From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2105 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Generalization of Browder's F.P. Theorem? Date: Thu, 16 Jan 2003 23:00:44 +0000 (GMT) Message-ID: References: <3E26BBF3.2D413C6A@cs.bham.ac.uk> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018415 2510 80.91.229.2 (29 Apr 2009 15:20:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:20:15 +0000 (UTC) To: CATEGORIES LIST , Original-X-From: rrosebru@mta.ca Fri Jan 17 11:43:39 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Jan 2003 11:43:39 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18ZYbR-0005iS-00 for categories-list@mta.ca; Fri, 17 Jan 2003 11:40:17 -0400 In-Reply-To: <3E26BBF3.2D413C6A@cs.bham.ac.uk> X-Scanner: exiscan for exim4 (http://duncanthrax.net/exiscan/) *18ZJ0C-0005sw-00*vyUCLebRbnw* Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 30 Original-Lines: 40 Xref: news.gmane.org gmane.science.mathematics.categories:2105 Archived-At: On Thu, 16 Jan 2003, Steven J Vickers wrote: > > I'm intrigued by Peter McBurney's question [below]. It looks rather like a > question of the constructive content of Brouwer's fixed point theorem. > > Suppose S is (homeomorphic to) an n-cell. Then in the internal logic of the > topos of sheaves over [0,1], f is just a continuous endomap of S. If > Brouwer's theorem were constructively true then f would have a fixpoint, > and that would come out as a continuous section of the projection [0,1]xS > -> [0,1]. More precisely, it would be a map g: [0,1] -> S such that f(x, > g(x)) = g(x) for all x. If this existed then the set A = {(x, g(x))| x in > [0,1]} would be as required. > > However, the proof of Brouwer that I've seen is not constructive - it goes > by contradiction. So maybe the requirements on A are a way of getting > constructive content in Brouwer's result. > > What is known constructively about Brouwer's fixed point theorem? > > Steve Vickers. > Similar thoughts had occurred to me. Brouwer's theorem is clearly not constructive, since it doesn't hold (even locally) continuously in parameters (consider a path in the space of endomaps of [0,1] passing through the identity, where the fixed point `flips' from one end of the interval to the other as it does so). However, Browder's result would seem to suggest that the `locale of fixed points of f' (that is, the equalizer of f and the identity in the category of locales) might be consistent (that is, `inhabited') in general, even though it may not have any points. It's certainly conceivable that that might be true constructively, though I can't see how to prove it -- but it isn't the full content of Browder's theorem. Peter Johnstone