Re: Stone duality for generalized Boolean algebras

Re: Stone duality for generalized Boolean algebras

[Jeff Egger]

Hi Andrej,

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".  [As it turns out: every (not necessarily unital) C*-algebra has an "approximate unit" (even a canonical one); and, for a *-homomorphism between unital  C*-algebras, preserving the approximate unit is equivalent to preserving the unit.]  

In any event, I have found it (paradoxically) illuminating to think of: locally compact Hausdorff spaces and proper maps as a subcategory, via the one-point compactification functor (here denoted ( )+1), of  compact Hausdorff spaces; and, (not necessarily unital) C*-algebras as a subcategory, via the free functor (also denoted ( )+1), of unital C*-algebras.  Since C(X+1)=C_0(X)+1 holds at the level of objects (where = means isomorphic), it remains to reverse-engineer the correct classes of arrows  in order to piggyback the desired statement off the usual duality theorem.  Of course, it's also possible to consider the b.o./f.f. factorisations of the two ( )+1 functors: that results in some class of partial maps on the topological side, as you suggest.

I expect that something similar happens in the case of Stone duality and GBAs.  Hope this helps!

Cheers,
Jeff.

--- On Fri, 1/21/11, Andrej Bauer <andrej.bauer@andrej.com>  wrote:

> From: Andrej Bauer <andrej.bauer@andrej.com>
> Subject: categories: Stone duality for generalized Boolean algebras
> To: "categories list" <categories@mta.ca>
> Received: Friday, January 21, 2011, 8:19 AM
> The well known Stone duality says
> that there is an equivalence between
> Boolean algebras (BA) and the opposite of Stone spaces and
> continuous
> maps. Here a Stone space is a Hausdorff zero-dimensional
> compact
> space. Furthermore, Boolean algebras correspond to Boolean
> rings with
> unit.
> 
> How exactly does this extend to generalized Boolean
> algebras? A
> generalized Boolean algebra (GBA) is an algebra with 0,
> binary meet,
> binary join, and relative complement in which meets
> distribute over
> joins. Equivalently it is a Boolean ring (possibly without
> a unit). I
> have seen it stated that the dual to these are (the
> opposite of)
> locally compact zero-dimensional Hausdorff spaces and
> proper maps,
> e.g., it is stated in Benjamin Steinberg: "A groupoid
> approach to
> discrete inverse semigroup algebras", Advances in
> Mathematics 223
> (2010) 689-727.
> 
> Another source to look at is Givant & Halmos
> "Introduction to Boolean
> algebras", but there  this material is covered in exercises
> and the
> duality is stated separately for objects and for morphisms,
> and I
> can't find an exercise that treats the morphisms, so I
> wouldn't count
> that as a reliable  reference. Stone's original work does
> not seem to
> speak about morphisms very clearly (to me).
> 
> Unless I am missing something very obvious, it cannot be
> the case that
> GBA's correspond to locally compact 0-dimensional Hausdorff
> spaces and
> proper maps, for the following  reason.
> 
> The space which corresponds to the GBA 2 = {0,1} is the
> singleton. The
> space which corresponds to the four-element GBA 2 x  2 is
> the two-point
> discrete space 2. There are _four_ GBA homomorphisms from 2
> to 2 x 2
> (because a GBA homomorphism preserves 0 but it need not
> preserve 1),
> but there is only one continuous map from 2  to 1. Or to put
> it another
> way, there are _four_ ring homomorphisms from Z_2 to Z_2 x
> Z_2
> (because they need not preserve 1), but there is only one
> continuous
> map from 2 to 1. So, either the spectrum of a GBA is not
> what I think
> it is (namely maximal ideals), or we should be taking a
> more liberal
> notion of maps on the topological side. For example, there
> are _four_
> partial maps from 2 to 1.
> 
> If someone knows of a reliable reference, one would be
> much
> appreciated. I won't object to a direct proof of duality
> either.
> 
> With kind regards,
> 
> Andrej
> 
> 
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
> 





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Re: Stone duality for generalized Boolean algebras

[Eduardo J. Dubuc]

We should all listen:

Bill Lawvere just illuminate us again with the following:

He wrote:

"Category theory can sometimes discern the germ of nontrivial in the trivial."

Categorically thinking mathematicians should be aware of this, if not, they
just remain in the surface of category theory

e.d.


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Re: Stone duality for generalized Boolean algebras

[F. William Lawvere]

Dear George You ask

would any categorically
thinking

mathematician say that the
category of pointed sets needs further description as the category of  sets  and partial maps?

 

In 1969-1970, recalling

a) the preorigins of sheaf
theory a hundred years ago in the still-non-trivial problem of extending
of partial maps in analysis and topology, and

b) desirous of an instrument for describing
sheafication in finitely algebraic terms

 

Myles and I proposed Ytilda->Omega as one of
the two axioms for an elementary theory of toposes (the other being the Pi
right adjoint to pullback; applying Grothendieck’s method of relativization
using any given model U of those axioms, the 2-category of U-Toposes was
obtained thus capturing precisely the original SGA4 notion by choice of U).  Of
course these axioms were soon shown to be deducible from special cases,

but the importance of classifying partial maps
X..->Y remains.

 

The fact that this construction reduces to
Y+1->1+1 in sets misled some recursion theorists to try to represent partial
recursive maps that way, but

the categorically thinking mathematician noted
that in their category, the complement of the domain is typically not a
subobject of X. As Phil Mulry showed with his Recursive Topos, subobjects  of
Omega provide a precise specification of degrees of complication for the
inclusion of the domain of definition by pulling back along X->Omega.
(Although it would seen that Hilbert schemes as subobjects of Omega might
provide similar representability, that apparently has not been pursued).

 

In the Boolean case Y+1 can be viewed as an action of the
two-element monoid of

idempotents (the instrument for analysis of  objects in in a protomodular category),
in other words the category of partial maps can be embedded in a topos.Over  a general topos, that can be replaced by actions of
Omega as a multiplicative monoid .

 

  Of course
partial maps are special binary relations, but of a qualitatively special  kind
that requires its own status. In Cat, if replace subobjects by discrete
opfibrations, the analogous “partial maps”  (”machines”) turn out  to be representable but give rise
analogously to special distributors.

 

Peter Freyd’s dictum has a dialectical companion. Category theory
can sometimes discern the germ of nontrivial in the trivial.

 





> From: janelg@telkomsa.net
> To: fejlinton@usa.net; categories@mta.ca
> Subject: categories: Re: Stone duality for generalized Boolean algebras
> Date: Mon, 24 Jan 2011 23:15:17 +0200
> 
> 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...)
> 

...


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Re: Stone duality for generalized Boolean algebras

[Eduardo J. Dubuc]

Dear George, I do not think that the answer to Andrej Bauer question is
trivial, as a matter of fact it is not trivial at all. Andrej or any other
able mathematician does not have to know that GBA is equivalent to BA/2.
Furthermore, I feel that Fred Linton bringing into consideration the analogy
with C* algebras is pointing to Andrej some relevant mathematical questions.

greetings  e.d.

George Janelidze wrote:
> 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.
>

...


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Re: Stone duality for generalized Boolean algebras

[George Janelidze]

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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Re: Stone duality for generalized Boolean algebras

[Fred E.J. Linton]

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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Re: Stone duality for generalized Boolean algebras

[Yoshihiro Maruyama]

Dear Andrej,

The following paper seems to provide some details of a duality between
generalized Boolean algebras and locally compact zero-dimensional
Hausdorff spaces:

H. P. Doctor, The categories of Boolean lattices, Boolean rings, and
Boolean spaces, Canadian Mathematical Bulletin 7 (1964) 245-252.

It would be better to be careful of the morphism part of the duality
in the paper, in which continuous proper maps of locally compact
zero-dimensional Hausdorff spaces correspond to "proper" homomorphisms
of generalized Boolean algebras (and, as you noted, do not correspond
to all homomorphisms).

Since the category of Boolean algebras is a full subcategory of GBA
and their proper homomorphisms, the duality in the above paper is a
generalization of Stone duality between Boolean algebras and compact
zero-dimensional Hausdorff spaces. As you suggested, another way would
be to extend morphisms of spaces (if we place emphasis on algebras
rather than spaces).

I wish this would be useful for you.
(Sorry if I misunderstand anything.)

With best regards,
Yoshihiro


**********************************************************
Yoshihiro Maruyama
Department of Humanistic Informatics
Kyoto University
E-mail: maruyama@i.h.kyoto-u.ac.jp
Webpage: http://researchmap.jp/ymaruyama/
**********************************************************



2011/1/21 Andrej Bauer <andrej.bauer@andrej.com>:
> The well known Stone duality says that there is an equivalence between
> Boolean algebras (BA) and the opposite of Stone spaces and continuous
> maps. Here a Stone space is a Hausdorff zero-dimensional compact
> space. Furthermore, Boolean algebras correspond to Boolean rings with
> unit.
>
> How exactly does this extend to generalized Boolean algebras? A
> generalized Boolean algebra (GBA) is an algebra with 0, binary meet,
> binary join, and relative complement in which meets distribute over
> joins. Equivalently it is a Boolean ring (possibly without a unit). I
> have seen it stated that the dual to these are (the opposite of)
> locally compact zero-dimensional Hausdorff spaces and proper maps,
> e.g., it is stated in Benjamin Steinberg: "A groupoid approach to
> discrete inverse semigroup algebras", Advances in Mathematics 223
> (2010) 689-727.
>
> Another source to look at is Givant & Halmos "Introduction to Boolean
> algebras", but there this material is covered in exercises and the
> duality is stated separately for objects and for morphisms, and I
> can't find an exercise that treats the morphisms, so I wouldn't count
> that as a reliable reference. Stone's original work does not seem to
> speak about morphisms very clearly (to me).
>
> Unless I am missing something very obvious, it cannot be the case that
> GBA's correspond to locally compact 0-dimensional Hausdorff spaces and
> proper maps, for the following reason.
>
> The space which corresponds to the GBA 2 = {0,1} is the singleton. The
> space which corresponds to the four-element GBA 2 x 2 is the two-point
> discrete space 2. There are _four_ GBA homomorphisms from 2 to 2 x 2
> (because a GBA homomorphism preserves 0 but it need not preserve 1),
> but there is only one continuous map from 2 to 1. Or to put it another
> way, there are _four_ ring homomorphisms from Z_2 to Z_2 x Z_2
> (because they need not preserve 1), but there is only one continuous
> map from 2 to 1. So, either the spectrum of a GBA is not what I think
> it is (namely maximal ideals), or we should be taking a more liberal
> notion of maps on the topological side. For example, there are _four_
> partial maps from 2 to 1.
>
> If someone knows of a reliable reference, one would be much
> appreciated. I won't object to a direct proof of duality either.
>
> With kind regards,
>
> Andrej
>


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Re: Stone duality for generalized Boolean algebras

[George Janelidze]

The answer is trivial: The category of Boolean rings without 1 (which means
"possibly without 1" of course) is equivalent to BA/2, which, by Stone
duality, is dually equivalent to the category of pointed Stone spaces.

However thinking of "partial maps" was not too bad since, say, the category
of pointed sets is equivalent to the category of sets with partial maps as
morphisms.

George Janelidze

--------------------------------------------------
From: "Andrej Bauer" <andrej.bauer@andrej.com>
Sent: Friday, January 21, 2011 3:19 PM
To: "categories list" <categories@mta.ca>
Subject: categories: Stone duality for generalized Boolean algebras

> The well known Stone duality says that there is an equivalence between
> Boolean algebras (BA) and the opposite of Stone spaces and continuous
> maps. Here a Stone space is a Hausdorff zero-dimensional compact
> space. Furthermore, Boolean algebras correspond to Boolean rings with
> unit.
>
> How exactly does this extend to generalized Boolean algebras? A
> generalized Boolean algebra (GBA) is an algebra with 0, binary meet,
> binary join, and relative complement in which meets distribute over
> joins. Equivalently it is a Boolean ring (possibly without a unit). I
> have seen it stated that the dual to these are (the opposite of)
> locally compact zero-dimensional Hausdorff spaces and proper maps,
> e.g., it is stated in Benjamin Steinberg: "A groupoid approach to
> discrete inverse semigroup algebras", Advances in Mathematics 223
> (2010) 689-727.
>
> Another source to look at is Givant & Halmos "Introduction to Boolean
> algebras", but there this material is covered in exercises and the
> duality is stated separately for objects and for morphisms, and I
> can't find an exercise that treats the morphisms, so I wouldn't count
> that as a reliable reference. Stone's original work does not seem to
> speak about morphisms very clearly (to me).
>
> Unless I am missing something very obvious, it cannot be the case that
> GBA's correspond to locally compact 0-dimensional Hausdorff spaces and
> proper maps, for the following reason.
>
> The space which corresponds to the GBA 2 = {0,1} is the singleton. The
> space which corresponds to the four-element GBA 2 x 2 is the two-point
> discrete space 2. There are _four_ GBA homomorphisms from 2 to 2 x 2
> (because a GBA homomorphism preserves 0 but it need not preserve 1),
> but there is only one continuous map from 2 to 1. Or to put it another
> way, there are _four_ ring homomorphisms from Z_2 to Z_2 x Z_2
> (because they need not preserve 1), but there is only one continuous
> map from 2 to 1. So, either the spectrum of a GBA is not what I think
> it is (namely maximal ideals), or we should be taking a more liberal
> notion of maps on the topological side. For example, there are _four_
> partial maps from 2 to 1.
>
> If someone knows of a reliable reference, one would be much
> appreciated. I won't object to a direct proof of duality either.
>
> With kind regards,
>
> Andrej


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Re: Stone duality for generalized Boolean algebras

[Fred E.J. Linton]

The Boolean rng counterpart to the Stone duality, identifying
Boolean algebras with the opposite of compact T2 0-dim'l spaces,
exploits the fact that the category of boolean rngs amounts to
the category of *augmented* Boolean algebras (the slice category
BA | 2 of 2-valued boolean homomorphisms from Boolean algebras) --
true because *kernel* gives an equivalence from latter to former --
hence is equivalent to the opposite of *pointed* compact T2 0-dim'l
spaces (and base-point-preserving continuous functions).

While the complement of the base point (in such a pointed space)
may be locally compact, that observation is far from functorial, 
so there's not much good any category of locally compact T2 0-dim'l
spaces will do you.

HTH. Cheers, -- Fred



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Stone duality for generalized Boolean algebras

[Andrej Bauer]

The well known Stone duality says that there is an equivalence between
Boolean algebras (BA) and the opposite of Stone spaces and continuous
maps. Here a Stone space is a Hausdorff zero-dimensional compact
space. Furthermore, Boolean algebras correspond to Boolean rings with
unit.

How exactly does this extend to generalized Boolean algebras? A
generalized Boolean algebra (GBA) is an algebra with 0, binary meet,
binary join, and relative complement in which meets distribute over
joins. Equivalently it is a Boolean ring (possibly without a unit). I
have seen it stated that the dual to these are (the opposite of)
locally compact zero-dimensional Hausdorff spaces and proper maps,
e.g., it is stated in Benjamin Steinberg: "A groupoid approach to
discrete inverse semigroup algebras", Advances in Mathematics 223
(2010) 689-727.

Another source to look at is Givant & Halmos "Introduction to Boolean
algebras", but there this material is covered in exercises and the
duality is stated separately for objects and for morphisms, and I
can't find an exercise that treats the morphisms, so I wouldn't count
that as a reliable reference. Stone's original work does not seem to
speak about morphisms very clearly (to me).

Unless I am missing something very obvious, it cannot be the case that
GBA's correspond to locally compact 0-dimensional Hausdorff spaces and
proper maps, for the following reason.

The space which corresponds to the GBA 2 = {0,1} is the singleton. The
space which corresponds to the four-element GBA 2 x 2 is the two-point
discrete space 2. There are _four_ GBA homomorphisms from 2 to 2 x 2
(because a GBA homomorphism preserves 0 but it need not preserve 1),
but there is only one continuous map from 2 to 1. Or to put it another
way, there are _four_ ring homomorphisms from Z_2 to Z_2 x Z_2
(because they need not preserve 1), but there is only one continuous
map from 2 to 1. So, either the spectrum of a GBA is not what I think
it is (namely maximal ideals), or we should be taking a more liberal
notion of maps on the topological side. For example, there are _four_
partial maps from 2 to 1.

If someone knows of a reliable reference, one would be much
appreciated. I won't object to a direct proof of duality either.

With kind regards,

Andrej


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]