From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7189 Path: news.gmane.org!not-for-mail From: David Spivak Newsgroups: gmane.science.mathematics.categories Subject: Re: question about discrete op-fibrations Date: Wed, 1 Feb 2012 23:51:54 -0600 Message-ID: References: <81982979-2217-4AC4-AEDD-154DB2EEAC7B@gmail.com> Reply-To: David Spivak NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1328236867 15236 80.91.229.3 (3 Feb 2012 02:41:07 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 3 Feb 2012 02:41:07 +0000 (UTC) Cc: categories list To: Mark Weber Original-X-From: majordomo@mlist.mta.ca Fri Feb 03 03:41:06 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 1Rt95E-0006Hu-Ph for gsmc-categories@m.gmane.org; Fri, 03 Feb 2012 03:41:04 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46993) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1Rt94F-0007Pe-4Z; Thu, 02 Feb 2012 22:40:03 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Rt94G-0003fq-B7 for categories-list@mlist.mta.ca; Thu, 02 Feb 2012 22:40:04 -0400 In-Reply-To: <81982979-2217-4AC4-AEDD-154DB2EEAC7B@gmail.com> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7189 Archived-At: Hi Mark, Nice work; thank you for the simple answer and good explanation. 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=3Dempty-category we get DOF(C)=3DCat, which does have a terminal object. Thanks, David On Wed, Feb 1, 2012 at 5:29 PM, Mark Weber wrote: > Dear David > > I'll assume by the coslice DOF_{C/} you mean the category whose objects a= re discrete opfibrations C --> D, and whose arrows are strictly commuting t= riangles 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 b= etween them, and DOF lacks a terminal object. For suppose that D is termina= l in DOF. Then for any set X, there is a discrete opfibration I(X) --> D, w= here I(X) is the category obtained from X by freely adding an initial objec= t. That is, the objects of I(X) are the elements of X together with one add= itional object 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 su= ch that the cardinality of the set of all arrows with source F(0) is that o= f X. Thus 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. This contradicts the smallness of D. > > With best regards, > > Mark Weber > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]