Re: Generalization of Browder's F.P. Theorem?

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

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