From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7855 Path: news.gmane.org!not-for-mail From: Zhen Lin Low Newsgroups: gmane.science.mathematics.categories Subject: Re: A category internal to itself Date: Thu, 5 Sep 2013 12:30:33 +0100 Message-ID: References: Reply-To: Zhen Lin Low NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1378385962 15546 80.91.229.3 (5 Sep 2013 12:59:22 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 5 Sep 2013 12:59:22 +0000 (UTC) Cc: categories list To: Andrej Bauer Original-X-From: majordomo@mlist.mta.ca Thu Sep 05 14:59:25 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 1VHZ9g-0005qo-2K for gsmc-categories@m.gmane.org; Thu, 05 Sep 2013 14:59:24 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:36143) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1VHZ8K-00006P-Fs; Thu, 05 Sep 2013 09:58:00 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1VHZ8J-0003ar-3y for categories-list@mlist.mta.ca; Thu, 05 Sep 2013 09:57:59 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7855 Archived-At: Dear 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, then C must be an elementary topos. But then the existence of el U implies that the internal logic of C is inconsistent, so C must be the degenerate topos. It appears we need to relax the notion of "internal" to get something more reasonable. Here is one idea: instead of taking just one internal category, we take a (large) filtered diagram of them. More precisely, let U be a diagram of shape J in the category of internal categories in C, where J is filtered and the transition functors are (internally) fully faithful, and define a C-indexed category whose fibre over X is the (external) category colim Hom(X, U). When C is an elementary topos, there exists a diagram U such that this construction yields a C-indexed category that is equivalent to the self-indexing of C: take J to be the poset of all finite subsets of ob C, and take as the internal category at a finite set {X_1, ..., X_n} of objects of C to be the internal full subcategory whose objects are the subobjects of the power object P(X_1 + ... + X_n). In the converse direction, if such a diagram of internal categories exists, then one can still deduce (from the condition on transition functors) that the self-indexing of C is locally small. But perhaps there is a strange locally cartesian closed category out there that is self-internal in the naive sense. Best regards, -- Zhen Lin On 4 September 2013 10:23, Andrej Bauer wrote: > Chatting at a conference, the question came up why there is no > (non-trivial) category which is "internal to itself" (interpret this > in some sensible sense). And over coffee we thought this must be well > known, but not to us. Can somene shed some light on the matter? > > With kind regards, > > Andrej > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]