From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7188 Path: news.gmane.org!not-for-mail From: Mark Weber Newsgroups: gmane.science.mathematics.categories Subject: Re: question about discrete op-fibrations Date: Thu, 2 Feb 2012 10:29:27 +1100 Message-ID: References: Reply-To: Mark Weber NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1328236782 14744 80.91.229.3 (3 Feb 2012 02:39:42 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 3 Feb 2012 02:39:42 +0000 (UTC) Cc: categories list To: David Spivak Original-X-From: majordomo@mlist.mta.ca Fri Feb 03 03:39:41 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 1Rt93s-0005kP-7q for gsmc-categories@m.gmane.org; Fri, 03 Feb 2012 03:39:40 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46987) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1Rt92V-00075e-6D; Thu, 02 Feb 2012 22:38:15 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Rt92W-0003eI-KV for categories-list@mlist.mta.ca; Thu, 02 Feb 2012 22:38:16 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7188 Archived-At: Dear David I'll assume by the coslice DOF_{C/} you mean the category whose objects are d= iscrete opfibrations C --> D, and whose arrows are strictly commuting triang= les under C. In that case the answer to your question is no. When C is empty, DOF_{C/} is= just your category DOF, of small categories and discrete opfibrations betwe= en them, and DOF lacks a terminal object. For suppose that D is terminal in D= OF. Then for any set X, there is a discrete opfibration I(X) --> D, where I(= X) is the category obtained from X by freely adding an initial object. That i= s, the objects of I(X) are the elements of X together with one additional ob= ject 0, and one has a unique arrow 0 --> x for all x in X. If F:I(X) --> D is a discrete opfibration, then F(0) is an object of D such t= hat the cardinality of the set of all arrows with source F(0) is that of X. T= hus since D is terminal in DOF, for any set X there is an object x of D such= that the cardinality of the set of all arrows with source x is that of X. T= his contradicts the smallness of D. With best regards, Mark Weber On 01/02/2012, at 11:03 AM, David Spivak wrote: > Hi all, >=20 > Here's a quick question perhaps someone here can answer easily. >=20 > 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. >=20 > Question: Does there exist a terminal object in DOF_{C/}? >=20 > Thanks! > David >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]