From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6279 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Sun, 3 Oct 2010 15:10:29 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1286194086 24763 80.91.229.12 (4 Oct 2010 12:08:06 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 4 Oct 2010 12:08:06 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Mon Oct 04 14:08:05 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P2jps-0008Cz-Rp for gsmc-categories@m.gmane.org; Mon, 04 Oct 2010 14:08:05 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58703) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P2jp4-0004uu-TP; Mon, 04 Oct 2010 09:07:14 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P2jp0-0001fC-5Y for categories-list@mlist.mta.ca; Mon, 04 Oct 2010 09:07:10 -0300 In-Reply-To: <20101001092434.GA9359@mathematik.tu-darmstadt.de> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6279 Archived-At: On Fri, Oct 1, 2010 at 2:24 AM, Thomas Streicher wrote: > Well, but then we can work with categories with a chosen structure, e.g. > chosen products; that's what is recommended in the Elephant and it looks to > me as close to practice; it's very unlikely to have a nonconstructive proof > of existence of products. Unlikely, but it happens occasionally. For instance, if the morphisms of a category are equivalence classes, then a "construction" of equalizers or pullbacks might require choosing representatives; this actually happens at one point in the Elephant. The "small complete categories" in realizability topoi are also generally "weakly complete," in the sense that "every small diagram has a limit" is true in the internal logic, but not "strongly complete" in the sense that there exist internal limit-assigning (non-ana) functors. The property of "strong completeness" is also not in general preserved or reflected by weak equivalence functors. One can of course develop a theory which distinguishes between weak and strong equivalence and weak and strong completeness, but I think it's reasonable to call it "unfamiliar" to most category theorists. It feels to me like trying to do work constructively with topological spaces and therefore having to talk about [0,1] being complete and totally bounded but not compact, instead of realizing that when working constructively, one should really replace topological spaces with locales. Just as in set theory, no axiom of choice is necessary to define a function whose values are individually uniquely determined, it seems to me that no axiom of choice should be necessary in category theory to define a functor whose values are individually uniquely determined up to unique isomorphism. But obviously this is a subjective judgement. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]