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