exponentials of perfect maps and local homeomorphisms

exponentials of perfect maps and local homeomorphisms

[claudio pisani]

Dear categorists (and topologists),

It is known (Clementino, Hofmann, Tholen, Richter, Niefield...) that perfect
maps p:X->Y are exponentiable in Top/Y, and that the same holds for local
homeomorphisms h:X->Y.

Question: is it true that 1) p=>h is a local homeomorphism) h=>p is a
perfect map?

The conjecture is suggested by the following observations: A) it holds
both for Y = 1 (compact and discrete space) and for subspaces = inclusion
(closed and open parts) B) the analogy between perfect maps and local
homeomorphisms with discrete (op)fibrations (via convergence or other
considerations) and the fact that 1) and 2) above hold in Cat/Y: if p
is a discrete fibration and h is a discrete opfibration then p=>h is
itself a discrete opfibration (and conversely).

Best regards,
Claudio


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Re: exponentials of perfect maps and local homeomorphisms

[Steven Vickers]

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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