From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6287 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: reverting religious terminology Date: Mon, 4 Oct 2010 11:49:00 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1286280885 9633 80.91.229.12 (5 Oct 2010 12:14:45 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 5 Oct 2010 12:14:45 +0000 (UTC) Cc: categories@mta.ca To: Thomas Streicher Original-X-From: majordomo@mlist.mta.ca Tue Oct 05 14:14:43 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P36Pm-0000Hd-Jt for gsmc-categories@m.gmane.org; Tue, 05 Oct 2010 14:14:38 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42005) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P36P9-0001Mz-BB; Tue, 05 Oct 2010 09:13:59 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P36P4-0000ED-8y for categories-list@mlist.mta.ca; Tue, 05 Oct 2010 09:13:54 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6287 Archived-At: On Mon, Oct 4, 2010 at 3:36 AM, Thomas Streicher wrote: > But for such weak fibrations one looses the important property that for every > u : J -> I one can transport X over I to u^*X over J along u. As I pointed out in my message to the list on September 16, all that one has to do to remedy this situation is consider "essential fibers" rather than strict fibers. In other words, the notion of "X over I" is itself evil and needs to be replaced by "X equipped with an isomorphism from P(X) to I". The category of all so-equipped Xs is called the "essential fiber" of P over I, and in a weak fibration there is indeed a functor u^* from the essential fiber over I to the essential fiber over J. In this way, any weak fibration also gives rise to an indexed category, and the 2-category of weak fibrations is biequivalent to that of indexed categories (whereas the 2-category of strict fibrations is strictly 2-equivalent to that of indexed categories). Also, if P is a strict Grothendieck fibration (indeed, an isofibration suffices), then its essential fibers are equivalent to its strict fibers, so the two constructions are compatible. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]