From: "Martin H. Escardo" <Martin.H.Escardo@ens.fr>
To: categories@mta.ca
Subject: Re: Freyd's couniversal characterization of [0,1]
Date: Wed, 26 Jan 2000 15:32:49 +0100 (MET) [thread overview]
Message-ID: <200001261432.PAA00407@agaric.ens.fr> (raw)
In-Reply-To: <vkaaelt766d.fsf@localhost.localdomain>
Andrej Bauer writes:
> I can answer this for equilogical spaces.
>
> The functor F: Bi[Equ] ---> Bi[Equ] has a final coalgebra. It
> is the equilogical space (C, ~) where C is the Cantor space
>
> C = 2^N = infinite sequences of 0's and 1's
>
> and ~ is the equivalence relation defined by
>
> a ~ b iff r(a) = r(b)
>
> where r: C --> [0,1] is defined by
>
> r(a) = \sum_{k=0}^\infty a_k / 2^{k+1}
This is interesting in connection with some previous discussion in
this list on "definability" of the mid-point operation "by
coinduction".
As it is well known, the mid-point operation is not continuously
realizable via binary expansions with the Cantor topology.
(As Andrej Bauer and other people have mentioned signed-digit binary
expansions in this discussion, let me emphasize that, in contrast to
the above situation, all continuous functions [-1,1]^n->[-1,1] are
continuously realizable via signed-digit binary expansions with the
Cantor topology. Put in another way, the space 3^N = (3^N)^n is
projective over (the regular epimorphism) s:3^N->[-1,1], but the space
2^N is not projective over r:2^N->[0,1].)
Martin Escardo
next prev parent reply other threads:[~2000-01-26 14:32 UTC|newest]
Thread overview: 4+ messages / expand[flat|nested] mbox.gz Atom feed top
2000-01-24 19:14 Martin H. Escardo
2000-01-26 5:28 ` Andrej Bauer
2000-01-26 14:32 ` Martin H. Escardo [this message]
2000-01-27 12:25 ` Martin H. Escardo
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