Re: "Kantor dust"

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

Vaughan Pratt wrote in part:

>Toby Bartels claims it's obvious, in which case there should be a short
>construction of a nonconstant function from N^N (as a final coalgebra)
>to 2 (or to N, the codomain Toby spoke of) in the topos of sheaves on R.
> Toby, what is it?

An element s of N^N as a final coalgebra of X |-> N x X
decomposes into an element s0 of N and an element s+ of N^N.
In this notation, the easiest of the desired functions is s |-> s0.
Other functions N^N -> N include s |-> s+0, s++0, ...; there are more.
This can be done in any topos (actually in more categories than that).
If any of these is constant, then N is a terminal object,
so if it's an NNO (as it should be to justify the notation N^N)
we are in the terminal category (the inconsistent topos).


--Toby Bartels