Re: "Kantor dust"

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

On Sat, 31 Jan 2009, Vaughan Pratt wrote:

> I'm not aware of any reason why a topos with a Cantor set object K has
> to also have a natural number object N, though I'm not enough of a topos
> hacker myself to know how to produce one with K but without N (but would
> be happy to learn).  Does such a topos exist in nature?  And what can be
> said of the free topos with Cantor set object?
>
A topos with a Cantor set object (i.e. a final coalgebra for FX = X+X)
necessarily has a natural number object. Observe that the Cantor set K
necessarily has a point (since 1 has an F-coalgebra structure), so the
isomorphism K+K \cong K yields a monomorphism K \to K and a point
disjoint from its image. From there on, use Corollary D5.1.3 in the
Elephant.

Peter Johnstone