From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2968 Path: news.gmane.org!not-for-mail From: Eduardo Dubuc Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology re fibrations and opfibrations of categories Date: Mon, 26 Dec 2005 18:57:48 -0300 (ART) Message-ID: References: <001901c606ce$ee6d2dc0$c8cb4c51@brown1> 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 1241019012 6645 80.91.229.2 (29 Apr 2009 15:30:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:30:12 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Dec 29 13:12:26 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 29 Dec 2005 13:12:26 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Es1GM-0002wf-7z for categories-list@mta.ca; Thu, 29 Dec 2005 13:08:26 -0400 In-Reply-To: <001901c606ce$ee6d2dc0$c8cb4c51@brown1> from "Ronald Brown" at Dec 22, 2005 08:07:43 AM X-Mailer: ELM [version 2.5 PL2] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 43 Original-Lines: 49 Xref: news.gmane.org gmane.science.mathematics.categories:2968 Archived-At: Concerning Ronnie wanderings about terminology around the word FINAL, the following is pertinent: I am just writing a paper with Luis Espannol where we need to develop (the basic part of the theory of cartesian and cocartesian arrows) for families we use the following terminology: consider a functor U: C ---> S, then: 1) a family in C Z _i ---> X over R_i ---> S is FINAL iff: given S ---> T = UY such that there exists Z_i --->Y over R_i ---> S ---> T (that is, R_i ---> S ---> T lifts), then there exists a unique X ---> Y over S ---> T (that is, S ---> T lifts). For topological spaces this is the usual Bourbaki notion of final topology. When U is not understood, we call this "U-FINAL" Notice that for single arrows, we have (proved in the SGA on fibered categories) Z ---> X is final iff it is cocartesian and cocartesian arrows compose 2) a family in C Z _i ---> X over R_i ---> S is SURJECTIVE iff: the family R_i ---> S is an strict (or regular) epimorphic family in S Our aim is to prove under some natural and minimal assumptions: Z _i ---> X is strict epimorphic iff it is final surjective All this is already done Here the leading examples are the topological spaces and the quasitopological spaces in the sense of Spanier (and the whole theory of concrete quasitopoi over S = Sets)