Re: Stone duality for generalized Boolean algebras

["George Janelidze" <janelg@telkomsa.net>]

Dear Fred,

Please forgive me, but let us distinguish between serious questions and
trivialities:

Andrej Bauer asked:

"...How exactly does this extend to generalized Boolean algebras?..."

And the answer is trivial (without quotation marks): The category GBA of
what he called generalized Boolean algebras is dually equivalent to the
category 1\STONE of pointed Stone spaces. This follows from Stone duality
(since GBA is equivalent to BA/2), but also extends it: just as BA is a
non-full subcategory of GBA, STONE can be considered as a non-full
subcategory of 1\STONE via the functor that adds base points. And this way
the dual equivalence between GBA and 1\STONE indeed extends the Stone
duality.

Although your first message about BA/2 was written after mine, I am sure you
know these things (you probably knew them before I knew the definition of a
category...)

Anyway, what I called the trivial answer is the full answer and we don't
need the Gelfand duality to motivate or explain it (even though the analogy
is correct).

Thinking further about partial maps simply means not thinking categorically:
look at the finite sets (or just sets) - would any categorically thinking
mathematician say that the category of pointed sets needs further
description as the category of finite sets and partial maps?

On the other hand, the "partial-map-version" of pointed Stone spaces is a
serious question even though it would not do any good to the question above.
Well, maybe the answer is known, but not to me. Naively, I don't think it is
as hopeless as you say. The reason is:

Let us take a pointed Stone space (X,x), and try to recover it from X-{x} (I
write "-" for the set-theoretic difference since I used "\" for something
else). Let us think about this in terms of ultrafilter convergence. Apart
from the principal ultrafilter generated by {x} every ultrafilter on X is of
the form T(i)(U), where T is the ultrafilter monad, i the inclusion map from
X-{x} to X, and U an ultrafilter on X-{x}. Knowing the topology of X-{x}, I
can recover the topology on X by requiring T(i)(U) to converge to the same
point in X-{x} as U and to converge to x if U does not converge to any
point. This indeed recovers X since every ultrafilter on a compact Hausdorff
space converges to a unique point (note also that T(i) is injective since i
is a split mono in SETS whenever X-{x} is non-empty).

I hope somebody on this mailing list will tell us that what I am saying is a
part of a well-known story and will give a reference, or am I missing
something?

What do you say?

Greetings - George

--------------------------------------------------
From: "Fred E.J. Linton" <fejlinton@usa.net>
Sent: Sunday, January 23, 2011 5:38 PM
To: <categories@mta.ca>
Cc: "Jeff Egger" <jeffegger@yahoo.ca>
Subject: categories: Re: Stone duality for generalized Boolean algebras

> On Sat, 22 Jan 2011 05:11:26 PM EST, Jeff Egger <jeffegger@yahoo.ca>
> responded
> to Andrej Bauer <andrej.bauer@andrej.com> as follows:
>
>> There is an analogous problem when trying to "extend" Gelfand duality to
>> locally compact Hausdorff spaces and (not necessarily unital)
>> C*-algebras:
>> everything works fine on the object level, but there are many  (not
>> necessarily unital) *-homomorphisms C_0(1) --> C_0(2), but only one
>> continuous map 2 --> 1.  The standard solution, IIRC, is to restrict the
>> class of *-homomorphisms to those which "preserve the approximate
> unit". ...
>
> Another approach more closely resembles the "solution" that George
> Janelidze and I have pointed out for the Boolean problem. To sketch it,
> let me temporarily borrow the old Gelfand-Naimark terminology "normed
> ring"
> for commutative C*-algebras with unit, and use "normed rng" for their
> not-necessarily-unital counterparts.
>
> As in the Boolean setting, then, "normed rngs" is, to within equivalence,
> augmented "normed rings" (that is, the slice category "normed rings"|'C',
> where C is the "coefficient ring" -- probably the real or the complex
> field in most applications), whence as opposite to "normed rngs" one
> immediately deduces the category of pointed compact Hausdorff spaces
> (and *all* continuous base-point-preserving functions).
>
> And, as there also, while the complement of the base point in such a
> space may be locally compact, the passage to that complement is, again,
> far from functorial -- unless one is willing either to restrict one's
> attention, among maps of pointed compact spaces, to those that send
> *only* the base point to the base point, or to extend one's attention to
> certain only partially defined functions as maps of locally compact
> spaces.
>
> Cheers, -- Fred
>
>


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