preservation of exponentials

preservation of exponentials

[Thomas Streicher]

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?

Thomas Streicher

Re: preservation of exponentials

[Prof. Peter Johnstone]

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