From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7858 Path: news.gmane.org!not-for-mail From: Alex Simpson Newsgroups: gmane.science.mathematics.categories Subject: Re: A category internal to itself Date: Thu, 05 Sep 2013 14:44:28 +0100 Message-ID: References: Reply-To: Alex Simpson NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; DelSp="Yes"; format="flowed" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1378473165 25207 80.91.229.3 (6 Sep 2013 13:12:45 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 6 Sep 2013 13:12:45 +0000 (UTC) To: Zhen Lin Low , Original-X-From: majordomo@mlist.mta.ca Fri Sep 06 15:12:46 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1VHvq9-0006j2-DL for gsmc-categories@m.gmane.org; Fri, 06 Sep 2013 15:12:45 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37743) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1VHvng-0007cj-OJ; Fri, 06 Sep 2013 10:10:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1VHvnf-0000os-EZ for categories-list@mlist.mta.ca; Fri, 06 Sep 2013 10:10:11 -0300 In-Reply-To: Content-Disposition: inline Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7858 Archived-At: A comment on Zhen Lin's reply to Andrej: > To begin, consider a category C with finite limits. Suppose C has an > internal category U such that the externalisation of U as a C-indexed > category (or category fibred over C) is equivalent to the > self-indexing of C. Since U is locally small as a C-indexed category, > the self-indexing of C has the same property, so we deduce that C is > locally cartesian closed. > > We have a universal fibration el U -> ob U (by restricting the > fibration mor U -> ob U x ob U), so it follows that every object X > admits a monomorphism X -> el U. Now, if we add the assumption that C > (or U) is well-powered as a C-indexed category ... Just to remark that there is no need to add any assumption here. If a locally cartesian closed category has a "universal fibration" as you call it (called "generic family" in the reference below), then it is degenerate. This appears in: Pitts, Andrew M.; Taylor, Paul. A note on Russell's paradox in locally Cartesian closed categories. Studia Logica 48 (1989), no. 3, 377?387. (MR1059248) With best wishes, Alex -- Alex Simpson, LFCS, School of Informatics, Univ. of Edinburgh, UK Email: Alex.Simpson@ed.ac.uk Tel: +44 (0)131 650 5113 Web: http://homepages.inf.ed.ac.uk/als Fax: +44 (0)131 651 1426 -- The University of Edinburgh is a charitable body, registered in Scotland, with registration number SC005336. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]