which the canonical map A ---> B x_D C is epic? I had always called it > a weak pullback, but Peter Freyd claims that that phrase is reserved for > the case that it satisfies the existence, but not necessarily the > uniqueness of the definition of pullback. In fact, he claims it means > that Hom(E,-) converts it to the kind of square I am talking about. > What is interesting is that in an abelian category, it satisfies > this condition iff it satisfies the dual condition iff the evident > sequence A ---> B x C ---> D is exact. Putting a zero at the left end > characterizes a genuine pullback and at the other end a pushout. > > Michael > > >