From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7190 Path: news.gmane.org!not-for-mail From: Thorsten Palm Newsgroups: gmane.science.mathematics.categories Subject: Re: question about discrete op-fibrations Date: Thu, 2 Feb 2012 11:22:56 +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 1328236915 15458 80.91.229.3 (3 Feb 2012 02:41:55 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 3 Feb 2012 02:41:55 +0000 (UTC) Cc: categories list To: David Spivak Original-X-From: majordomo@mlist.mta.ca Fri Feb 03 03:41:55 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 1Rt962-0006aE-Bk for gsmc-categories@m.gmane.org; Fri, 03 Feb 2012 03:41:54 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46999) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1Rt95E-0007WW-07; Thu, 02 Feb 2012 22:41:04 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Rt95F-0003hX-Cl for categories-list@mlist.mta.ca; Thu, 02 Feb 2012 22:41:05 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7190 Archived-At: David Spivak hat am 31.01.12 geschrieben: > > Let DOF denote the category whose objects are small categories C,D, > etc. and in which Hom(C,D) is the set of discrete op-fibrations C-->D. > For a category C, let DOF_{C/} denote the coslice over C. > > Question: Does there exist a terminal object in DOF_{C/}? Answer: No. The "op-" is irrelevant; let us look at DF_{C/} instead. There is not even a weakly terminal object, for the same reason for which there isn't one in DF itself. (Suppose (T,z) is one. Let A be an arbitrary small category having a terminal object a. From the object (C+A,incl) of DF_{C/} we get a discrete fibration f : A-->T. But then for t = f(a) the slice category T_{/t} is isomorphic to A. There are not enough t to accommodate all possible A.) Thorsten Palm [For admin and other information see: http://www.mta.ca/~cat-dist/ ]