Re: preservation of exponentials

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

On Tue, 18 Feb 2003, Thomas Streicher wrote:

> Recently when rereading an old paper I came across a passage insinuating
> that every finite limit preserving full and faithful functor between toposes
> does also preserve exponentials.
> I am sceptical because I don't see any obvious reason for it. It is certainly
> wrong for ccc's (a counterexample is the inclusion of open sets of reals into
> powersets of reals). On the other hand Yoneda functors and direct image parts
> of injective geom morphs do preserve exponentials.
> So I was thinking of inverse image parts of connected geom.morph.'s.
> Of course, \Delta : Set -> Psh(C) for a connected C does preserve exponentials.
> What about Delta : Set -> Sh(X) for X connected but not locally connected,
> e.g. take for X Cantor space with a focal point added?
>
If a full and faithful functor between ccc's has a left adjoint which
preserves binary products, then it preserves exponentials (Elephant,
A1.5.9(ii)). In the absence of a left adjoint, the result is not true
in general: Set --> Sh(X) for X connected but not locally connected
gives a counterexample, as you suggest, and so does the inclusion
(continuous G-sets) --> (arbitrary G-sets) for a topological group G.

Peter Johnstone