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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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...)
... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (7)
-
Andrej Bauer -
Eduardo J. Dubuc -
F. William Lawvere -
Fred E.J. Linton -
George Janelidze -
Jeff Egger -
Yoshihiro Maruyama