From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7191 Path: news.gmane.org!not-for-mail From: Thorsten Palm Newsgroups: gmane.science.mathematics.categories Subject: Re: question about discrete op-fibrations Date: Fri, 3 Feb 2012 15:07:21 +0100 (MET) Message-ID: References: Reply-To: Thorsten Palm NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: dough.gmane.org 1328318751 3224 80.91.229.3 (4 Feb 2012 01:25:51 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 4 Feb 2012 01:25:51 +0000 (UTC) Cc: Mark Weber , categories list To: David Spivak Original-X-From: majordomo@mlist.mta.ca Sat Feb 04 02:25:50 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RtUNx-0007ek-RI for gsmc-categories@m.gmane.org; Sat, 04 Feb 2012 02:25:50 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:48769) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1RtUMm-0003bw-AZ; Fri, 03 Feb 2012 21:24:36 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RtUMn-0000nX-Oi for categories-list@mlist.mta.ca; Fri, 03 Feb 2012 21:24:37 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7191 Archived-At: [ To all, moderator in particular, and with apologies to David: Please ignore my previous message. It contains a seriously wrong piece of information. ] Dear David, This time the answer is "yes" for discrete C, "no" otherwise. If C is discrete, the functor !_C : C-->1 is a discrete op-fibration, so that the terminal objet (1,!_C) of Cat_{C/} belongs to DOF(C). Now let C contain a non-trivial morphism u : c_0-->c_1 and suppose that (T,z) is terminal in DOF(C). Construct a category C_{c_1} by freely adding to C an object d and a morphism v : d-->c_1. Then we have an inclusion j : C-->C_{c_1}, which is a discrete op-fibration. We obtain two morphisms (C_{c_1},j)-->(T,z) in DOF(C) by applying z on the subcategory C and sending v to z(u) and the identity of z(c_1), respectively. By terminality of (T,z) they have to be the same. But since z is an op-fibration, z(u) cannot be an identity --- contradiction. (I hope Mark hasn't beaten me to it again.) Thorsten David Spivak hat am 01.02.12 geschrieben: > > I hope this isn't annoying, but what if I change the problem somewhat > and take DOF(C) to be the full subcategory of Cat_{C/} spanned by the > discrete opfibrations C-->D? Again I want to know whether DOF(C) has a > terminal object. Under this definition, by setting C=empty-category we > get DOF(C)=Cat, which does have a terminal object. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]