Re: The internal logic of a topos

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

Dear Vaughan,

I don't think one can give a straight answer to this question: it all
depends on what you mean by `the logic of a topos'. I presume you're
thinking of the fact that, in Set, any function 2^n --> 2 is a polynomial
in the Boolean operations (i.e., is the interpretation of some n-ary term
in the theory of Boolean algebras). One could ask the same question about
a general topos, with `Heyting' replacing `Boolean'; but the answer
would mostly be `no', even for Boolean toposes. On the other hand, one
might well *define* `the internal logic of a topos' as meaning the
collection of all natural operations on subobjects -- that is, the
collection of all morphisms \Omega^n --> \Omega.

Incidentally, there is nothing unnatural or counterintuitive about `the
notion of internal Heyting algebra: it is a very natural consequence of
the definition of a subobject classifier, see A1.6.3 in the Elephant.

Peter

On Sun, 16 Mar 2008, Vaughan Pratt wrote:

> As I understand the internal logic of a topos it consists of certain
> morphisms from finite powers of Omega to Omega.  In the case of Set it
> consists of all such morphisms.  For which toposes is this not the case,
> and for those how are the morphisms that do belong to the internal logic
> best characterized?
>
> I do hope it's not necessary to start from the notion of an internal
> Heyting algebra, that sounds so counter to mathematical practice and
> intuition.
>
> If the internal logic consists of precisely those morphisms preserved by
> geometric morphisms this will give me the necessary motivation to go to
> the mats with geometry.
>
> Vaughan
>
>
>