Re: exponentials of perfect maps and local homeomorphisms

[Steven Vickers <S.J.Vickers@cs.bham.ac.uk>]

Dear Claudio,

For locales, the questions about bundles can be reduced to results about
spaces (= locales) internal in the topos of sheaves over the base space.

By "bundle" I just mean "map" (always assumed continuous), but it is
useful to have the alternative word because we take an alternative
perspective in which the bundle is thought of as a space (the fibre)
continuously parametrized by a base point. Then there are correspondences
between properties of the bundle as map and its properties "fibrewise",
the
latter corresponding to properties of the internal space.

   local homeomorphism <-> fibrewise discrete
   perfect <-> fibrewise compact regular (or completely regular? They
become inequivalent in a topos.)

Now the localic version of your questions can be reduced to the
topos-validity of questions about exponentiation of locales.

If X compact regular and Y discrete,

(1) Is Y^X discrete?
(2) Is X^Y compact regular?

I'm reasonably sure (2) is known, but I can't think of the references
offhand. The hard part is Tychonoff, which is topos-valid localically.

As for (1), I'm inclined to believe it but can't think of reasons or
references at the moment.

The "localic bundle theorem" that relates bundles over a topos to locales
in it is fundamental and goes back - I believe - to Fourman, Scott, Joyal,
Tierney. It is a result that depends on using point-free rather than
point-set topology. All the same, I guess there's a chance of exploiting
it
to prove results about classical point-set bundles.

"Topos-valid locale" sounds scary - Do we have to use lattices instead of
spaces? Do we have to know all about toposes? A lot of my work has been
about the beneficial effect of using the "geometric" fragment of
topos-valid logic, using which one can reason with the points of a
point-free space.

All the best,

Steve Vickers.

On Mon, 15 Aug 2011 20:03:22 +0100 (BST), claudio pisani
<pisclau@yahoo.it> wrote:
> Dear Steve, by "perfect" I mean proper and separated (proper = closed
with compact fibers
> = stably closed). I prefer the topological setting because I am not very
acquainted with locales,
> but any result which seems to confirm the conjecture is good to me.
> So you are saying that (apart from the separation condition) the second
question has an
> affirmative answer in Loc. Any idea about the first one?
>
> Regards, Claudio


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