From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2964 Path: news.gmane.org!not-for-mail From: "Ronald Brown" Newsgroups: gmane.science.mathematics.categories Subject: Terminology re fibrations and opfibrations of categories Date: Thu, 22 Dec 2005 08:07:43 -0000 Message-ID: <001901c606ce$ee6d2dc0$c8cb4c51@brown1> 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 1241019010 6617 80.91.229.2 (29 Apr 2009 15:30:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:30:10 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Mon Dec 26 11:29:11 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Dec 2005 11:29:11 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EquA4-00039B-RJ for categories-list@mta.ca; Mon, 26 Dec 2005 11:21:20 -0400 X-ME-UUID: 20051222081357410.641D25800084@mwinf3108.me.freeserve.com Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 39 Original-Lines: 30 Xref: news.gmane.org gmane.science.mathematics.categories:2964 Archived-At: To add to my previous email, I'd like reactions to the following terminology: Let P: X \to B be a functor. A morphism u: x \to y in X is cofinal w.r.t. P, and y is the P-final object w,r,t u and P , if ... (and here we have the usual notion of cocartesian). Dually, u is coinitial, and x is the initial object w.r.t u and P if ... (and here we have the usual notion of cartesian). In situations where P is understood, we can then talk about cofinal and coinitial morphisms, and structures or objects or (in my case, groupoids). An advantage is that the direction of the notion and its dual should be clear. If f=P(u), I would then write \bar{f}: x \to f_*(x) in the first case, and \underline{f}: f^*(y) \to y in the second. I would also call f_*(x) the object induced by f. What is a handy name for f^*(y)? The restriction of y by f? All these notions occur for modules, crossed modules, ...... and relate to change of base. Ronnie www.bangor.ac.uk/r.brown