Categorical Extension of Dualities: From Stone to de Vries and Beyond, I
aa r X i v : . [ m a t h . GN ] J un Extensions of dualities and a newapproach to the de Vries duality
G. Dimov, E. Ivanova-Dimova and W. Tholen ∗ Faculty of Mathematics and Inf., Sofia University, 5 J. Bourchier Blvd., 1164 Sofia, BulgariaDept. of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada
Abstract
We prove a general categorical theorem for the extension of dualities. Apply-ing it, we present new proofs of the de Vries Duality Theorem for the category
CHaus of compact Hausdorff spaces and continuous maps, and of the recentBezhanishvili-Morandi-Olberding Duality Theorem which extends the de Vriesduality to the category
Tych of Tychonoff spaces and continuous maps. In theprocess of doing so we obtain new duality theorems for the categories
CHaus and
Tych . The celebrated Stone Duality Theorem [36] shows that the entire information abouta zero-dimensional compact Hausdorff space (=
Stone space ) X is, up to homeomor-phism, contained in its Boolean algebra CO( X ) of all clopen (= closed and open)subsets of X . Likewise, all information about the continuous maps between two such ∗ The first author acknowledges the support by Bulgarian National Science Fund, contract no.DN02/15/19.12.2016. The second author acknowledges the support by the Bulgarian Ministry ofEducation and Science under the National Research Programme Young scientists and postdoctoralstudents approved by DCM Keywords: compact Hausdorff space, continuous map, irreducible map, projective cover, Stonespace, (complete) Boolean algebra, (normal) contact algebra, Stone duality, de Vries duality, Fe-dorchuk homomorphism, de Vries’ transformation, duality . E-mail addresses: [email protected]fia.bg, [email protected]fia.bg, [email protected]. X and Y is encoded by the Boolean homomorphisms between the Booleanalgebras CO( Y ) and CO( X ). It is natural to ask whether a similar result holds for allcompact Hausdorff spaces and continuous maps between them. The first candidatefor the role of the Boolean algebra CO( X ) under such an extension seems to be theBoolean algebra RC( X ) of all regular closed subsets of a compact Hausdorff space X (or, its isomorphic copy RO( X ), which collects all regular open subsets of X ), butit fails immediately since, as is well-known, RC( X ) is isomorphic to RC( EX ), where EX is the absolute of X . However, in 1962, de Vries [15] showed that, if we regardthe Boolean algebra RC( X ) together with the relation ρ X on RC( X ), defined by F ρ X G ⇔ F ∩ G = ∅ , then the pair (RC( X ) , ρ X ) determines uniquely (up to homeomorphism) the com-pact Hausdorff space X . Moreover, with the help of some special maps between(RC( X ) , ρ X ) and (RC( Y ) , ρ Y ), where X and Y are compact Hausdorff spaces, onecan reconstruct all continuous maps between Y and X . De Vries gave an algebraicdescription of the pairs (RC( X ) , ρ X ) as pairs ( A, C ), formed by a complete Booleanalgebra A and a relation C on A , satisfying some axioms, and he also described alge-braically the needed special maps of such pairs. In this way he obtained the category DeV and its dual equivalence with the category
CHaus of compact Hausdorff spacesand continuous maps. In fact, de Vries did not use the relation ρ X as mentioned above,but its “dual”, that is, the relation F ≪ X G , defined by ( F ≪ X G ⇔ F ( − ρ X ) G ∗ )(with − ρ X complementary to ρ X and G ∗ denoting the Boolean negation of G inRC( X )) and called the non-tangential inclusion ; equivalently, F ≪ X G ⇔ F ⊆ int X ( G ) . Now known as de Vries algebras , he originally called the abstract pairs ( A, ≪ ) compin-gent algebras . The axioms for the relation C (respectively, ≪ C ) on A are preciselythe axioms for Efremoviˇc proximities [23], with only one exception: instead of Efre-moviˇc’s separation axiom, which refers to the points of the space in question, de Vriesintroduced what is now called the extensionality axiom (see [20, Lemma 2.2, p.215]for a motivation for this terminology). Since Efremoviˇc proximities are relations onthe Boolean algebra (P( X ) , ⊆ ) of all subsets of a set X , de Vries algebras may beregarded as point-free generalizations of the Efremoviˇc proximities.Nowadays the pairs ( A, C ), where A is a Boolean algebra and C is a proximity-type relation on A , attract the attention not only of topologists, but also of logiciansand theoretical computer sciencists. Amongst the many generalizations of de Vriesalgebras, the most popular ones are the so-called RCC systems (Region ConnectionCalculus) of Randell-Cui-Cohn [34]. Their generalizations include the contact algebras (introduced in [20, 21]), which are point-free analogues of the ˇCech proximity spaces,and precontact algebras , defined independently and almost simultaneously, but in acompletely different form, by S. Celani [12] (for the needs of logic) and by I. D¨untschand D. Vakarelov [22] (for the needs of theoretical computer science). These andthe RCC systems are very useful notions in the foundations of artificial intelligence,geographic information systems, robot navigation, computer-aided design, and more2see [13], [27] or [39] for details), as well as in logic, specificly in spatial logics [3](called sometimes logics of space ).A relation C on a Boolean algebra A , which satisfies the de Vries axioms (cor-responding to the relation ρ X above), is called a normal contact relation , and thepair ( A, C ) then becomes a normal contact algebra (briefly, an NCA, [20]). In otherwords, the de Vries algebras “in ρ X -form” are precisely the complete NCAs. De Vries[15] noted that his dual equivalence Ψ a : DeV −→ CHaus is an extension of therestriction T : CBool −→ ECH of Stone’s dual equivalence S a : Bool −→ Stone ;here
Bool denotes the category of Boolean algebras and Boolean homomorphisms,and
CBool is its full subcategory of complete Boolean algebras;
Stone is the cat-egory of Stone spaces and continuous maps, and
ECH denotes its full subcategoryof extremally disconnected compact Hausdorff spaces. Therefore, the objects of thecategory
DeV are precisely “ the structured CBool -objects ( A, C ) ”. Using the de Vries duality, in [7, Theorem 8.1(1)] Bezhanishvili proved that, if A is a complete Boolean algebra, then there exists a bijective correspondence betweenthe set of all normal contact relations C on A and the set of all (up to homeomorphism)Hausdorff irreducible images of the Stone dual S a ( A ) of A . Hence, the objects of deVries’ category DeV may be regarded as pairs ( A, p ) , where A is a CBool -object and p : S a ( A ) −→ X is an irreducible map onto a Hausdorff space X , so that p is a special CHaus -morphism; in fact, p is a projective cover of X . With the structure of theobjects presented in map form, we are ready to formulate the principal problem ofthis paper in categorical terms.Let T : A −→ B be a dual equivalence between two categories A and B , and B be a full subcategory of a category C . Then it is not at all surprising that one canconstruct a category D containing A as a full subcategory, and a dual equivalence˜ T : D −→ C extending T along the inclusion functors I and J , as in the diagram D ˜ T / / CA ?(cid:31) J O O T / / B . ?(cid:31) I O O Inside C , one may simply replace B by A and adjust the composition using the dualequivalence T to obtain the category D ! This ad-hoc procedure, however, does notmake for a naturally described category D , since the definition of the hom-sets of D changes with the two types of objects involved. The principal goal of this paperis therefore to model the objects of a suitable extension category D of A duallyequivalent to C in a natural way, as A -objects provided with a structure that givesthem a strong algebraic flavour. Our comments on the de Vries duality suggest toconsider as objects of D the pairs ( A, p ), with A an A -object and p a “special” C -morphism with domain T ( A ). Being “special” may be described as lying in a givenclass P of C -morphisms satisfying suitable axioms, which suffice to establish a category D with a dual equivalence ˜ T : D −→ C .In [19] we presented a set of axioms on the class P and a construction of thecategory D , which allowed us to reproduce the Fedorchuk duality [25] from the generalsetting. The same construction will, however, not work for the principal target of this3aper, the de Vries duality [15]. The general reason for this lies in the fact that theformation of the projective cover, or of the injective hull of an object, cannot beextended functorially, in such a way that the chosen essential morphism from thecover, or into the hull, becomes part of a natural transformation (see [2]). However,by slightly weakening the axioms on P and substantially modifying the constructionof D we succeed in providing a general categorical template from which to derive thede Vries duality, and other known or new dualities.The paper is organized as follows. Section 2 summarizes all notions and termsthat are needed for our exposition. Section 3 gives the categorical extension result,as indicated in the previous paragraph. In Section 4 we give a review of mostlyknown facts concerning the de Vries duality, augmented by some results from [19]on which we rely heavily in our exposition. In Section 5, we derive the de VriesDuality Theorem from our general categorical Theorem 3.1. Introducing the notionsof Fedorchuk homomorphism and de Vries transformation (see Definitions 5.5 and 5.6),we define a new category
StoneDeV ; it has the same objects as the category
DeV ,but its morphisms are equivalence classes of Fedorchuk homomorphisms, where twoFedorchuk homomorphisms between the same two objects are equivalent when theirde Vries transformations coincide. With the help of Theorem 3.1, we prove that thecategory
StoneDeV is dually equivalent to the category
CHaus (see Theorem 5.9).Note that, in contrast to the de Vries category
DeV , the morphisms of the category
StoneDeV are equivalence classes of Boolean homomorphisms which preserve thenon-tangential inclusion ≪ , and their composition is a very natural one; however theyare sets of special Boolean homomorphisms. (Let us mention that in [18] anothercategory dually equivalent to the category CHaus was constructed; it has the sameobjects as the category
DeV , and its morphisms are multi-valued maps which may becomposed in a natural way.) After that, in Theorem 5.13, we show that the categories
StoneDeV and
DeV are isomorphic. Obviously, Theorems 5.13 and 5.9 imply thatthe categories
CHaus and
DeV are dually equivalent, obtaining in this way a newproof of de Vries’ Duality Theorem [15].In Section 6, we apply Theorem 3.3 (a dualization of Theorem 3.1) for presentinga new proof of the Bezhanishvili-Morandi-Olberding Duality Theorem [9]. Adaptingour general categorical result to the concrete situation, we first obtain a new dualitytheorem which extends the de Vries Duality Theorem to the category
Tych (seeTheorem 6.6). A crucial role in this process of adaptation plays our Proposition6.2 (a more general and still unpublished version of it was presented in [17]). Itpermits to define in a new way de Vries’ dual equivalence Ψ a (see Proposition 6.3).We obtain Bezhanishvili-Morandi-Olberding Duality Theorem from our new dualitytheorem using the Tarski duality between the category Set of sets and functions andthe category
CaBa of complete atomic Boolean algebras and suprema-preservingBoolean homomorphisms (see Theorem 6.13, Corollary 6.14 and Theorem 6.15).Our general references for unexplained notation are [1] for category theory, [24]for topology, and [29] for Boolean algebras.4
Preliminaries
Below we first recall the notions of contact algebra and normal contact algebra . Theycan be regarded as algebraic analogues of proximity spaces (see [23, 35, 11, 5, 31] forproximity spaces). Generally speaking, in this paper we work mainly with Booleanalgebras with supplementary structures on them. In all cases, we will say that thestructured Boolean algebra in question is complete if the underlying Boolean algebrais complete. Our standard notation for the operations of a Boolean algebra B isindicated by B = ( B, ∧ , ∨ , ∗ , , B isdenoted by ∗ , and that 0 and 1 denote the least and largest element in B , not excludingthe case 0 = 1. Definition 2.1. ([20]) A
Boolean contact algebra , or, simply, contact algebra (abbre-viated as CA), is a structure (
B, C ), where B is a Boolean algebra, and C a binaryrelation on B , called a contact relation , which satisfies the following axioms:(C1). If a = 0 then aCa .(C2). If aCb then a = 0 and b = 0.(C3). aCb implies bCa .(C4). aC ( b ∨ c ) if, and only if, aCb or aCc .Two contact algebras ( B, C ) and ( B ′ , C ′ ) are said to be isomorphic if there exists a CA-isomorphism between them, i.e. , a Boolean isomorphism ϕ : B −→ B ′ such that,for all a, b ∈ B , aCb if and only if ϕ ( a ) C ′ ϕ ( b ).With − C denoting the set complement of C in B × B , we shall consider twomore properties of contact algebras:(C5). If a ( − C ) b then a ( − C ) c and b ( − C ) c ∗ for some c ∈ B .(C6). If a = 1 then there exists b = 0 such that b ( − C ) a .A contact algebra ( B, C ) is called a
Boolean normal contact algebra or, briefly, a normal contact algebra (abbreviated as NCA) [15, 25] if it satisfies (C5) and (C6).(Note that if 0 = 1, then (C2) follows from the axioms (C4), (C3), and (C6).)The notion of normal contact algebra was introduced by Fedorchuk [25] underthe name Boolean δ -algebra , as an equivalent expression of the notion of compingentBoolean algebra by de Vries (see the definition below). We call such algebras normalbecause they form a subclass of the class of contact algebras which naturally arise inthe context of normal Hausdorff spaces (see [20]). Definition 2.2.
For a contact algebra (
B, C ) we define a binary relation ≪ C on B ,called non-tangential inclusion , by a ≪ C b if, and only if, a ( − C ) b ∗ . If C is understood, we shall simply write ≪ instead of ≪ C .5he relations C and ≪ are inter-definable. For example, normal contact alge-bras may be defined equivalently – and exactly in this way they were introduced underthe name of compingent Boolean algebras by de Vries in [15] – as a pair consisting ofa Boolean algebra B and a binary relation ≪ on B satisfying the following axioms:( ≪ a ≪ b implies a ≤ b .( ≪ ≪ ≪ a ≤ b ≪ c ≤ t implies a ≪ t .( ≪ a ≪ c and b ≪ c implies a ∨ b ≪ c .( ≪ a ≪ c then a ≪ b ≪ c for some b ∈ B .( ≪ a = 0 then there exists b = 0 such that b ≪ a .( ≪ a ≪ b implies b ∗ ≪ a ∗ .Indeed, if ( B, C ) is an NCA, then the relation ≪ C satisfies the axioms ( ≪
1) –( ≪ B, ≪ ), where B is a Boolean algebra and ≪ is abinary relation on B which satisfies ( ≪
1) – ( ≪ C ≪ by aC ≪ b if, and only if, a b ∗ ; then ( B, C ≪ ) is an NCA. Note that the axioms (C5) and (C6)correspond to ( ≪
5) and ( ≪ B and a binaryrelation ≪ on B subject to the axioms ( ≪
1) – ( ≪
4) and ( ≪ X . Let us start with some standard notations andconventions that we use throughout the paper. For a subset M of X , we denoteby cl X ( M ) (or simply cl( M )) the closure of M in X , and by int( M ) its interior.CO( X ) denotes the set of all clopen (= closed and open) subsets of X ; trivially,(CO( X ) , ∪ , ∩ , \ , ∅ , X ) is a Boolean algebra. RC( X ) (resp., RO( X )) denotes the set ofall regular closed (resp., regular open) subsets of X ; recall that a subset F of X is saidto be regular closed (resp., regular open ) if F = cl(int( F )) (resp., F = int(cl( F )))).Note that in this paper (unlike in [24]) compact spaces are not assumed to beHausdorff. Example 2.3.
For a topological space X , the collection RC( X ) becomes a completeBoolean algebra under the operations F ∨ G df = F ∪ G, F ∧ G df = cl(int( F ∩ G )) , F ∗ df = cl( X \ F ) , df = ∅ , df = X. The infinite operations are given by the formulas _ { F γ | γ ∈ Γ } = cl( [ γ ∈ Γ F γ ) (= cl( [ γ ∈ Γ int( F γ )) = cl(int( [ γ ∈ Γ F γ ))) , ^ { F γ | γ ∈ Γ } = cl(int( \ { F γ | γ ∈ Γ } )) . ρ X on RC( X ) by setting, for each F, G ∈ RC( X ), F ρ X G if, and only if, F ∩ G = ∅ . Clearly, ρ X is a contact relation on RC( X ), called the standard contact relation of X . The complete CA (RC( X ) , ρ X ) is called a standard contact algebra . Note that,for F, G ∈ RC( X ), F ≪ ρ X G if, and only if, F ⊆ int X ( G ) . Thus, if X is a normal Hausdorff space then the standard contact algebra (RC( X ) , ρ X )is a complete NCA.Instead of looking at regular closed sets, we may, equivalently, consider regularopen sets. The collection RO( X ) of regular open sets becomes a complete Booleanalgebra by setting U ∨ V df = int(cl( U ∪ V )) , U ∧ V df = U ∩ V, U ∗ df = int( X \ U ) , df = ∅ , df = X, and ^ i ∈ I U i df = int(cl( \ i ∈ I U i )) (= int( \ i ∈ I U i )) , _ i ∈ I U i df = int(cl( [ i ∈ I U i )) , see [29, Theorem 1.37]. We define a contact relation D X on RO( X ) as follows: U D X V if, and only if, cl( U ) ∩ cl( V ) = ∅ . Then (RO( X ) , D X ) is a complete CA.The contact algebras (RC( X ) , ρ X ) and (RO( X ) , D X ) are CA-isomorphic viathe mapping ν : RC( X ) −→ RO( X ) defined by the formula ν ( F ) df = int( F ), for every F ∈ RC( X ). Example 2.4.
Let B be a Boolean algebra. Then there exist a largest and a smallestcontact relation on B ; the largest one, ρ l , is defined by aρ l b ⇐⇒ ( a = 0 and b = 0) , and the smallest one, ρ s , by aρ s b ⇐⇒ a ∧ b = 0 . Note that, for a, b ∈ B , a ≪ ρ s b ⇐⇒ a ≤ b ;hence a ≪ ρ s a , for any a ∈ B . Thus ( B, ρ s ) is a normal contact algebra.We will need the following definition and assertion from [20]:7 efinition 2.5. ([20]) For a contact algebra ( B, C ) one defines the relation R ( B,C ) on the set of all filters on B by f R ( B,C ) g if, and only if, f × g ⊆ C, (1)for all filters f, g on B . Proposition 2.6. (a) ([20, Lemma 3.5, p. 222])
Let ( B, C ) be a contact algebra.Then, for all a, b ∈ B , one has aCb if, and only if, there exist ultrafilters u, v in B such that a ∈ u , b ∈ v and uR ( B,C ) v . (b) ([20, 22]) If ( B, C ) is a normal contact algebra, then R ( B,C ) is an equivalencerelation. Definition 2.7.
For CA (
B, C ), a non–empty subset σ of B is called a cluster if forall x, y ∈ B ,(CL1). If x, y ∈ σ then xCy ;.(CL2). If x ∨ y ∈ σ then x ∈ σ or y ∈ σ .(CL3). If xCy for every y ∈ σ , then x ∈ σ .The set of all clusters in an NCA ( B, C ) is denoted by Clust(
B, C )The next theorem is used later on and may be proved exactly as Theorem 5.8of [31]:
Theorem 2.8.
A subset σ of a normal contact algebra ( B, C ) is a cluster if, andonly if, there exists an ultrafilter u in B such that σ = { a ∈ B | aCb for every b ∈ u } . (2) Moreover, given σ and a ∈ σ , there exists an ultrafilter u in B satisfying (2) andcontaining a . Corollary 2.9.
Let ( B, C ) be a normal contact algebra and u be an ultrafilter in B .Then there exists a unique cluster σ u in ( B, C ) containing u , namely σ u = { a ∈ B | aCb for every b ∈ u } . (3) The following simple result can be proved exactly as Lemma 5.6 of [31]: Fact 2.10.
Let ( B, C ) be a normal contact algebra and σ , σ clusters in ( B, C ) . If σ ⊆ σ , then σ = σ . Notation 2.11.
For a topological space (
X, τ ) and x ∈ X , we set σ Xx = { F ∈ RC ( X ) | x ∈ F } (4)and often write just σ x .The next assertion is obvious: 8 act 2.12. For a regular topological space X , σ x is a cluster in the CA ( RC ( X ) , ρ X ) ,called a point-cluster . For a category C , we denote by | C | its class of objects, by Mor( C ) its class ofmorphisms, and by C ( X, Y ) the set of all C -morphisms X −→ Y . Let us fix the notation for the Stone Duality ([36, 29]). We denote by
Stone thecategory of all zero-dimensional compact Hausdorff spaces (=
Stone spaces ) and theircontinuous mappings, and by
Bool the category of Boolean algebras and Booleanhomomorphisms. The contravariant functors furnishing the Stone duality are denotedby S a : Bool −→ Stone and S t : Stone −→ Bool . Hence, for A ∈ | Bool | , S a ( A ) is the set Ult( A ) of all ultrafilters in A endowed withthe topology whose open base is the family { s A ( a ) | a ∈ A } , where s A ( a ) df = { u ∈ Ult( A ) | a ∈ u } for all a ∈ A . For X ∈ | Stone | , one sets S t ( X ) df = CO( X ) , and for morphisms f ∈ Stone ( X, Y ) and ϕ ∈ Bool ( B , B ) one puts S t ( f )( F ) = f − ( F ) and S a ( ϕ )( u ) = ϕ − ( u )for all F ∈ CO( Y ) and u ∈ Ult( B ). Now, for every Boolean algebra A , the map s A : A −→ S t ( S a ( A )) , a s A ( a ) , is a Boolean isomorphism, and for every Stone space X , the map t X : X −→ S a (CO( X )) , x u x , is a homeomorphism; here, for every x ∈ X , u x df = { P ∈ CO( X ) | x ∈ P } . (5)Moreover, s A and t X are natural in A and X . Let us recall some standard properties for a continuous map of topologicalspaces: f : X −→ Y is • closed if the image of each closed set is closed; • perfect if it is closed and has compact fibres; • quasi-open ([30]) if int( f ( U )) = ∅ for every non-empty open subset U of X ; • irreducible if f ( X ) = Y and if, for every proper closed subset F of X , f ( F ) = Y .9ecall that, for a regular space X , a space EX is called an absolute of X if thereexists a perfect irreducible mapping π X : EX −→ X and every perfect irreduciblepreimage of EX is homeomorphic to EX (see, e.g. , [6, 33]). It is well-known that:(a) the absolute is unique up to homeomorphism;(b) a space Y is an absolute of a regular space X if, and only if, Y is an extremallydisconnected Tychonoff space for which there exists a perfect irreducible mapping π : Y −→ X ; such mappings π are called projective covers of X ;(c) if X is a compact Hausdorff space, then it is well-known (see, e.g., [38]) that EX = S a (RC( X )) and the projective cover π X of X is defined by π X ( u ) df = \ u, for every u ∈ Ult(RC( X )) (= S a (RC( X )) (here S a : Bool −→ Stone is the Stonecontravariant functor).
Let C be a subcategory of the category Top of all topological spaces and allcontinuous mappings between them. Recall that a C -object P is called a projectiveobject in C if for every g ∈ C ( P, Y ) and every perfect surjection f ∈ C ( X, Y ), thereexists h ∈ C ( P, X ) such that f ◦ h = g .A. M. Gleason [26] proved: In the category
CHaus of compact Hausdorff spaces and continuous mappings,the projective objects are precisely the extremally disconnected spaces.
Given a dual equivalence T : A −→ B and an embedding I of B as a fullsubcategory of a category C , we wish to give a natural construction for a category D into which A may be fully embedded via J , such that T extends to a dual equivalence˜ T : D −→ C : D ˜ T / / CA T / / J O O B I O O Our construction depends on a class P of morphisms of C satisfying certain conditions,which are closely related to certain properties of the full embedding I . It turns outthat, when B is projective in C , such a class P always exists.We call a class P of morphisms in C a weak ( B , C ) -covering class if it satisfiesthe following conditions:(P1) ∀ ( p : B −→ C ) ∈ P : B ∈ | B | ;(P2) ∀ B ∈ | B | : 1 B ∈ P ; 10P3) P ◦ Iso( B ) ⊆ P ;(P4) ∀ C ∈ | C | ∃ ( p : B −→ C ) ∈ P ;(P5 ◦ ) for morphisms in C , there is an assignment B p (cid:15) (cid:15) B ′ p ′ (cid:15) (cid:15) C v / / C ′ B ˆ v / / p (cid:15) (cid:15) B ′ p ′ (cid:15) (cid:15) C v / / C ′ (( p : B → C ) ∈ P , v : C → C ′ , ( p ′ : B ′ → C ′ ) ∈ P ) (ˆ v : B → B ′ with v ◦ p = p ′ ◦ ˆ v ) , Note that in the given assignment, ˆ v depends not only on v , but also on p and p ′ .In condition (P4) we tacitly assume that, for every C ∈ | C | , we have a chosen mor-phism p ∈ P with codomain C . In the presence of (P2), that morphism may be takento be an identity morphism whenever C ∈ | B | . To emphasize the choice, we mayreformulate (P4), as follows:(P4 ′ ) ∀ C ∈ | C | ∃ ( π C : EC −→ C ) ∈ P (with π C = 1 C when C ∈ | B | ) .As a precursor to the category D as envisaged at the beginning of 3.1, weconsider the comma category ( IT ↓ P C ), defined as follows: • objects in ( IT ↓ P C ) are pairs ( A, p ) with A ∈ | A | and p : T A −→ C in theclass P ; • morphisms ( ϕ, f ) : ( A, p ) −→ ( A ′ , p ′ ) in ( IT ↓ P C ) are given by morphisms ϕ : A −→ A ′ in A and f : C ′ −→ C in C , such that p ◦ T ϕ = f ◦ p ′ : T A p (cid:15) (cid:15) T A ′ T ϕ o o p ′ (cid:15) (cid:15) C C ′ f o o • composition is as in A and C ; that is, ( ϕ, f ) as above gets composed with( ϕ ′ , f ′ ) : ( A ′ , p ′ ) −→ ( A ′′ , p ′′ ) by the horizontal pasting of diagrams, that is,( ϕ ′ , f ′ ) ◦ ( ϕ, f ) df = ( ϕ ′ ◦ ϕ, f ◦ f ′ ) . • the identity morphism of a ( IT ↓ P C )-object ( A, p ) is the ( IT ↓ P C )-morphism(1 A , cod( p ) ).On the hom-sets of ( IT ↓ P C ) we define a compatible equivalence relation by( ϕ, f ) ∼ ( ψ, g ) ⇐⇒ f = g, ϕ, f ) , ( ψ, g ) : ( A, p ) −→ ( A ′ , p ′ ) . We denote the equivalence class of ( ϕ, f ) by[ ϕ, f ] (or [ ϕ, f ] ( A,p ) , ( A ′ ,p ′ ) , if clarity demands it), and let D be the quotient category( IT ↓ P C ) / ∼ . Thanks to (P2), we have the functor J : A −→ D , defined by( ϕ : A −→ A ′ ) ( J ϕ df = [ ϕ, T ϕ ] : ( A, T A ) −→ ( A ′ , T A ′ ) ) , which is easily seen to be a full embedding.Given a dual equivalence ( S, T, η, ε ) with contravariant functors T : A −→ B and S : B −→ A and natural isomorphisms η : Id B −→ T ◦ S and ε : Id A −→ S ◦ T , satisfying thetriangular identities T ε ◦ ηT = 1 T and Sη ◦ εS = 1 S , it is now straightforward to establish a dual equivalence of D with C , as follows: Theorem.
There is a dual equivalence ˜ T : D ←→ C : ˜ S extending the given dualequivalence T : A ←→ B : S , in the sense that that ˜ T J = IT and ˜ SI ∼ = J S : D ˜ T / / CA T / / J O O B I O O D ∼ = C ˜ S o o A J O O B I O O S o o The unit ˜ η : Id C −→ ˜ T ˜ S and the counit ˜ ε : Id D −→ ˜ S ˜ T of the extended adjunctionand the natural isomorphism γ : J S −→ ˜ SI may be chosen to satisfy the identities ˜ η = 1 Id C , ˜ T ˜ ε = 1 ˜ T , ˜ ε ˜ S = 1 ˜ S , and ˜ T γ = Iη , γT ◦ J ε = ˜ εJ .Proof. ˜ T is given by the projection [ ϕ, f ] f ; this trivially gives a faithful functor.With (P5 ◦ ) it is easy to see that ˜ T is full since T is. To define ˜ S on objects, onechooses for every C ∈ | C | a morphism π C : EC −→ C in P (as in (P4 ′ )), with π B = 1 B for all B ∈ | B | (according to (P2)), and then puts ˜ SC df = ( SEC, π C ◦ η − EC ). For amorphism f : C ′ −→ C in C , again, (P5 ◦ ) and the fullness of T allow one to choosea morphism ϕ f : SEC −→ SEC ′ in A with π C ◦ η − EC ◦ T ϕ f = f ◦ π C ′ ◦ η − EC ′ ; we thenput ˜ Sf df = [ ϕ f , f ]. Checking that ˜ S is a functor with ˜ T ˜ S = Id C is straightforward.For ( A, p : T A −→ C ) ∈ | D | one puts ˜ ε ( A,p ) df = [ ϕ ( A,p ) , C ], with any A -morphism ϕ ( A,p ) : A −→ SEC satisfying p ◦ T ϕ ( A,p ) = π C ◦ η − EC . Clearly, ˜ ε is, like ˜ η df = 1 Id C , anatural isomorphism satisfying the claimed identities. Also, with γ B df = [1 SB , η B ] forall B ∈ | B | , one obtains a natural isomorphism γ satisfying ˜ T γ = Iη , γT ◦ J ε = ˜ εJ .12 .2. Recall that, for a class Q of morphisms in C , an object B ∈ | C | is Q - projective if,for all ( q : C −→ D ) ∈ Q , the map C ( B, q ) : C ( B, C ) −→ C ( B, D ) , h q ◦ h, is surjective. Since this map is trivially bijective when q is an isomorphism, withoutloss of generality we may assume that Q contain all isomorphisms and be closed undercomposition with them. We call a full subcategory B in C projective if there is a sucha class Q satisfying(Q1) ∀ C ∈ | C | ∃ ( q : B −→ C ) ∈ Q with B ∈ | B | ;(Q2) ∀ B ∈ | B | : B is Q -projective. Proposition.
A full subcategory B of a category C is projective if, and only if, thereis a weak ( B , C ) -covering class P .Proof. Having a class Q containing all C -isomorphisms, being closed under composi-tion with them, and satisfying (Q1-2), one lets P be the subclass of those morphismsin Q whose domains lie in B . Then, trivially (P1-3) hold, and (Q1) coincides with(P4). Given C -morphisms ( p : B → C ) ∈ P , v : C → C ′ , ( p ′ : B ′ → C ′ ) ∈ P , oneexploits the Q -projectivity of B (by (Q2)) to find ˆ v : B −→ B ′ with v ◦ p = p ′ ◦ ˆ v, which confirms (P5 ◦ ).Conversely, having a class P satisfying (P1-4), (P5 ◦ ), we consider its closure Q under isomorphisms in C and trivially obtain (Q1) from (P4). To confirm (Q2), welet B ∈ | B | , ( q : C −→ D ) ∈ Q , and f : B −→ D in C , and may, for simplicity,assume q ∈ P . Since 1 B ∈ P by (P2), condition (P5 ◦ ) provides us with a morphism h = ˆ f with f = f ◦ B = q ◦ h , thus confirming the surjectivity of the map C ( B, q ).Note that conditions (Q1-2) imply in particular that the following conditionholds:(Q1 ◦ ) C has enough Q -projectives : ∀ C ∈ | C | ∃ ( q : B −→ C ) ∈ Q and B is Q -projective.If we strengthen (Q2) to(Q2*) ∀ B ∈ | C | : ( B ∈ | B | ⇐⇒ B is Q -projective),then, in the presence of (Q2*), condition (Q1 ◦ ) is a weakening of (Q1). The conjunc-tion of (Q1 ◦ ) and (Q2*) is equivalent to (Q1-2) if the full subcategory B is retractive in C , that is: if, for all s : C −→ B, r : B −→ C in C with r ◦ s = 1 C , B ∈ | B | implies C ∈ | B | . Since retracts of Q -projective objects are Q -projective, one obtainsthe following modification of Proposition 3.2: Corollary.
For a full subcategory B of a category C , there is a class Q satisfying (Q1 ◦ ) and (Q2*) if, and only if, B is retractive and C admits a weak ( B , C ) -coveringclass P . .3. In [19] we noted that B is a coreflective subcategory of C if, and only if, thereexists a class P of C -morphisms satisfying properties (P1-4) and the following strength-ening of (P5 ◦ ):(P5 ∗ ) for all v : C −→ C ′ in C and p : B −→ C, p ′ : B ′ −→ C ′ in P , there isprecisely one morphism ˆ v : B −→ B ′ with v ◦ p = p ′ ◦ ˆ v . Note that if, in the notation of 3.1, the class P satisfies conditions (P1-4) and(P5 ∗ ), then the equivalence relation ∼ is just the equality relation. Thus, in this case,the category D coincides with the category ( IT ↓ P C ). In the sequel, we will also usethe dualization of this special form of Theorem 3.1. To be able to refer to it later on,next we formulate this dualization explicitly.Let A be a full subcategory of a category D with inclusion functor J . We call aclass J of morphisms in D a strong ( A , D ) - insertion class if it satisfies the followingconditions (J1-4) and (J5 ∗ ):(J1) ∀ ( j : D −→ A ) ∈ J : A ∈ | A | ;(J2) ∀ A ∈ | A | : 1 A ∈ J ;(J3) Iso( A ) ◦ J ⊆ J ;(J4) ∀ D ∈ | D | ∃ ( j : D −→ A ) ∈ J ;(J5 ∗ ) for all v : D −→ D ′ in D and j : D −→ A, j ′ : D ′ −→ A ′ in J , there isprecisely one morphism v : A −→ A ′ with j ′ ◦ v = v ◦ j . Again, we point out that, in the given assignment, v depends not only on v , but alsoon j and j ′ , so that, whenever needed, we will write v ( j, j ′ ) instead of just v . Next,we note that, in the presence of (J3), condition (J2) means equivalently(J2 ′ ) Iso( A ) ⊆ J .In condition (J4) we tacitly assume that, for every D ∈ | D | , we have a chosen morphism j ∈ J with domain D . In the presence of (J2), that morphism may betaken to be an identity morphism whenever D ∈ | A | . To emphasize the choice, wemay reformulate (J4), as follows:(J4 ′ ) ∀ D ∈ | D | ∃ ( ρ D : D −→ F D ) ∈ J (with ρ D = 1 D when D ∈ | A | ) .It is now clear that (J5 ∗ ) enables us to make F a functor D −→ A and ρ anatural transformation Id C −→ J F .Dualizing an observation made in [19], we obtain the following proposition:
Proposition.
The full subcategory A of D is reflective in D if, and only if, there isa strong ( A , D ) -insertion class J of morphisms in D . In addition to the full subcategory A of D with inclusion functor J and a strong( A , D )-insertion class J (giving us the reflector F : D −→ A and a natural transfor-mation ρ : Id D −→ J F , with ρ A an isomorphism for all A ∈ | A | ), as in Theorem 3.1we consider again a dual equivalence ( S, T, η, ε ) with contravariant functors T : A −→ B and S : B −→ A η : Id B −→ T ◦ S and ε : Id A −→ S ◦ T satisfying thetriangular identities. We then construct the category C , as follows: • objects in C are pairs ( B, j ) with B ∈ | B | and j : D −→ SB in the class J ; • morphisms ( ϕ, f ) : ( B, j ) −→ ( B ′ , j ′ ) in C are given by morphisms ϕ : B −→ B ′ in B and f : D ′ −→ D in D , such that, in the notation of (J5 ∗ ), Sϕ = f : SB SB ′ Sϕ = f o o D j O O D ′ f o o j ′ O O • composition is as in B and D ; that is, ( ϕ, f ) as above gets composed with( ϕ ′ , f ′ ) : ( B ′ , j ′ ) −→ ( B ′′ , j ′′ ) by the horizontal pasting of diagrams, that is,( ϕ ′ , f ′ ) ◦ ( ϕ, f ) df = ( ϕ ′ ◦ ϕ, f ◦ f ′ ) . • the identity morphism of a C -object ( B, j ) is the C -morphism (1 B , dom( j ) ).Of course, the fact that the composition and the identity morphisms of C are welldefined, relies heavily on (P5 ∗ ). Since S is fully faithful, we note that, for a morphism( ϕ, f ) in C , the B -morphism ϕ is determined by f and, hence, by f, j, and j ′ . With(J2) one obtains the full embedding I : B −→ C , defined by( ϕ : B −→ B ′ ) ( Iϕ df = ( ϕ, Sϕ ) : ( B, SB ) −→ ( B ′ , SB ′ ) ) . A dual equivalence S : C ←→ D : T with natural isomorphisms ε : Id D −→ S ◦ T and η : Id C −→ T ◦ S may now beestablished, as follows: • S : (( ϕ, f ) : ( B, j ) −→ ( B ′ , j )) ( f : dom( j ′ ) −→ dom( j )); • T : ( f : D ′ −→ D ) (( ϕ f , f ) : ( T F D, ε
F D ◦ ρ D ) −→ ( T F D ′ , ε F D ′ ◦ ρ D ′ )) , where ϕ f : T F D −→ T F D ′ is the unique B -morphism to make the diagram ST F D ST F D ′ Sϕ f o o F D ε F D O O F D ′ F f o o ε F D ′ O O D ρ D O O D ′ f o o ρ D ′ O O commutative (so that Sϕ f = f ( j ′ , j ) with j = ε F D ◦ ρ D and j ′ = ε F D ′ ◦ ρ D ′ );15 ε D df = 1 D : D −→ S T D = D , for every D ∈ | D | ; • η ( B,j ) df = ( ψ B,j , D ) : ( B, j ) −→ T S ( B, j ) = (
T F D, ε
F D ◦ ρ D ), for every C -object( B, j : D −→ SB ), where the B -isomorphism ψ B,j : B −→ T F D is determinedby the commutative diagram
SB ST F D Sψ B,j o o F D ε D O O D j O O D . D o o ρ D O O The dualization of Theorem 3.1 now reads as follows:
Theorem. ( T , S, ε, η ) is a dual equivalence with S T = Id D , extending the given dualequivalence ( T, S, ε, η ) , so that SI = J S and
T J ∼ = IT : D C S o o A J O O B I O O S o o D ˜ T / / ∼ = CA T / / J O O B I O O Furthermore, with a natural isomorphism δ : IT −→ T J , the unit and co-unit of theadjunction satisfy ε = 1 Id D , Sη = 1 S , ηT = 1 T , Sδ = J ε, δS ◦ Iη = ηI. We note that, as A is reflective in D , B is coreflective in C , with the coreflectionsatisfying some easily established identities involving the reflection and the units andcounits of the dual equivalences. In this section we recall and extend various facts leading up to the de Vries DualityTheorem [15]. Our alternative proof of it follows in the next section.We will now formulate and sketch a proof of the de Vries Duality Theorem.
Definition 4.1. (De Vries [15]) We denote by
DeV the category of complete normalcontact algebras (see 2.1); its morphisms ϕ : ( A, C ) −→ ( A ′ , C ′ ) are maps A −→ A ′ satisfying the conditions:(DV1) ϕ (0) = 0;(DV2) ϕ ( a ∧ b ) = ϕ ( a ) ∧ ϕ ( b ), for all a, b ∈ A ;(DV3) If a, b ∈ A and a ≪ C b , then ( ϕ ( a ∗ )) ∗ ≪ C ′ ϕ ( b );(DV4) ϕ ( a ) = W { ϕ ( b ) | b ≪ C a } , for every a ∈ A ;16he composition “ ⋄ ” of ϕ : ( A , C ) −→ ( A , C ) with ϕ : ( A , C ) −→ ( A , C ) in DeV is defined by ϕ ⋄ ϕ = ( ϕ ◦ ϕ )ˇ , (6)where, for objects ( A, C ) , ( A ′ , C ′ ) in DeV and any function ψ : A −→ A ′ , one defines ψ ˇ : ( A, C ) −→ ( A ′ , C ′ ) for all a ∈ A by ψ ˇ( a ) df = _ { ψ ( b ) | b ≪ C a } . (7) We call the morphisms of the category DeV de Vries morphisms . Fact 4.2. ([15])
Let ϕ : ( A, C ) −→ ( A ′ , C ′ ) be a de Vries morphism. Then: (a) ϕ (1 A ) = 1 A ′ ; (b) for every a ∈ A , ϕ ( a ∗ ) ≤ ( ϕ ( a )) ∗ ; (c) for every a, b ∈ A , a ≪ C b implies ϕ ( a ) ≪ C ′ ϕ ( b ) ; (d) if ϕ ′ : ( A ′ , C ′ ) −→ ( A ′′ , C ′′ ) is a de Vries morphism, such that ϕ ′ is a suprema-preserving Boolean homomorphism, then ϕ ′ ⋄ ϕ = ϕ ′ ◦ ϕ ; (e) for every a ∈ A , a = W { b | b ∈ A, b ≪ a } . De Vries [15] proved the following duality theorem:
Theorem 4.3. ([15])
The categories
CHaus and
DeV are dually equivalent.Sketch of the proof.
One defines contravariant functorsΨ t : CHaus −→ DeV , Ψ a : DeV −→ CHaus , by • Ψ t ( X, τ ) df = (RC( X, τ ) , ρ X ) , for all X ∈ | CHaus | ; • Ψ t ( f )( G ) df = cl( f − (int( G ))) , for all f ∈ CHaus ( X, Y ) and G ∈ RC( Y ); • Ψ a ( A, C ) df = (Clust( A, C ) , T ) , for all ( A, C ) ∈ | DeV | , where T is the topologyon Clust( A, C ) having the family { υ ( A,C ) ( a ) | a ∈ A } with υ ( A,C ) ( a ) = { σ ∈ Clust(
A, C ) | a ∈ σ } as a base of closed sets; • Ψ a ( ϕ )( σ ′ ) df = { a ∈ A | ∀ b ∈ A ( b ≪ C a ∗ = ⇒ ( ϕ ( b )) ∗ ∈ σ ′ ) } , for all ϕ ∈ DeV (( A, C ) , ( A ′ , C ′ )) and σ ′ ∈ Clust( A ′ , C ′ ).Then one shows that, for every ( A, C ) ∈ | DeV | , υ ( A,C ) : ( A, C ) −→ Ψ t (Ψ a ( A, C )) isa
DeV -isomorphism, producing the natural isomorphism υ : Id DeV −→ Ψ t ◦ Ψ a . Likewise, t ′ : Id CHaus −→ Ψ a ◦ Ψ t , t ′ X ( x ) df = σ x , for every X ∈ | CHaus | and all x ∈ X , is a natural isomorphism.Thus, the categories CHaus and
DeV are dually equivalent.We note that, in [15], de Vries used regular open sets, rather than regular closed sets,as we do here. Hence, above we have paraphrased his definitions in terms of regularclosed sets.
Remarks 4.4. (a) As it is noted in [15], for any complete Boolean algebra B ,Ψ a (( B, ρ s )) = S a ( B ) , where S a is the Stone dual equivalence.(b) If B is a complete atomic Boolean algebra, then for every x ∈ At( B ), ↑ ( x ) df = { b ∈ B | x ≤ b } is an ultrafilter in B and, thus, by (a), ↑ ( x ) ∈ Ψ a ( B, ρ s ).(c) If B is a complete atomic Boolean algebra, then the set {↑ ( x ) | x ∈ At( B ) } isdense in S a ( B ) ( = Ψ a (( B, ρ s )) ). Indeed, if b ∈ B + then there exists x ∈ At( B ) such that x ≤ b . Then b ∈↑ ( x )and thus ↑ ( x ) ∈ s B ( b ). Therefore, the set {↑ ( x ) | x ∈ At( B ) } is dense in S a ( B ).(d)([15] If ( A, C ) ∈ | DeV | and Y df = Ψ a ( A, C ), then, for every a ∈ A ,int Y ( υ ( A,C ) ( a )) = Y \ υ ( A,C ) ( a ∗ ) . We will need the following result as well:
Theorem 4.5. ([15])
A de Vries morphism α is an injection if, and only if, , themapping Ψ a ( α ) is a surjection. Of great importance to our investigations is the following beautiful theoremby Alexandroff [4], which follows easily from Ponomarev’s results [32] on irreduciblemappings:
Theorem 4.6. ([4, Corollary, p. 346])
Let p : X −→ Y be a closed irreduciblemapping. Then the map ϕ p : RC( X ) −→ RC( Y ) , H p ( H ) , is a Boolean isomorphism, and one has ϕ − p ( K ) = cl X ( p − (int Y ( K ))) , for all K ∈ RC( Y ) . Denote by
CBool the category of complete Boolean algebras and Boolean ho-momorphisms. The following assertions, proved in [19], are very important in thispaper. 18 emma 4.7. ([19])
Let A ∈ | CBool | , X ∈ | CHaus | , π : S a ( A ) −→ X be anirreducible mapping and for every a, b ∈ A , define aC ( A,π ) b ⇔ π ( s A ( a )) ∩ π ( s A ( b )) = ∅ . Then ( A, C ( A,π ) ) is a complete normal contact algebra. Clearly, the definition of the relation C ( A,π ) as in Lemma 4.7 may be givenequivalently, as follows: for all a, b ∈ A , aC ( A,π ) b ⇐⇒ ∃ u, v ∈ Ult( A ) : a ∈ u, b ∈ v and π ( u ) = π ( v ) . Lemma 4.8. ([19])
Let ( A, C ) be a CNCA and R ( A,C ) be the equivalence relation of Definition 2.5 (see also
Proposition 2.6(b) ), i.e. , for all u, v ∈ S a ( A ) , uR ( A,C ) v ⇔ u × v ⊆ C. Then the natural quotient mapping π ( A,C ) : S a ( A ) −→ S a ( A ) /R ( A,C ) is an irreduciblemapping, and S a ( A ) /R ( A,C ) is a compact Hausdorff space. Following [8], we call a closed equivalence relation R on a compact Hausdorffspace X irreducible if the natural quotient mapping π R : X −→ X/R is irreducible.
Proposition 4.9. ([19])
For a complete Boolean algebra A , let NCRel( A ) be the setof all normal contact relations on A and IRel( T ( A )) the set of all closed irreducibleequivalence relations on S a ( A ) . Then the function f : NCRel( A ) −→ IRel( S a ( A )) , C R ( A,C ) , is a bijection, and f − ( R ) = C ( A,π R ) , for every R ∈ IRel( T ( A )) . Note that Lemmas 4.7 and 4.8 and Proposition 4.9 reveal the topological natureof CNCAs, i.e. , of the objects of the category
DeV . Proposition 4.9 implies alsoBezhanishvili’s Theorem [7, Theorem 8.1] mentioned in the Introduction: for anycomplete Boolean algebra B there is a bijection between the set of all normal contactrelations on B and the set of all (up to homeomorphism) Hausdorff irreducible imagesof the Stone dual S a ( B ) of B . In [7] this result is obtained with the help of the deVries Duality Theorem, while our proof is direct and therefore topologically moreinformative.Let ( A, C ) and ( A ′ , C ′ ) be contact algebras, and ϕ : ( A, C ) −→ ( A ′ , C ′ ) be amap. Following Fedorchuk [25], we consider the following condition(F) ϕ ( a ) C ′ ϕ ( b ) implies aCb , for all a, b ∈ A .If ϕ preserves the negation, we see immediately that condition (F) is equivalent toasking that(F ′ ) a ≪ C b implies ϕ ( a ) ≪ C ′ ϕ ( b ), for all a, b ∈ A .19 roposition 4.10. ([19]) For objects ( A, C ) , ( A ′ , C ′ ) in DeV , a Boolean homomor-phism ψ : A −→ A ′ satisfies condition (F) (or, equivalently, condition (F ′ ) ) if, andonly if, u ′ R ( A ′ ,C ′ ) v ′ implies S a ( ψ )( u ′ ) R ( A,C ) S a ( ψ )( v ′ ) , for all u ′ , v ′ ∈ S a ( A ′ ) . Proposition 4.11. ([19])
For all ( A, C ) ∈ | DeV | , the mapping h ( A,C ) : S a ( A ) /R ( A,C ) −→ Ψ a ( A, C ) , [ u ] σ u , is well-defined and is a homeomorphism (see Corollary 2.9 for σ u and Lemma 4.8 for R ( A,C ) ). Note that Proposition 4.11 clarifies the definition of the contravariant functorΨ a on the objects of the category DeV . In view of Section 3, throughout this section we use the following notation: A df = CBool , B df = ECH , C df = CHaus , with I : B ֒ → C denoting the inclusion functor; P denotes the class of all irreduciblecontinuous maps between compact Hausdorff spaces with domain in | B | . (Recall thatwe denote by CBool the category of complete Boolean algebras and Boolean homo-morphisms, and by
ECH the category of extremally disconnected compact Hausdorffspaces and continuous maps.)Trivially, B is a full subcategory of C that is closed under C -isomorphisms. Bythe results of Gleason [26] (see 2.15), the class P satisfies conditions (P1-4), (P5 ◦ ) ofSection 3 (and B is a projective subcategory of C ).With the restrictions T df = S a ↾ A and S df = S t ↾ B of the functors furnishing the Stone Duality, using the well-known Stone’s result [36],we obtain the contravariant functors T : A −→ B and S : B −→ A . Together withthe restrictions η df = t ↾ B and ε df = s ↾ A of Stone’s natural isomorphisms (so that onehas natural isomorphisms η : Id B −→ T ◦ S and ε : Id A −→ S ◦ T ), they realize a dualequivalence between the categories A and B .Defining the category D as in Theorem 3.1, we obtain the full embedding J : A −→ D and the dual equivalence ˜ T : D −→ C which extends the dual equivalence T : A −→ B , so that I ◦ T = ˜ T ◦ J , as given by Theorem 3.1. We now prove thatthe categories DeV and D are equivalent, thus completing our alternative proof ofde Vries Duality Theorem. This will be done in several steps. In one of them, we willobtain a new category dual to the category CHaus . Let us start by recalling the definition of the category D . In our concretesituation, following Theorem 3.1, we obtain that | D | df = { ( A, π ) | A ∈ A , π ∈ P , dom( π ) = T ( A ) } ;20urther, for every ( A, π ) , ( A ′ , π ′ ) ∈ | D | , D (( A, π ) , ( A ′ , π ′ )) df = { [ ϕ, f ] | ϕ ∈ A ( A, A ′ ) , f ∈ C (cod( π ′ ) , cod( π )) , f ◦ π ′ = π ◦ T ( ϕ ) } , where [ ϕ, f ] is the equivalence class of ( ϕ, f ) under the equivalence relation ≃ in theset { ( ψ, g ) | ψ ∈ A ( A, A ′ ) , g ∈ C (cod( π ′ ) , cod( π )) , g ◦ π ′ = π ◦ T ( ψ ) } defined by( ϕ, f ) ≃ ( ψ, g ) ⇔ f = g ; the composition law is the following one:[ ϕ ′ , f ′ ] ◦ [ ϕ, f ] df = [ ϕ ′ ◦ ϕ, f ◦ f ′ ] , where [ ϕ, f ] , [ ϕ ′ , f ′ ] are any two composable D -morphisms; finally, for any D -object( A, π ), 1 ( A,π ) df = [1 A , cod( π ) ].We will need the following assertion: Proposition.
The full subcategory D nqm of D , where | D nqm | df = { ( A, π ) ∈ | D | | π is a natural quotient mapping } , is equivalent to D .Proof. Denote by J ′ : D nqm −→ D the inclusion functor. Obviously, it is full andfaithful. We have to show that it is essentially surjective on objects. Let ( A, π ) ∈ | D | , X df = cod( π ), R π be the equivalence relation on X determined by the fibres of π , and q : T ( A ) −→ T ( A ) /R π be the natural quotient mapping. Since π is a closed mapping,the map f π : T ( A ) /R π −→ X , ∀ u ∈ T ( A ) , q ( u ) π ( u ), is a homeomorphism and π = f π ◦ q . Hence, q is an irreducible mapping and ( A, q ) ∈ | D nqm | . Then, clearly,[1 A , f π ] : ( A, π ) −→ ( A, q ) and [1 A , f − π ] : ( A, q ) −→ ( A, π ) are D -isomorphisms.Therefore, J ′ ( A, q ) is D -isomorphic to ( A, π ). Thus, J ′ is essentially surjective onobjects. All this shows that J ′ is an equivalence. Lemma 5.3. ([19])
The correspondence F : | DeV | −→ | D nqm | , ( A, C ) ( A, π ( A,C ) ) ,is a bijection.Proof. Let (
A, C ) ∈ | DeV | . Then, by Lemma 4.8, π ( A,C ) ∈ P and ( A, π ( A,C ) ) ∈ | D | .This makes the correspondence F well-defined. Now, with the notation of Lemma4.7, we consider G : | D nqm | −→ | DeV | , ( A, π ) ( A, C ( A,π ) ) . Clearly, Lemma 4.7 confirms that G is well-defined. We show that F and G areinverse to each other.For ( A, C ) ∈ | DeV | one has G ( F ( A, C )) = G ( A, π ( A,C ) ) = ( A, C ( A,π ( A,C ) ) ). ByProposition 4.9, C = f − ( f ( C )) = C ( A,π R ( A,C ) ) follows. Since π ( A,C ) = π R ( A,C ) (seeLemma 4.8), we obtain G ( F ( A, C )) = (
A, C ).For (
A, π ) ∈ | D nqm | one has F ( G ( A, π )) = F ( A, C ( A,π ) ) = ( A, π ( A,C ( A,π ) ) ). De-note by R π the relation on T ( A ) determined by the fibers of π ; then R π ∈ IRel ( T ( A )).Using once more Proposition 4.9, we obtain R π = f ( g ( R π )) = R ( A,C ( A,π ( Rπ )) ) . Since π ( A,C ( A,π ) ) = π R ( A,C ( A,π )) and π = π ( R π ) (because π is a natural quotient map), weobtain F ( G ( A, π )) = (
A, π ). 21et us note that in this paper, by a T -space (resp., T -space) we will understanda regular (resp., normal) Hausdorff space. For proving our new duality theorem forthe category CHaus , we will need the following lemma:
Lemma 5.4.
Let π : X −→ Y and π ′ : X ′ −→ Y ′ be two closed irreducible mappings, Y be a T -space, f : Y ′ −→ Y and ˆ f : X ′ −→ X be continuous maps such that π ◦ ˆ f = f ◦ π ′ . Then, for every G ∈ RC( X ) , cl( f − (int( π ( G )))) = _ { π ′ ( ˆ f − ( H )) | H ∈ RC( X ) and π ( H ) ⊆ int( π ( G )) } . Proof.
By Alexandroff’s Theorem 4.6, the map ϕ π : RC( X ) −→ RC( Y ) , H π ( H ), is a Boolean isomorphism, and one has ϕ − π ( K ) = cl X ( π − (int Y ( K ))), forall K ∈ RC( Y ). From here, using the fact that Y is a T -space, we obtain thatint( π ( G )) = S { π ( H ) | H ∈ RC( X ) and π ( H ) ⊆ int( π ( G )) } . Since π ′ is a surjection,we have that f − ( M ) = π ′ ( ˆ f − ( π − ( M ))), for every M ⊆ Y . Hence, f − (int( π ( G ))) = [ { π ′ ( ˆ f − ( π − ( π ( H )))) | H ∈ RC( X ) and π ( H ) ⊆ int( π ( G )) } . Since Y is a T -space, the theorem of Alexandroff cited above implies that for every H ∈ RC( X ) such that π ( H ) ⊆ int( π ( G )), there exists H ′ ∈ RC( X ) with π ( H ) ⊆ int( π ( H ′ )) ⊆ π ( H ′ ) ⊆ int( π ( G )). Using again Alexandroff’s theorem, we obtain that H ′ = cl( π − (int( π ( H ′ )))). Therefore, π − ( π ( H )) ⊆ π − (int( π ( H ′ ))) ⊆ H ′ . On theother hand, it is obvious that H ⊆ π − ( π ( H )). Thus we obtain that f − (int( π ( G ))) = [ { π ′ ( ˆ f − ( H )) | H ∈ RC( X ) and π ( H ) ⊆ int( π ( G )) } . Then, by Example 2.3,cl( f − (int( π ( G )))) = _ { π ′ ( ˆ f − ( H )) | H ∈ RC( X ) and π ( H ) ⊆ int( π ( G )) } . The next two definitions are of great importance for our investigations.
Definition 5.5.
Let (
A, C ) , ( A ′ , C ′ ) be two normal contact algebras. Then a Booleanhomomorphism ϕ : A −→ A ′ will be called a Fedorchuk homomorphism (briefly,
F-homomorphism ) if a ≪ C b implies ϕ ( a ) ≪ C ′ ϕ ( b ), for all a, b ∈ A .Note that Fedorchuk [25] defined a category Fed such that | Fed | df = | DeV | ,the morphisms of the category Fed are the complete
Fedorchuk homomorphisms andtheir compositions are the usual set-theorethic compositions of functions; he provedthat
Fed is a subcategory of the category
DeV which is dually equivalent to thecategory
CHaus qop of compact Hausdorff spaces and their quasi-open mappings.
Definition 5.6.
Let (
A, C ) be a contact algebra, B be a complete Boolean algebraand ϕ : A −→ B be a function. Then the function V ( ϕ ) : A −→ B , defined by( V ( ϕ ))( a ) df = _ { ϕ ( b ) | b ≪ a } , for every a ∈ A , will be called a de Vries transformation of the function ϕ .22e need two more lemmas. The first of them is a generalization of [15, Propo-sition 1.5.4]. Lemma 5.7.
Let ( A, C ) , ( A ′ , C ′ ) , ( A ′′ , C ′′ ) be complete normal contact algebras and ϕ : ( A, C ) −→ ( A ′ , C ′ ) , ψ : ( A ′ , C ′ ) −→ ( A ′′ , C ′′ ) be Fedorchuk homomorphisms.Then V ( ψ ◦ ϕ ) = V ( V ( ψ ) ◦ V ( ϕ )) .Proof. Set α df = V ( ϕ ) and β df = V ( ψ ). Then, for every a ∈ A ,( V ( β ◦ α ))( a ) = W { β ( α ( b )) | b ≪ a } = W { β ( W { ϕ ( c ) | c ≪ b } ) | b ≪ a } = W { W { ψ ( d ) | d ≪ W { ϕ ( c ) | c ≪ b }} | b ≪ a } .Also, ( V ( ψ ◦ ϕ ))( a ) = _ { ψ ( ϕ ( b )) | b ≪ a } . Let d ≪ W { ϕ ( c ) | c ≪ b } . Then ϕ ( c ) ≪ ϕ ( b ), for every c ≪ b . Thus, W { ϕ ( c ) | c ≪ b } ≤ ϕ ( b ). Hence d ≪ ϕ ( b ). This implies that ψ ( d ) ≪ ψ ( ϕ ( b )).Therefore, W { ψ ( d ) | d ≪ W { ϕ ( c ) | c ≪ b }} ≤ ψ ( ϕ ( b )) and we obtain that( V ( β ◦ α ))( a ) ≤ ( V ( ψ ◦ ϕ ))( a )for every a ∈ A .Conversely, let a, b ∈ A and b ≪ a . Then there exist a ′ , b ′ ∈ A such that b ≪ a ′ ≪ b ′ ≪ a . Now we obtain that ϕ ( b ) ≪ ϕ ( a ′ ) ≤ W { ϕ ( c ) | c ≪ b ′ } . Set d df = ϕ ( b ). Then d ≪ W { ϕ ( c ) | c ≪ b ′ } . Hence, ψ ( ϕ ( b )) = ψ ( d ) ≤ ( V ( β ◦ α ))( a ).Therefore, ( V ( ψ ◦ ϕ ))( a ) ≤ ( V ( β ◦ α ))( a ) for every a ∈ A .All this shows that V ( ψ ◦ ϕ ) = V ( V ( ψ ) ◦ V ( ϕ )). Lemma 5.8.
Let X be a topological space, Y be a T -space and f, g : X −→ Y be two continuous mappings such that cl( f − (int( G ))) = cl( g − (int( G ))) for every G ∈ RC( Y ) . Then f = g .Proof. Suppose that f = g . Then there exists x ∈ X such that f ( x ) = g ( x ). Since Y is a T -space, there exists G ∈ RC( Y ) such that f ( x ) ∈ int( G ) and g ( x ) G .Then x ∈ f − (int( G ). Thus x ∈ cl( f − (int( G ))) = cl( g − (int( G ))). Then g ( x ) ∈ g (cl( g − (int( G )))) ⊆ cl( g ( g − (int( G )))) ⊆ cl(int( G )) = G , a contradiction. Therefore, f = g . We are now ready to define a new category
StoneDeV and to prove that it isdually equivalent to the category
CHaus . We set | StoneDeV | df = | DeV | . Further, for every (
A, C ) , ( A ′ , C ′ ) ∈ | StoneDeV | , we define StoneDeV (( A, C ) , ( A ′ , C ′ )) df = {h ϕ i | ϕ : ( A, C ) −→ ( A ′ , C ′ ) is F-homomorphism } , h ϕ i is the equivalence class of ϕ under the equivalence relation ≃ in the set ofall Fedorchuk homomorphisms between ( A, C ) and ( A ′ , C ′ ) defined by ϕ ≃ ψ ⇔ V ( ϕ ) = V ( ψ ) . The
StoneDeV -composition between two
StoneDeV -morphisms h ϕ i : ( A, C ) −→ ( A ′ , C ′ ) and h ψ i : ( A ′ , C ′ ) −→ ( A ′′ , C ′′ ) is defined as follows: h ψ i ◦ h ϕ i df = h ψ ◦ ϕ i . Finally, for every
StoneDeV -object (
A, C ), its
StoneDeV -identity is1 ( A,C ) df = h A i . Let us prove that the composition in
StoneDeV is well-defined. Indeed, let ϕ, ϕ ′ : ( A, C ) −→ ( A ′ , C ′ ) and ψ, ψ ′ : ( A ′ , C ′ ) −→ ( A ′′ , C ′′ ) be Fedorchuk homo-morphisms, ϕ ≃ ϕ ′ and ψ ≃ ψ ′ . Then ψ ◦ ϕ ≃ ψ ′ ◦ ϕ ′ . Indeed, we have that V ( ϕ ) = V ( ϕ ′ ) and V ( ψ ) = V ( ψ ′ ); then, using twice Lemma 5.7, we obtain that V ( ψ ◦ ϕ ) = V ( V ( ψ ) ◦ V ( ϕ )) = V ( V ( ψ ′ ) ◦ V ( ϕ ′ )) = V ( ψ ′ ◦ ϕ ′ ) which means that ψ ◦ ϕ ≃ ψ ′ ◦ ϕ ′ .Consequently, StoneDeV is a well-defined category.
Proposition.
The categories D nqm and StoneDeV are isomorphic.Proof.
Since | StoneDeV | df = | DeV | = | Fed | , Corollary 5.3 shows that the correspon-dence I V : | StoneDeV | −→ | D nqm | , ( A, C ) ( A, π ( A,C ) ) , is a bijection (see Lemma4.8 for π ( A,C ) ). We will extend this bijection to an isomorphism I V : StoneDeV −→ D nqm . Let (
A, C ) , ( A ′ , C ′ ) ∈ | StoneDeV | and h ϕ i ∈ StoneDeV (( A, C ) , ( A ′ , C ′ )). Then ϕ : ( A, C ) −→ ( A ′ , C ′ ) is a Fedorchuk homomorphism. Thus ϕ ∈ A ( A, A ′ ). For π df = π ( A,C ) , π ′ df = π ( A ′ ,C ′ ) , X df = cod( π ) and X ′ df = cod( π ′ ), we will define a continuousfunction f h ϕ i : X ′ −→ X ) such that f h ϕ i ◦ π ′ = π ◦ T ( ϕ ).Since ϕ satisfies condition (F), using Proposition 4.10, we obtain that, if u ′ , v ′ ∈ T ( A ′ ) and π ′ ( u ′ ) = π ′ ( v ′ ), then π ( T ( α )( u ′ )) = π ( T ( α )( v ′ )) . (8)To define f h ϕ i , since π ′ is a surjection, given x ′ ∈ X ′ , one has some u ′ ∈ T ( A ′ ) suchthat x ′ = π ′ ( u ′ ), and with (8) we can put f h ϕ i ( x ′ ) df = π ( T ( α )( u ′ )) . Then f h ϕ i ◦ π ′ = π ◦ T ( ϕ ). Since π ′ is a quotient mapping, we obtain that f h ϕ i : X ′ −→ X is continuous.We have to show that if ψ ∈ h ϕ i ( i.e. , h ψ i = h ϕ i ) then f h ϕ i = f h ψ i . Set f df = f h ϕ i , g df = f h ψ i , ˆ f df = T ( ϕ ) and ˆ g df = T ( ψ ). Then f ◦ π ′ = π ◦ ˆ f and g ◦ π ′ = π ◦ ˆ g .24ince T ( A ) , T ( A ′ ) ∈ | ECH | , we have that RC( T ( A )) = CO( T ( A )) and RC( T ( A ′ )) =CO( T ( A ′ )). Then, by Theorem 4.6, the maps ϕ π : CO( T ( A )) −→ RC( X ) , H π ( H ) , and ϕ π ′ : CO( T ( A ′ )) −→ RC( X ′ ) , H π ′ ( H ) , are Boolean isomorphisms.We will show that for every G ∈ RC( X ),cl( f − (int( G ))) = ( ϕ π ′ ◦ s A ′ )(( V ( ϕ ))( s − A ( ϕ − π ( G ))))(9)and cl( g − (int( G ))) = ( ϕ π ′ ◦ s A ′ )(( V ( ψ ))( s − A ( ϕ − π ( G ))))(10)(see 2.13 for s A ). It is enough to prove the first equality since the proof of the secondone is analogous.Let us first recall that, according to Proposition 4.9, Lemma 4.7 and Lemma4.8, we have that for every a, b ∈ A , aCb ⇔ aC ( A,π ) b ⇔ π ( s A ( a )) ∩ π ( s A ( b )) = ∅ ;thus, a ≪ b ⇔ π ( s A ( a )) ⊆ int( π ( s A ( b ))). Recall as well that, by the Stone DualityTheorem, s A ′ ◦ ϕ = S ( T ( ϕ )) ◦ s A . Now, using also Lemma 5.4, we obtain that forevery G ∈ RC( X ),cl( f − (int( G ))) = W { π ′ ( ˆ f − ( H )) | H ∈ CO( T ( A )) , π ( H ) ⊆ int( G ) } = W { ϕ π ′ ( S ( T ( ϕ ))( ϕ − π ( F ))) | F ∈ RC( X ) , F ⊆ int( G ) } = ϕ π ′ ( W { s A ′ ( ϕ ( s − A ( ϕ − π ( F )))) | F ∈ RC( X ) , F ⊆ int( G ) } = ( ϕ π ′ ◦ s A ′ )( W { ϕ ( b ) | b ∈ A, b ≪ s − A ( ϕ − π ( G )) } = ( ϕ π ′ ◦ s A ′ )(( V ( ϕ ))( s − A ( ϕ − π ( G )))).Since V ( ϕ ) = V ( ψ ), we obtain that cl( f − (int( G ))) = cl( g − (int( G ))), for every G ∈ RC( X ). According to Lemma 5.8, this implies that f = g . All this shows that f h ϕ i is well-defined. We now set I V ( h ϕ i ) df = [ ϕ, f h ϕ i ] . As it follows from the above considerations, I V is well defined on the objects andmorphisms of the category StoneDeV . I V : StoneDeV −→ D nqm is obviously afunctor. As it is bijective on objects, we need to show only that it is full and faithful.Let h ϕ i , h ψ i ∈ StoneDeV (( A, C ) , ( A ′ , C ′ )) and I V ( h ϕ i ) = I V ( h ψ i ). Then f h ϕ i = f h ψ i .Using (9) and (10), we obtain that V ( ϕ ) = V ( ψ ). Thus h ϕ i = h ψ i . Therefore, I V is faithful. Let now [ ϕ, f ] ∈ D nqm ( I V ( A, C ) , I V ( A ′ , C ′ )). Setting π df = π ( A,C ) and π ′ df = π ( A ′ ,C ′ ) , we obtain that f ◦ ϕ ′ = π ◦ T ( ϕ ). Then Proposition 4.10 implies that ϕ : ( A, C ) −→ ( A ′ , C ′ ) is a Fedorchuk homomorphism. Now we obtain easily that f = f h ϕ i . Thus, I V ( h ϕ i ) = [ ϕ, f ]. Therefore, I V is full. All this shows that I V is anisomorphism. Theorem.
The categories
CHaus and
StoneDeV are dually equivalent.Proof.
Composing the dual equivalence ˜ T : D −→ C from 5.1 with the equivalence J ′ : D nqm ֒ → D from Proposition 5.2, and with the isomorphism I V : StoneDeV −→ D nqm from the above Proposition, we obtain a dual equivalence ˜ T ′ df = ˜ T ◦ J ′ ◦ I V : StoneDeV −→ CHaus . 25n what follows, we will prove that the categories
StoneDeV and
DeV areisomorphic. We start with some lemmas. The first one is a particular case of [16,Lemma 3.9] but, for completeness of our exposition, we outline its proof.
Lemma 5.10.
Let ( A, C ) and ( A ′ , C ′ ) be complete normal contact algebras and ϕ : A −→ B be a function between them. Then:(a) If ϕ satisfies condition (DV2) , then V ( ϕ ) satisfies conditions (DV2) and (DV4) ;(b) If ϕ satisfies condition (DV4) , then V ( ϕ ) = ϕ ;(c) If ϕ satisfies condition (DV2) , then V ( V ( ϕ )) = V ( ϕ ) ;(d) If ϕ is a monotone function then, for every a ∈ A , ( V ( ϕ ))( a ) ≤ ϕ ( a ) .Proof. Properties (b) and (d) are clearly fulfilled, and (c) follows from (a) and (b).Hence, we need to prove only (a).Let a ∈ A . If c ∈ A and c ≪ a then there exists d c ∈ A such that c ≪ d c ≪ a and we fix such a one; hence ϕ ( c ) ≤ ( V ( ϕ ))( d c ). Also, by (d), for every a ∈ A ,( V ( ϕ ))( a ) ≤ ϕ ( a ). Now we obtain that( V ( ϕ ))( a ) = W { ϕ ( c ) | c ∈ A, c ≪ a }≤ W { ( V ( ϕ ))( d c ) | c ∈ A, c ≪ a }≤ W { ( V ( ϕ ))( e ) | e ∈ A, e ≪ a }≤ W { ϕ ( e ) | e ∈ A, e ≪ a } = ( V ( ϕ ))( a ).Thus, ( V ( ϕ ))( a ) = W { ( V ( ϕ ))( e ) | e ∈ A, e ≪ a } . So, V ( ϕ ) satisfies (DV4).Further, let a, b ∈ A . Then( V ( ϕ ))( a ) ∧ ( V ( ϕ ))( b ) = W { ϕ ( d ) ∧ ϕ ( e ) | d, e ∈ A, d ≪ a, e ≪ b } = W { ϕ ( d ∧ e ) | d, e ∈ A, d ≪ a, e ≪ b } = W { ϕ ( c ) | c ∈ A, c ≪ a ∧ b } = ( V ( ϕ ))( a ∧ b ).So, V ( ϕ ) satisfies condition (DV2). Lemma 5.11.
Let ( A, C ) and ( A ′ , C ′ ) be two complete normal contact algebras and ϕ : ( A, C ) −→ ( A ′ , C ′ ) be a Fedorchuk homomorphism. Then V ( ϕ ) is a de Vriesmorphism.Proof. Clearly, ( V ( ϕ ))(0) = 0; thus, condition (DV1) is satisfied. Since ϕ satisfies(DV2), Lemma 5.10(a) implies that V ( ϕ ) satisfies conditions (DV2) and (DV4). So,we need only to prove that V ( ϕ ) satisfies condition (DV3).Let a, b ∈ A and a ≪ b . There exist c, d ∈ A such that a ≪ c ≪ d ≪ b . Nowwe have that(( V ( ϕ ))( a ∗ )) ∗ = ( W { ϕ ( d ) | d ∈ A, d ≪ a ∗ } ) ∗ = ( W { ϕ ( e ∗ ) | e ∈ A, e ∗ ≪ a ∗ } ) ∗ = ( W { ( ϕ ( e )) ∗ | e ∈ A, a ≪ e } ) ∗ = V { ϕ ( e ) | e ∈ A, a ≪ e }≤ ϕ ( c ) ≪ ϕ ( d ) ≤ ( V ( ϕ ))( b ).26ence, V ( ϕ ) satisfies condition (DV3).All this shows that V ( ϕ ) is a de Vries morphism. We will now recall a result of de Vries [15]. Since de Vries works with ends andwe work with clusters, we will present here a proof of his result.Let (
A, C ) be a complete normal contact algebra. Set Y df = Ψ a ( A, C ) (see theproof of Theorem 4.3 for Ψ a ( A, C )). Setting π df = π ( A,C ) and X df = T ( A ) /R ( A,C ) (seeLemma 4.8 for the notation), Proposition 4.11 tell us that the mapping h ( A,C ) : X −→ Y, [ u ] σ u , (see (3) for σ u ) is a homeomorphism. Set h df = h ( A,C ) . Then, clearly, themap ψ h : RC( X ) −→ RC( Y ) , G h ( G ) , is a Boolean isomorphism. Recall that for every a ∈ A , υ ( A,C ) ( a ) df = { σ ∈ Y | a ∈ σ } (see the proof of Theorem 4.3) and the family { υ ( A,C ) ( a ) | a ∈ A } is a closed base for Y . Hence, setting υ ( A,C ) ( a ) df = Y \ υ ( A,C ) ( a ) , we obtain that the family { υ ( A,C ) ( a ) | a ∈ A } is an open base for Y . Further, fromthe proof of Proposition 4.11 we know that for every a ∈ A , υ ( A,C ) ( a ) = h ( π ( s A ( a ))) . Hence, with υ ( A,C ) : A −→ RC( Y ) , a υ ( A,C ) ( a ) , we obtain that υ ( A,C ) = ψ h ◦ ϕ π ◦ s A (see Theorem 4.6 for ϕ π ). Thus, υ ( A,C ) : A −→ RC( Y )is a Boolean isomorphism. We are now ready to prove the result of de Vries mentionedabove. Lemma. ([15])
Let α : ( A, C ) −→ ( A ′ , C ′ ) be a de Vries morphism, Y df = Ψ a ( A, C ) , Y ′ df = Ψ a ( A ′ , C ′ ) and g α : Y ′ −→ Y be defined by the formula g α ( σ ′ ) df = { a ∈ A | ∀ b ∈ A, ( b ≪ a ∗ ) ⇒ (( α ( b )) ∗ ∈ σ ′ ) } . Then g α is a continuous function and for every a ∈ A , cl( g − α (int( υ ( A,C ) ( a )))) = υ ( A ′ ,C ′ ) ( α ( a )) . Proof.
We first show that the function g α is well-defined. Let σ ′ ∈ Y ′ . We haveto prove that σ df = g α ( σ ′ ) satisfies conditions (CL1)-(CL3) of Definition 2.7. Clearly,1 ∈ σ , i.e., σ = ∅ .(CL1): Let a, b ∈ σ . Suppose that a ( − C ) b . Then a ≪ b ∗ . There exist c, d ∈ A suchthat a ≪ c ≪ d ∗ ≪ b ∗ . Then b ≪ d ≪ c ∗ ≪ a ∗ and, by the definition of σ , ( α ( d ∗ )) ∗ ∈ σ ′ and ( α ( c ∗ )) ∗ ∈ σ ′ . Thus ( α ( d ∗ )) ∗ C ( α ( c ∗ )) ∗ . Since c ≪ d ∗ , condition (DV3) from27efinition 4.1 implies that ( α ( c ∗ )) ∗ ≪ α ( d ∗ ). Therefore, ( α ( d ∗ )) ∗ ( − C )( α ( c ∗ )) ∗ , acontradiction. Hence, aCb .(CL2): Let a ∨ b ∈ σ . Suppose that a σ and b σ . Then there exist c, d ∈ A suchthat c ≪ a ∗ , d ≪ b ∗ and ( α ( c )) ∗ σ ′ , ( α ( d )) ∗ σ ′ . We have that c ∧ d ≪ a ∗ ∧ b ∗ =( a ∨ b ) ∗ . Then ( α ( c ∧ d )) ∗ ∈ σ ′ . Thus ( α ( c ) ∧ α ( d )) ∗ ∈ σ ′ , i.e., ( α ( c )) ∗ ∧ ( α ( d )) ∗ ∈ σ ′ .This implies that ( α ( c )) ∗ ∈ σ ′ or ( α ( d )) ∗ ∈ σ ′ , a contradiction.(CL3): Let aCb for every b ∈ σ . Suppose that a σ .Then there exists c ∈ A suchthat c ≪ a ∗ and ( α ( c )) ∗ σ ′ . Thus α ( c ) ∈ σ ′ . We will show that c ∈ σ . Indeed, let d ∈ A and d ≪ c ∗ . Then, by Fact 4.2(c), α ( d ) ≪ α ( c ∗ ). Hence, using Fact 4.2(b), weobtain that α ( c ) ≤ ( α ( c ∗ )) ∗ ≪ ( α ( d )) ∗ . Thus ( α ( d )) ∗ ∈ σ ′ . Therefore, c ∈ σ . Since c ≪ a ∗ , we have that a ( − C ) c , a contradiction.This proves that g α is well defined. Now we will show that g α is continuous.Clearly, RO( Y ) = { int( F ) | F ∈ RC( Y ) } and RO( Y ) is an open base for Y . Fromthe above considerations we know that RC( Y ) = { υ ( A,C ) ( a ) | a ∈ A } . Thus, it isenough to show that for every a ∈ A , g − α (int( υ ( A,C ) ( a ))) is an open subset of Y ′ .Let a ∈ A and set G df = υ ( A,C ) ( a ). Then, by Example 2.3 and above considera-tions, int( G ) = Y \ cl( Y \ G ) = Y \ G ∗ = Y \ υ ( A,C ) ( a ∗ ) = υ ( A,C ) ( a ∗ ) = { σ ∈ Y | a ∗ σ } .Thus, g − α (int( G )) = { σ ′ ∈ Y ′ | a ∗ g α ( σ ′ ) } . We have that for any σ ′ ∈ Y ′ , a ∗ g α ( σ ′ ) ⇔ ∃ b ∈ A such that b ≪ a and ( α ( b )) ∗ σ ′ ⇔ ∃ b ∈ A such that b ≪ a and σ ′ ∈ υ ( A ′ ,C ′ ) (( α ( b )) ∗ ). Therefore, g − α (int( G )) = [ { υ ( A ′ ,C ′ ) (( α ( b )) ∗ ) | b ≪ a } . This shows that g α is a continuous function. Further, using Example 2.3, (DV4) andabove considerations, we obtain that g − α (int( υ ( A,C ) ( a ))) = g − α (int( G )) = [ { int( υ ( A ′ ,C ′ ) ( α ( b )) | b ≪ a } and thus,cl( g − α (int( υ ( A,C ) ( a )))) = cl( S { int( υ ( A ′ ,C ′ ) ( α ( b )) | b ≪ a } )= W { υ ( A ′ ,C ′ ) ( α ( b )) | b ≪ a } = υ ( A ′ ,C ′ ) ( W { α ( b ) | b ≪ a } )= υ ( A ′ ,C ′ ) ( α ( a )). Theorem 5.13.
The categories
StoneDeV and
DeV are isomorphic.Proof.
We will define a functor J V : StoneDeV −→ DeV and will prove that it isbijective on objects, full and faithful.For every
StoneDeV -object (
A, C ), we set J V ( A, C ) df = ( A, C ) . Further, for every h ϕ i ∈ StoneDeV (( A, C ) , ( A ′ , C ′ )), we put J V ( h ϕ i ) df = V ( ϕ ) . J V is well-defined on the morphisms of the category StoneDeV . From Fact 4.2 we obtain that J V preserves identities. Let h ψ i ∈ StoneDeV (( A ′ , C ′ ) , ( A ′′ , C ′′ )) and h ϕ i ∈ StoneDeV (( A, C ) , ( A ′ , C ′ )). Then, usingLemma 5.7, we obtain that J V ( h ψ i◦h ϕ i ) = J V ( h ψ ◦ ϕ i ) = V ( ψ ◦ ϕ ) = V ( V ( ψ ) ◦ V ( ϕ )) = V ( J V ( h ψ i ) ◦ J V ( h ϕ i )) = J V ( h ψ i ) ⋄ J V ( h ϕ i ). Hence, J V is a functor.Since | StoneDeV | = | DeV | , we obtain that J V is bijective on objects.Let ( A, C ) , ( A ′ , C ′ ) ∈ | StoneDeV | . If h ϕ i , h ψ i ∈ StoneDeV (( A, C ) , ( A ′ , C ′ ))and J V ( h ϕ i ) = J V ( h ψ i ), then V ( ϕ ) = V ( ψ ); this implies that h ϕ i = h ψ i . Thus, J V isfaithful. Hence, it is only left to show that J V is full.Let α ∈ DeV ( J V ( A, C ) , J V ( A ′ , C ′ )). Then α ∈ DeV (( A, C ) , ( A ′ , C ′ )). Set Y df = Ψ a ( A, C ) and Y ′ df = Ψ a ( A ′ , C ′ ). Using Lemma 5.12, we obtain that the function g α : Y ′ −→ Y, σ ′
7→ { a ∈ A | ∀ b ∈ A, ( b ≪ a ∗ ) ⇒ (( α ( b )) ∗ ∈ σ ′ ) } , is continuous. Set π df = π ( A,C ) , π ′ df = π ( A ′ ,C ′ ) , X df = cod( π ) and X ′ df = cod( π ′ ) (seeLemma 4.8 for the notation π ( A,C ) ). Then Proposition 4.11 implies that the mappings h : X −→ Y, [ u ] σ u , and h ′ : X ′ −→ Y ′ , [ v ] σ v , (see (3) for the notation σ u )are homeomorphisms. Put f α df = h − ◦ g α ◦ h ′ . Then f α : X ′ −→ X is a continuous function. Now, using the Gleason Theorem 2.15,we obtain that there exists a continuous function f : T ( A ′ ) −→ T ( A )such that π ◦ f = f α ◦ π ′ . By the Stone Duality Theorem, there exists a uniqueBoolean homomorphism ϕ : A −→ A ′ such that f = T ( ϕ ) . Proposition 4.10 shows that ϕ is a Fedorchuk homomorphism. Further, recall thatby the Alexandroff Theorem 4.6, the map ϕ π : CO( T ( A )) −→ RC( X ) , U π ( U ) , is a Boolean isomorphism. Hence the map ϕ π ◦ s A : A −→ RC( X ) is a Booleanisomorphism (see 2.13 for the notation s A ). Clearly, the map ψ h : RC( X ) −→ RC( Y ) , G h ( G ) , is a Boolean isomorphism. As we have shown in 5.12, themap υ ( A,C ) : A −→ RC( Y ) , a υ ( A,C ) ( a ) , is a Boolean isomorphism (see theproof of Theorem 4.3 for υ ( A,C ) ( a )) and υ ( A,C ) = ψ h ◦ ϕ π ◦ s A . From Lemma 5.12we have that for every a ∈ A , cl( g − α (int( υ ( A,C ) ( a )))) = υ ( A ′ ,C ′ ) ( α ( a )) . Since g a = h ◦ f α ◦ ( h ′ ) − , we obtain that for every M ⊆ Y , g − a ( M ) = h ′ ( f − α ( h − ( M ))) andthus cl(( h ′ ◦ f − α ◦ h − )(int(( ψ h ◦ ϕ π ◦ s A )( a )))) = ( ψ h ′ ◦ ϕ π ′ ◦ s A ′ )( α ( a )) . This impliesthat h ′ (cl( f − α ( h − (int( h ( π ( s A ( a )))))))) = h ′ ( π ′ ( s A ′ ( α ( a )))) . Hence,cl( f − α (int( π ( s A ( a ))))) = π ′ ( s A ′ ( α ( a ))) . Set G df = π ( s A ( a )). Then G ∈ RC( X ), a = s − A ( ϕ − π ( G )) and we obtain thatcl( f − α (int( G ))) = π ′ ( s A ′ ( α ( a ))) . Further, by (9),cl( f − α (int( G ))) = ( ϕ π ′ ◦ s A ′ )(( V ( ϕ ))( s − A ( ϕ − π ( G )))) . ϕ π ′ ( s A ′ (( V ( ϕ ))( a ))) = ϕ π ′ ( s A ′ ( α ( a ))) . Thus, α ( a ) = ( V ( ϕ ))( a ) for every a ∈ A . This implies that α = J V ( h ϕ i ). So, J V is full.All this shows that J V is an isomorphism.In conclusion we obtain a new proof of the de Vries Duality Theorem: Corollary 5.14. ([15])
The categories
CHaus and
DeV are dually equivalent.Proof.
By Theorem 5.9, there is a dual equivalence ˜ T ′ : StoneDeV −→ CHaus .Composing it with the isomorphism ( J V ) − : DeV −→ StoneDeV from Theorem5.13, we obtain a dual equivalence ˜ T ′′ df = ˜ T ′ ◦ ( J V ) − : DeV −→ CHaus . In their recent paper [9], G. Bezhanishvili, P.J. Morandi and B. Olberding de-scribed a category
BMO and a dual equivalence of
BMO with the category
Tych of Tychonoff spaces and continuous maps which extends de Vries’ dual equivalenceΨ a : DeV −→ CHaus . In this section we will derive the Bezhanishvili-Morandi-Olberding Duality Theorem ([9]) from our Theorem 3.3.We set (and we will keep this notation throughout this section ) A df = CHaus , B df = DeV , D df = Tych , S df = Ψ a , J df = { j : X → Y | X ∈ | Tych | , Y ∈ | CHaus | , j is a dense embedding, j ( X ) ⊆ C ∗ Y } , where j ( X ) ⊆ C ∗ Y means that j ( X ) is C ∗ -embedded in Y , and we denote by J : A ֒ → D the inclusion functor. Note that we regard as elements of the class J all representativesof the Stone- ˇCech compactifications of Tychonoff spaces. Obviously, the class J sat-isfies conditions (J1-4) and (J5 ∗ ) (and A is a reflective subcategory of D ). Therefore,we can apply Theorem 3.3. It gives us a category C , a dual equivalence S : C −→ D and a full embedding I : B −→ C such that S ◦ I = J ◦ S. The plan of the section is now as follows. Adapting the category C to theconcrete situation, we first describe a subcategory C ′ of C which is equivalent to C ,and after that we find a second category C ′′ isomorphic to the category C ′ . In thisway we obtain a new duality theorem which extends de Vries’ Duality Theorem to thecategory Tych . Finally, using the Tarski Duality between the category
Set of sets and30unctions and the category
CaBa of complete atomic Boolean algebras and suprema-preserving Boolean homomorphisms, we prove that our category C ′′ is equivalent tothe category BMO . All this shows that we obtain, as an application of our Theorem3.3, a new proof of the Bezhanishvili-Morandi-Olberding Duality Theorem.We start with the description of the category C ′ mentioned above. Proposition.
Let C ′ be the full subcategory of the category C with | C ′ | df = { ( B, j ) ∈| C | | j is an inclusion map } . Then the inclusion functor I ′ : C ′ ֒ → C is an equivalence.Proof. Clearly, I ′ is a full and faithful functor. For showing that it is essentiallysurjective on objects, let (( A, C ) , j ) ∈ | C | , i.e., ( A, C ) ∈ | DeV | and j : X −→ S ( A, C )is in J . Let j ′ : j ( X ) ֒ → S ( A, C ) be the inclusion mapping and let f df = j ↾ X ,where j ↾ X : X −→ j ( X ) is the restriction of j . Then f is a homeomorphismand j ′ = j ◦ f − . Since J satisfies condition (J3), we obtain that j ′ ∈ J . Hence,(( A, C ) , j ′ ) ∈ | C ′ | . Obviously, the map (1 ( A,C ) , f − ) : (( A, C ) , j ) −→ (( A, C ) , j ′ ) is a C -isomorphism. Therefore, I ′ is an equivalence.A more general version of the next proposition was proved in [17]. Since it wasnot published till now and since it plays an important role in the construction of ourcategory C ′′ , we will present its proof here. Proposition 6.2.
Let ( A, C ) be a CNCA. Then the clusters of ( A, C ) are preciselythose subsets of A which are of the form σ ϕ df = { a ∈ A | ϕ ( a ∗ ) = 0 } , where ϕ ∈ DeV (( A, C ) , ( , ρ s )) .Proof. We will show that the map ξ : DeV (( A, C ) , ( , ρ s )) −→ Clust(
A, C ) , ϕ σ ϕ is a bijection. First of all, we will prove that the map ξ is well defined.Let ϕ ∈ DeV (( A, C ) , ( , ρ s )). We will show that σ ϕ is a cluster in the CNCA( A, C ). Clearly, σ ϕ = ∅ because, by (DV1), ϕ (0) = 0 and thus 1 ∈ σ ϕ . We have toprove that σ ϕ satisfies the axioms (CL1), (CL2), (CL3).(CL1): Let a, b ∈ σ ϕ . Suppose that a ( − C ) b . Then a ≪ b ∗ . Thus, using (DV3),we obtain that ( ϕ ( a ∗ )) ∗ ≪ ϕ ( b ∗ ), i.e., 1 ≪
0, a contradiction. Hence, aCb .(CL2): Let a ∨ b ∈ σ ϕ . Then, using (DV2), we obtain that 0 = ϕ (( a ∨ b ) ∗ ) = ϕ ( a ∗ ∧ b ∗ ) = ϕ ( a ∗ ) ∧ ϕ ( b ∗ ). Hence, ϕ ( a ∗ ) = 0 or ϕ ( b ∗ ) = 0. Thus, a ∈ σ ϕ or b ∈ σ ϕ .(CL3): Let aCb , for every b ∈ σ ϕ . Suppose that a σ ϕ . Then ϕ ( a ∗ ) = 1. Now,using (DV4), we obtain that there exists b ∈ A such that b ≪ a ∗ and ϕ ( b ) = 1. Then a ( − C ) b . Hence b σ ϕ . Thus ϕ ( b ∗ ) = 1. Since 0 = ϕ ( b ∧ b ∗ ) = ϕ ( b ) ∧ ϕ ( b ∗ ), we obtainthat ϕ ( b ) = 0, a contradiction. Therefore, a ∈ σ ϕ .So, σ ϕ ∈ Clust(
A, C ) and thus, the map ξ is well defined. Setting M ∗ = { b ∗ | b ∈ M } , M of A , we can rewrite the definition of ξ ( ϕ ), i.e., of σ ϕ , as follows: ξ ( ϕ ) = ( ϕ − (0)) ∗ . This shows that ξ is an injection. We are now going to prove that ξ is a surjection.Let σ ∈ Clust(
A, C ). Let ϕ σ : A −→ be defined by ϕ σ ( a ) = 0 ⇐⇒ a ∗ ∈ σ. (11)Then, clearly, σ = { a ∈ A | ϕ σ ( a ∗ ) = 0 } , i.e., σ = ξ ( ϕ σ ). We will show that ϕ σ ∈ DeV (( A, C ) , ( , ρ s )), i.e., we will prove that ϕ σ satisfies axioms (DV1)-(DV4).(DV1): Since 0 ∗ = 1 ∈ σ , we obtain that ϕ σ (0) = 0.(DV2): Let ϕ σ ( a ∧ b ) = 0. Then ( a ∧ b ) ∗ ∈ σ , i.e., a ∗ ∨ b ∗ ∈ σ . Hence, by (CL2), a ∗ ∈ σ or b ∗ ∈ σ . Therefore, ϕ σ ( a ) = 0 or ϕ σ ( b ) = 0. Thus ϕ σ ( a ) ∧ ϕ σ ( b ) = 0 = ϕ σ ( a ∧ b ).Let ϕ σ ( a ∧ b ) = 1. Then ( a ∧ b ) ∗ σ , i.e., a ∗ ∨ b ∗ σ . Now, using (CL3)and (C4), we obtain that a ∗ σ and b ∗ σ . Therefore, ϕ σ ( a ) = 1 = ϕ σ ( b ). Thus ϕ σ ( a ) ∧ ϕ σ ( b ) = 1 = ϕ σ ( a ∧ b ).(DV3): Let a, b ∈ A and a ≪ b . Let ϕ σ ( a ∗ ) = 0. Then a ∈ σ . Since a ( − C ) b ∗ ,we obtain that b ∗ σ . Hence ϕ σ ( b ) = 1. Thus ( ϕ σ ( a ∗ )) ∗ ≪ ϕ σ ( b ). If ϕ σ ( a ∗ ) = 1then, clearly, ( ϕ σ ( a ∗ )) ∗ ≪ ϕ σ ( b ). Therefore, ϕ σ satisfies the axiom (DV3).(DV4): Let a ∈ A . If ϕ σ ( a ) = 0 then, using the facts that ϕ σ is a monotonefunction (since, as we have shown, ϕ σ satisfies (DV2)), 0 ≪ a and ϕ σ (0) = 0, weobtain that ϕ σ ( a ) = W { ϕ σ ( b ) | b ∈ A, b ≪ a } . If ϕ σ ( a ) = 1 then a ∗ σ . Thus, by(CL3), there exists c ∈ σ such that a ∗ ( − C ) c . Hence c ≪ a . Then there exists b ∈ A such that c ≪ b ≪ a . Since c ( − C ) b ∗ , we obtain that b ∗ σ . Therefore ϕ σ ( b ) = 1.This implies that ϕ σ ( a ) = W { ϕ σ ( b ) | b ≪ a } . Hence, ϕ σ satisfies the axiom (DV4).So, ϕ σ ∈ DeV (( A, C ) , ( , ρ s )) and σ = ξ ( ϕ σ ). All this shows that ξ is abijection. Also, we have seen that ξ − ( σ ) = ϕ σ , for every σ ∈ Clust(
A, C ). The definition of de Vries’ dual equivalence Ψ a is given on the language of clusters(see the proof of Theorem 4.3). The above Proposition 6.2 shows that we can use deVries’ morphisms from a CNCA to ( , ρ s ) instead of clusters. Transporting everythingfrom clusters to morphisms via the bijection ξ − from 6.2, we will here express thedefinition of Ψ a in a new much more natural and beautiful form. Although we haveset above S df = Ψ a , in order to distinguish between the old and new form of Ψ a , wewill use the symbol S when we have in mind the new form of Ψ a .Let us first introduce some notation. For every CNCA ( A, C ) and each a ∈ A ,we set X ( A,C ) df = DeV (( A, C ) , ( , ρ s )) and υ ′ ( A,C ) ( a ) df = ξ − ( υ ( A,C ) ( a )) . Thus, according to Proposition 6.2, we obtain that υ ′ ( A,C ) ( a ) = { ϕ ∈ X ( A,C ) | ϕ ( a ∗ ) = 0 } (12) 32nd therefore, X ( A,C ) \ υ ′ ( A,C ) ( a ) = { ϕ ∈ X ( A,C ) | ϕ ( a ∗ ) = 1 } . (13)Now we can prove the following assertion: Proposition.
The new form S of the dual equivalence Ψ a is the following one: • for any ( A, C ) ∈ | B | , S ( A, C ) df = ( X ( A,C ) , T ′ ) , where the topology T ′ is generated by the closed base { υ ′ ( A,C ) ( a ) | a ∈ A } , • for every α ∈ B (( A, C ) , ( A ′ , C ′ )) , S ( α ) : S ( A ′ , C ′ ) −→ S ( A, C ) is defined bythe formula S ( α )( ϕ ′ ) df = ϕ ′ ⋄ α, for any ϕ ′ ∈ X ( A ′ ,C ′ ) .Proof. The definition of S on the objects of B is obtained simply by transportingthe topological structure of Ψ a ( A, C ) from Clust(
A, C ) to X ( A,C ) via the bijection ξ − from Proposition 6.2. Thus Ψ a ( A, C ) and S ( A, C ) are homeomorphic topologicalspaces. Also, we obtain that the family {{ ϕ ∈ X ( A,C ) | ϕ ( a ∗ ) = 1 } | a ∈ A } is an open base for the topology T ′ .If α ∈ B (( A, C ) , ( A ′ , C ′ )), then we have that Ψ a ( α )( σ ′ ) = σ , for every σ ′ ∈ Clust( A ′ , C ′ ), where σ = { a ∈ A | if b ∈ A and b ≪ C a ∗ then ( α ( b )) ∗ ∈ σ ′ } (see4.3). Thus, in the notation of Proposition 6.2, the transportation of the clustersvia the bijection ξ − gives us the following formula: S ( α )( ϕ σ ′ ) df = ϕ σ . Now, usingagain Proposition 6.2, as well as the definition of the composition ⋄ (see Definition4.1), we obtain that, for every a ∈ A , ϕ σ ( a ) = 0 ⇔ a ∗ ∈ σ ⇔ [( b ∈ A and b ≪ a ) → ( ϕ σ ′ ( α ( b )) = 0)] ⇔ ( ϕ σ ′ ⋄ α )( a ) = 0. Hence, S ( α )( ϕ σ ′ ) = ϕ σ ′ ⋄ α . Since ξ is a bijection, we can rewrite this formula as follows: S ( α )( ϕ ′ ) df = ϕ ′ ⋄ α, for each ϕ ′ ∈ X ( A ′ ,C ′ ) .Let us also note that, setting Y df = S ( A, C ) and using Remark 4.4(d) and (13),one has, for every a ∈ A ,int Y ( υ ′ ( A,C ) ( a )) = { y ∈ Y | y ( a ) = 1 } . (14) We are now almost ready for defining our category C ′′ . Let us start with thefollowing definition: Definition.
If (
A, C ) is an CNCA, X is a set and f ∈ Set ( X, X ( A,C ) ), then f isa t-injection (resp., t-inclusion ) if f is an injection (resp., an inclusion) and for each33 ∈ A there exists x ∈ X such that f ( x )( a ) = 1. We will also express the fact that f : X −→ X ( A,C ) is a t-inclusion by saying that X is t-included in X ( A,C ) .The following two assertions are obvious: Fact. If ( A, C ) is an CNCA, then a subset X of X ( A,C ) is t-included in X ( A,C ) if,and only if, X is a dense subset of S ( A, C ) . Proposition.
One may define a category C ′′ , as follows : • its objects are all pairs (( A, C ) , X ) , where ( A, C ) is a CNCA, X is t-included in X ( A,C ) , and for every CNCA ( A ′ , C ′ ) and every t-injection f : X −→ X ( A ′ ,C ′ ) for which { X ∩ υ ′ ( A,C ) ( a ) | a ∈ A } = { f − ( υ ′ ( A ′ ,C ′ ) ( a ′ )) | a ′ ∈ A ′ } , (15) there exists α ∈ B (( A ′ , C ′ ) , ( A, C )) such that f ( x ) = x ⋄ α for every x ∈ X ; • its morphisms are all pairs ( α, f ) : (( A, C ) , X ) −→ (( A ′ , C ′ ) , X ′ ) such that α ∈ B (( A, C ) , ( A ′ , C ′ )) , f ∈ Set ( X ′ , X ) and f = S ( α ) | X ′ ; • composition is as in B and Set ; that is, ( α, f ) as above gets composed with ( α ′ , f ′ ) : (( A ′ , C ′ ) , X ′ ) −→ (( A ′′ , C ′′ ) , X ′′ ) by the horizontal pasting of dia-grams, that is, ( α ′ , f ′ ) ◦ ( α, f ) df = ( α ′ ⋄ α, f ◦ f ′ ); • the identity morphism of an (( A, C ) , X ) ∈ | C ′′ | is the C ′′ -morphism (1 ( A,C ) , X ) . For brevity, the condition (15) will be written in the following form: X ∩ υ ′ ( A,C ) ( A ) = f − ( υ ′ ( A ′ ,C ′ ) ( A ′ )) . Remark.
Using Proposition 6.3, one can easily obtain that ((
A, C ) , X ) ∈ | C ′′ | if,and only if, X is a dense subspace of the compact Hausdorff space S ( A, C ) suchthat if f : X −→ S ( A ′ , C ′ ) is a compactification of X , then there exists a contin-uous map g : S ( A, C ) −→ S ( A ′ , C ′ ) with g | X = f ; at that, condition (15) meansthat the subspace X of S ( A, C ) is homeomorphic to the subspace f ( X ) of S ( A ′ , C ′ )(recall that υ ′ ( A,C ) ( A ) = RC( S ( A, C )) and υ ′ ( A ′ ,C ′ ) ( A ′ ) = RC( S ( A ′ , C ′ )) (see 4.3)).In other words, if β : X −→ S ( A, C ) is the inclusion map, then β is a Stone- ˇCechcompactification of X (we do not regard it here up to equivalence). Proposition 6.5.
The categories C ′ and C ′′ are isomorphic.Proof. We will define a functor E ′ : C ′ −→ C ′′ . Let (( A, C ) , j ) ∈ | C ′ | . Setting X df = dom( j ), we obtain that j : X ֒ → S ( A, C ), the subset X (= j ( X )) is dense in S ( A, C ) and X is C ∗ -embedded in S ( A, C ). Set E ′ (( A, C ) , j ) df = (( A, C ) , X ) . X is t-included in X ( A,C ) ; also, thefact that X is C ∗ -embedded in S ( A, C ) shows, as it is well known, that the requiredextensions described in Remark 6.4 can be obtained. Hence, E ′ (( A, C ) , j ) ∈ | C ′′ | .Further, it is easy to see that setting for every C ′ -morphism ( α, f ), E ′ ( α, f ) df = ( α, f ) , we obtain that ( α, f ) is a C ′′ -morphism. Clearly, E ′ is a functor which is full, faithfuland injective on objects. For showing that it is surjective on objects, let (( A, C ) , X ) ∈| C ′′ | . Then, using Remark 6.4, we obtain that the inclusion j : X −→ S ( A, C ) is adense embedding and X is C ∗ -embedded in S ( A, C ), i.e., ((
A, C ) , j ) ∈ | C ′ | . Since E ′ (( A, C ) , j ) = (( A, C ) , X ), we obtain that E ′ is surjective on objects. Therefore, E ′ is an isomorphism.Now we obtain the following result: Theorem 6.6.
There is a dual equivalence between the categories C ′′ and Tych whichextends de Vries’ dual equivalence Ψ a between the categories DeV and
CHaus .Proof.
Setting S ′ df = S ◦ I ′ ◦ ( E ′ ) − , we obtain, using 6.1 and Propositions 6.5, 6.1,that S ′ is a dual equivalence. Now define a functor I ′′ : DeV −→ C ′′ by I ′′ (( A, C )) df =(( A, C ) , X ( A,C ) ) and I ′′ ( α ) df = ( α, S ( α )). Clearly, it is a full embedding and I = I ′ ◦ ( E ′ ) − ◦ I ′′ . Hence, S ′ ◦ I ′′ = J ◦ S .Now we are going to prove that the category C ′′ is equivalent to the category BMO . First we need to prove some lemmas and to recall some definitions and factsabout the Tarski duality and the Bezhanishvili-Morandi-Olberding Duality Theorem.
We recal that the Tarski duality between the categories
Set and
CaBa is givenby the contravariant functors T s : Set −→ CaBa and T a : CaBa −→ Set which are defined as follows. For every set X , T s ( X ) df = ( P ( X ) , ⊆ ) . If f ∈ Set ( X, Y ), then T s ( f ) : T s ( Y ) −→ T s ( X ) is defined by the formula T s ( f )( M ) df = f − ( M ) , for every M ∈ P ( Y ). Further, for every B ∈ | CaBa | , T a ( B ) df = At( B );if ϕ ∈ CaBa ( A, B ), then T a ( ϕ ) : T a ( B ) −→ T a ( A ) is defined by the formula T a ( ϕ )( x ) df = ^ { a ∈ A | x ≤ ϕ ( a ) } , x ∈ T a ( B ). For each set X , we have a natural isomorphism η TX : X −→ T a ( T s ( X )) (= At( P ( X )) = {{ x } | x ∈ X } ) , given by η TX ( x ) = { x } for each x ∈ X . For each B ∈ | CaBa | we have a naturalisomorphism ε TB : B −→ T s ( T a ( B )) (= P (At( B ))) , given by ε TB ( b ) df = { x ∈ At( B ) | x ≤ b } for each b ∈ B .The following assertion is well-known (because T a ( ϕ ) is the restriction on At( B ′ )of the lower (or, left) adjoint for ϕ (see [28, Theorem 4.2]), but we will present hereits short proof. Lemma 6.8.
Let ϕ ∈ CaBa ( B, B ′ ) . Then, for every b ∈ B and each x ′ ∈ At( B ′ ) , ( x ′ ≤ ϕ ( b )) ⇔ ( T a ( ϕ )( x ′ ) ≤ b ) .Proof. Since T a ( ϕ )( x ′ ) = V { b ∈ B | x ′ ≤ ϕ ( b ) } , we obtain immediately that ( x ′ ≤ ϕ ( b )) ⇒ ( T a ( ϕ )( x ′ ) ≤ b ). Suppose now that T a ( ϕ )( x ′ ) ≤ b . Then ϕ ( T a ( ϕ )( x ′ )) ≤ ϕ ( b ). Since ϕ ( T a ( ϕ )( x ′ )) = ϕ ( V { c ∈ B | x ′ ≤ ϕ ( c ) } ) = V { ϕ ( c ) | c ∈ B, x ′ ≤ ϕ ( c ) } ≥ x ′ , we obtain that x ′ ≤ ϕ ( b ). The following assertion is well-known:
Fact.
For every complete atomic Boolean algebra B , there is a bijection m B betweenthe sets At( B ) and CaBa ( B, ) , namely, • for every x ∈ At( B ) , we set m B ( x ) df = u x , where u x ∈ CaBa ( B, ) is defined by u x ( b ) df = 1 ⇔ x ≤ b, for every b ∈ B ; • to every u ∈ CaBa ( B, ) corresponds x u ∈ At( B ) defined by x u df = ^ u − (1) (= ^ { b ∈ B | u ( b ) = 1 } ); at that, x = x u x and u = u x u , for every u ∈ CaBa ( B, ) and for every x ∈ At( B ) . Note that the notation “ u x ” in the above assertion was already used in (5),but we hope that it will be clear from the context which of the two meanings of thisnotation is used. We now recall the Bezhanishvili-Morandi-Olberding Duality Theorem, startingwith the main definitions of [9]: • if ( A, C ) is a CNCA, B ∈ | CaBa | and γ : ( A, C ) −→ ( B, ρ s ) is an injective deVries morphism, then γ is called a de Vries extension provided that each atomof B is a meet from γ ( A ); 36 two de Vries extensions γ : ( A, C ) −→ ( B, ρ s ) and γ ′ : ( A ′ , C ′ ) −→ ( B, ρ s ) aresaid to be compatible if γ ( A ) = γ ′ ( A ′ ). • a de Vries extension γ : ( A, C ) −→ ( B, ρ s ) is called maximal if for every com-patible de Vries extension γ ′ : ( A ′ , C ′ ) −→ ( B, ρ s ) there is a de Vries morphism α ′ : ( A ′ , C ′ ) −→ ( A, C ) such that γ ⋄ α ′ = γ ′ .Now we are ready to recall the definition of the category BMO : • its objects are all maximal de Vries extensions; • its morphisms are all pairs ( α, ς ) : γ −→ γ ′ , where γ : ( A, C ) −→ ( B, ρ s ) and γ ′ : ( A ′ , C ′ ) −→ ( B ′ , ρ s ) are de Vries extensions, α ∈ DeV (( A, C ) , ( A ′ , C ′ )) , ς ∈ CaBa ( B, B ′ ) and ς ◦ γ = γ ′ ⋄ α ; • composition is as in DeV and
CaBa ; that is, ( α, ς ) as above gets composedwith ( α ′ , ς ′ ) : γ ′ −→ γ ′′ as follows: ( α ′ , ς ′ ) ◦ ( α, ς ) df = ( α ′ ⋄ α, ς ′ ◦ ς ); • the identity morphism of a BMO -object γ : ( A, C ) −→ ( B, ρ s ) is the BMO -morphism (1 ( A,C ) , B ) . Let us recall the following assertions from [9]:
Theorem. ([9, Theorem 4.5])
Let c : X −→ Y be a Hausdorff compactification of aTychonoff space X . Then the map γ : (RO( Y ) , D Y ) −→ ( P ( X ) , ρ s ) , U c − ( U ) , isa de Vries extension. Lemma. ([9, Lemma 6.2])
Two de Vries extensions γ : ( A, C ) −→ ( B, ρ s ) and γ ′ : ( A ′ , C ′ ) −→ ( B, ρ s ) are compatible if, and only if, the initial topologies on the set X B df = {↑ ( x ) | x ∈ At( B ) } generated by the map Ψ a ( γ ) | X B and the map Ψ a ( γ ′ ) | X B ,respectively, are equal. Proposition. ([9, Theorem 6.4((3) → (1))]) Let X be a Tychonoff space, c : X −→ Y be a Hausdorff compactification of X which is equivalent to the Stone- ˇCech compact-ification of X . Then the de Vries extension γ c : ( RC ( Y ) , ρ Y ) −→ ( P ( X ) , ρ s ) ismaximal. Since we work with the Boolean algebra RC( Y ), we have to restate Theo-rem 6.10. By Example 2.3, there exists a CA-isomorphism ν : (RC( Y ) , ρ Y ) −→ (RO( Y ) , D Y ) , F int Y ( F ) . Thus, using Theorem 6.10, it is easy to see that themap γ ⋄ ν : (RC( Y ) , ρ Y ) −→ ( P ( X ) , ρ s ) is a de Vries extension. We will show, how-ever, that even the map γ ◦ ν is a de Vries extension. Obviously, this will imply that γ ⋄ ν = γ ◦ ν . Proposition.
Let c : X −→ Y be a Hausdorff compactification of a Tychonoff space X . Then the map γ c : (RC( Y ) , ρ Y ) −→ ( P ( X ) , ρ s ) , F c − (int Y ( F )) , is a de Vriesextension. roof. Since the map r : RO( Y ) −→ RO( X ) , U c − ( U ) is a Boolean isomorphism(see [14, p. 271] or [37, Lemma 44]), we obtain that γ c is an injection. Let us show thatit is a de Vries morphism. Obviously, condition (DV1) is satisfied. For showing that(DV2) is fulfilled, let F, G ∈ RC( Y ). We have to prove that γ c ( F ∧ G ) = γ c ( F ) ∩ γ c ( G ),i.e., that c − (int Y ( F ∧ G )) = c − (int Y ( F )) ∩ c − (int Y ( F )). Obviously, it is enough toshow that int Y ( F ∧ G ) = int Y ( F ∩ G ). Since int Y ( F ∧ G ) = int Y (cl Y (int Y ( F ∩ G )))and, as it is well known, int Y ( F ∩ G ) ∈ RO( Y ), we conclude that (DV2) is satisfied.For proving (DV3), let F ≪ G , i.e., F ⊆ int Y ( G ). We have to show that ( γ c ( F ∗ )) ∗ ⊆ γ c ( G ), i.e., that X \ c − (int Y ( F ∗ )) ⊆ c − (int Y ( G )). We have that int Y ( F ∗ ) = Y \ F .Thus X \ c − (int Y ( F ∗ )) = c − ( F ) ⊆ c − (int Y ( G )). So, (DV3) is also satisfied. Since Y is a regular space, we have that int Y ( F ) = S { int Y ( G ) | G ∈ RC( Y ) , G ⊆ int Y ( F ) } ,for every F ∈ RC( Y ). This implies that (DV4) is fulfilled. Hence, γ c is a de Vriesmorphism. Since { int Y ( F ) | F ∈ RC( Y ) } is a base for Y , we obtain that γ c is a deVries extension. Lemma 6.12. If γ : ( A, C ) −→ ( B, ρ s ) is a de Vries extension, then there exists abijection µ between the sets { u x | x ∈ At( B ) } and { u x ◦ γ | x ∈ At( B ) } (see Fact 6.9for notation). (When it is needed, we will write µ γ instead of µ .)Proof. For every x ∈ At( B ), set µ ( u x ) df = u x ◦ γ . Let x, y ∈ At( B ) and u x = u y . Then x = y . There exists a subset A x of A such that x = V γ ( A x ). Thus 1 = u x ( x ) = u x ( V γ ( A x )) = V { u x ( γ ( a )) | a ∈ A x } . Therefore, u x ( γ ( a )) = 1 for every a ∈ A x .There exists a ∈ A x such that y (cid:2) γ ( a ). Then u y ( γ ( a )) = 0 = 1 = u x ( γ ( a )).Hence µ is an injection. Clearly, µ is a surjection. Therefore, µ is a bijection. Theorem 6.13.
The categories
BMO and C ′′ are equivalent.Proof. We define a functor Θ :
BMO −→ C ′′ as follows: • for every de Vries’ extension γ : ( A, C ) −→ ( B, ρ s ), we setΘ( γ ) df = (( A, C ) , X γ ) , where X γ df = { u x ◦ γ | x ∈ At( B ) } ; • for every ( α, ς ) ∈ BMO ( γ, γ ′ ), where γ : ( A, C ) → ( B, ρ s ) and γ ′ : ( A ′ , C ′ ) → ( B ′ , ρ s ), we set Θ(( α, ς )) df = ( α, f ς ) , where f ς : X γ ′ −→ X γ is defined by f ς ( u x ′ ◦ γ ′ ) df = u T a ( ς )( x ′ ) ◦ γ, ∀ x ′ ∈ At( B ′ ) . Now we have to show that the functor Θ is well-defined. Let us start by provingthat it is well-defined on objects.First of all, note that ( u ∈ CaBa ( B, )) ⇒ ( u ∈ DeV (( B, ρ s ) , ( , ρ s ))) and u ◦ γ = u ⋄ γ (see Fact 4.2(d)). Thus, if γ : ( A, C ) −→ ( B, ρ s ) is a de Vries’ extension,then X γ ⊆ X ( A,C ) . Moreover, X γ is t-included in X ( A,C ) . Indeed, if a ∈ A + then38 ( a ) = 0 (because γ (0) = 0 and γ is an injection); hence, there exists x ∈ At( B ) suchthat x ≤ γ ( a ); this means that ( u x ◦ γ )( a ) = 1. Therefore, X γ is t-included in X ( A,C ) .Let now f : X γ −→ X ( A ′ ,C ′ ) be a t-injection for which X γ ∩ υ ′ ( A,C ) ( A ) = f − ( υ ′ ( A ′ ,C ′ ) ( A ′ )) . (16)We have to show that there exists α ∈ DeV (( A ′ , C ′ ) , ( A, C )) such that f ( u x ◦ γ ) =( u x ◦ γ ) ⋄ α for every x ∈ At( B ).Set X df = { u x | x ∈ At( B ) } , Y df = S ( A, C ), Z df = S ( A ′ , C ′ ) and let i γ : X γ ֒ → Y be the inclusion map. Note that, by Remark 4.4 and Proposition 6.3, X ⊆ S ( B, ρ s ) = S a ( B ). Now we define c df = µ ◦ i γ and c ′ df = f ◦ µ. We have, by Lemma 6.12, that µ ( X ) = X γ . Hence, c ( X ) is dense in Y and if T c is theinitial topology on X generated by the map c , then c : X −→ Y is a compactificationof ( X, T c ). Further, since f is a t-injection, we obtain that c ′ ( X ) is dense in Z and c ′ is an injection. Thus, if if T c ′ is the initial topology on X generated by the map c ′ , then c ′ : X −→ Z is a compactification of ( X, T c ′ ). Moreover, the topologies T c and T c ′ on X are equal. Indeed, µ − ( X γ ∩ υ ′ ( A,C ) ( A )) is a closed base for thetopology T c and µ − ( f − ( υ ′ ( A ′ ,C ′ ) ( A ′ ))) is a closed base for the topology T c ′ . Thus,by (16), T c = T c ′ . Also, note that Proposition 6.3) and Fact 6.9 show that the set X defined above plays the role of the set X B from Lemma 6.10. Since, by Proposition6.3, µ ( u x ) = S ( γ )( u x ) for every x ∈ At( B ), we obtain that the initial topology on X generated by the map S ( γ ) | X coincides with the topology T c . We will now define a de Vries extension γ ′ : ( A ′ , C ′ ) −→ ( B, ρ s ) such that γ ( A ) = γ ′ ( A ′ ).By Proposition 6.11, γ c ′ : ( RC ( Z ) , ρ Z ) −→ ( P ( X ) , ρ s ), G ( c ′ ) − (int Z ( G )), isa de Vries’ extension. Set, for brevity, ε B df = m B ◦ ε TB (see Fact 6.9 and 6.7 for thenotation). Now we define γ ′ df = ε − B ◦ ( γ c ⋄ υ ′ ( A ′ ,C ′ ) ) . Since ε B can be regarded as a DeV -isomorphism from (
B, ρ s ) to ( P ( X ) , ρ s ), weobtain that γ ′ : ( A ′ , C ′ ) −→ ( B, ρ s ) is a de Vries’ morphism. Obviously, by Theorem4.5, S ( γ ′ ) is a surjection (as a composition of surjections). Thus, applying once moreTheorem 4.5, we obtain that γ ′ is an injection. Now it is easy to see that γ ′ is a deVries’ extension. We will show that the initial topology on X generated by the map S ( γ ′ ) | X coincides with the initial topology on X generated by the map c ′ , i.e., withtopology T c on X . So, we have to prove that S ( γ ′ )( u x ) = c ′ ( u x ) for every x ∈ At( B ).We have that c ′ ( u x ) = f ( µ ( u x )) = f ( u x ◦ γ ) and, by Proposition 6.3 and Fact4.2(d), S ( γ ′ )( u x ) = u x ◦ γ ′ = u x ◦ ε − B ◦ ( γ c ′ ⋄ υ ′ ( A ′ ,C ′ ) ). Further, for any a ′ ∈ A ′ ,we obtain, using (14), that ( γ c ′ ⋄ υ ′ ( A ′ ,C ′ ) )( a ′ ) = S { γ c ′ ( υ ′ ( A ′ ,C ′ ) ( b )) | b ≪ a ′ } = S { µ − ( f − ((int Z ( υ ′ ( A ′ ,C ′ ) ( b ))))) | b ≪ a ′ } = S { µ − ( { u y ◦ γ | y ∈ At( B ) , f ( u y ◦ γ )( b ) =1 } ) | b ≪ a ′ } = S {{ u y | y ∈ At( B ) , f ( u y ◦ γ )( b ) = 1 } ) | b ≪ a ′ } . Since, for every M ⊆ X , ε − B ( M ) = W m − B ( M ), we obtain that ε − B (( γ c ′ ⋄ υ ′ ( A ′ ,C ′ ) )( a ′ )) = W { y ∈ At( B ) | f ( u y ◦ γ )( b ) = 1 , b ≪ a ′ } . Now we have, by Fact 6.9, that u x ( γ ′ ( a ′ )) = 1 ⇔ ≤ γ ′ ( a ′ ) ⇔ x ≤ ε − B (( γ c ′ ⋄ υ ′ ( A ′ ,C ′ ) )( a ′ )) ⇔ x ≤ W { y ∈ At( B ) | f ( u y ◦ γ )( b ) =1 , b ≪ a ′ } ⇔ ( ∃ b ≪ a ′ such that f ( u x ◦ γ )( b ) = 1) ⇔ f ( u x ◦ γ )( a ′ ) = 1. Hence, u x ◦ γ ′ = f ( u x ◦ γ ), for every x ∈ At( B ). Therefore, S ( γ ′ )( u x ) = c ′ ( u x ) for every x ∈ At( B ).All this shows that the initial topologies on X generated by the maps S ( γ ) | X and S ( γ ′ ) | X , respectively, are equal. Now, Lemma 6.10 implies that γ ( A ) = γ ′ ( A ′ ).Since γ is a maximal de Vries’ extension, there exists α ∈ DeV (( A ′ , C ′ ) , ( A, C )) suchthat γ ′ = γ ⋄ α . Hence, we obtain that for every x ∈ At( B ), f ( u x ◦ γ ) = S ( γ ′ )( u x ) = u x ◦ γ ′ = u x ◦ ( γ ⋄ α ) = ( u x ◦ γ ) ⋄ α . Therefore, Θ( γ ) ∈ | C ′′ | .We now show that Θ is well-defined on morphisms. Let ( α, ς ) ∈ BMO ( γ, γ ′ ),where γ : ( A, C ) −→ ( B, ρ s ) and γ ′ : ( A ′ , C ′ ) −→ ( B ′ , ρ s ) are de Vries’ extensions.Then α ∈ DeV (( A, C ) , ( A ′ , C ′ )), ς ∈ CaBa ( B, B ′ ) and ς ◦ γ = γ ′ ⋄ α . We havethat Θ( γ ) = (( A, C ) , X γ ) and Θ( γ ′ ) = (( A ′ , C ′ ) , X γ ′ ), where X γ = { u x ◦ γ | x ∈ At( B ) } and X γ ′ = { u x ′ ◦ γ ′ | x ′ ∈ At( B ′ ) } . Also, Θ(( α, ς )) = ( α, f ), where f : X γ ′ −→ X γ , u x ′ ◦ γ ′ u T a ( ς )( x ′ ) ◦ γ , for every x ′ ∈ At( B ′ ). We have to prove that f = S ( α ) | X γ ′ , i.e., that u T a ( ς )( x ′ ) ◦ γ = ( u x ′ ◦ γ ′ ) ⋄ α , for every x ′ ∈ At( B ′ ). Fix a x ′ ∈ At( B ′ ). Note that ( u x ′ ◦ γ ′ ) ⋄ α = u x ′ ⋄ γ ′ ⋄ α = u x ′ ◦ ( γ ′ ⋄ α ) = u x ′ ◦ ς ◦ γ .Also, for every a ∈ A , u x ′ ( ς ( γ ( a ))) = 1 ⇔ x ′ ≤ ς ( γ ( a )) and u T a ( ς )( x ′ ) ( γ ( a )) = 1 ⇔ T a ( ς )( x ′ ) ≤ γ ( a ). Now, applying Lemma 6.8, we obtain that u x ′ ◦ ς ◦ γ = u T a ( ς )( x ′ ) ◦ γ .Thus, f = S ( α ) | X γ ′ . Therefore, Θ(( α, ς )) ∈ C ′′ (Θ( γ ) , Θ( γ ′ )), i.e., Θ is well-definedon morphisms.Now, it is easy to see that Θ : BMO −→ C ′′ is a functor. Let us show that Θis full and faithful. Let γ : ( A, C ) −→ ( B, ρ s ) and γ ′ : ( A ′ , C ′ ) −→ ( B ′ , ρ s ) be deVries’ extensions. We have to prove that the restrictionΘ : BMO ( γ, γ ′ ) −→ C ′′ (Θ( γ ) , Θ( γ ′ ))is a bijection. For proving injectivity, we let ( α, ς ) , ( α ′ , ς ′ ) ∈ BMO ( γ, γ ′ ) and assume( α, ς ) = ( α ′ , ς ′ ). Then Θ(( α, ς )) = ( α, f ς ) and Θ(( α ′ , ς ′ )) = ( α ′ , f ς ′ ), where α, α ′ ∈ DeV (( A, C ) , ( A ′ , C ′ )), f ς , f ς ′ : X γ ′ −→ X γ , X γ = { u x ◦ γ | x ∈ At( B ) } , X γ ′ = { u x ′ ◦ γ ′ | x ′ ∈ At( B ′ ) } , f ς ( u x ′ ◦ γ ′ ) = u T a ( ς )( x ′ ) ◦ γ and f ς ′ ( u x ′ ◦ γ ′ ) = u T a ( ς ′ )( x ′ ) ◦ γ .If α = α ′ , then, clearly, Θ(( α, ς )) = Θ(( α ′ , ς ′ )). Let now α = α ′ and ς = ς ′ . Then,by the Tarski duality, T a ( ς ) = T a ( ς ′ ). Hence, there exists x ′ ∈ At( B ′ ) such that T a ( ς )( x ′ ) = T a ( ς ′ )( x ′ ). Using Fact 6.9 and Lemma 6.12, we obtain that f ς ( u x ′ ◦ γ ′ ) = f ς ′ ( u x ′ ◦ γ ′ ), i.e., f ς = f ς ′ and thus, Θ(( α, ς )) = Θ(( α ′ , ς ′ )). So, Θ is a faithful functor.We now prove that Θ is full, i.e., the above restriction of Θ is a surjection.Let ( α, f ) ∈ C ′′ (Θ( γ ) , Θ( γ ′ )). Then f : X γ ′ −→ X γ and f = S ( α ) | X γ ′ . As Fact6.9 and Lemma 6.12 show, the maps µ γ ◦ m B : At( B ) −→ X γ , x u x ◦ γ, and µ γ ′ ◦ m B ′ : At( B ′ ) −→ X γ ′ , x ′ u x ′ ◦ γ ′ , are bijections. Set λ df = µ γ ◦ m B , λ ′ df = µ γ ′ ◦ m B ′ and f df = λ − ◦ f ◦ λ ′ . Then f : At( B ′ ) −→ At( B ). Since T a is full (and faithful), there exists a (unique) ς ∈ CaBa ( B, B ′ ) such that T a ( ς ) = f .
40e will prove that ( α, ς ) ∈ BMO ( γ, γ ′ ) and Θ(( α, ς )) = ( α, f ). Let us first show that( α, ς ) ∈ BMO ( γ, γ ′ ). We need only to check that ς ◦ γ = γ ′ ⋄ α . Since S is faithful,it is enough to prove that S ( ς ◦ γ ) = S ( γ ′ ⋄ α ), i.e., that S ( γ ) ◦ S ( ς ) = S ( α ) ◦ S ( γ ′ ).Since CaBa ( B ′ , ) is a dense subset of S (( B ′ , ρ s )) (= S a ( B ′ )) (see Remark 4.4),we need only to prove that S ( γ ) ◦ S ( ς ) = S ( α ) ◦ S ( γ ′ ) on CaBa ( B ′ , ), i.e., on { u x ′ | x ′ ∈ At( B ′ ) } (see Fact 6.9). So, let x ′ ∈ At( B ′ ). Then S ( γ )( S ( ς )( u x ′ )) = S ( γ )( u x ′ ◦ ς ) = u x ′ ◦ ς ◦ γ and S ( α )( S ( γ ′ )( u x ′ )) = S ( α )( u x ′ ◦ γ ′ ) = f ( u x ′ ◦ γ ′ )(since f = S ( α ) | X γ ′ ). Hence, we have to show that f ( u x ′ ◦ γ ′ ) = u x ′ ◦ ς ◦ γ . Since f = λ ◦ f ◦ ( λ ′ ) − , we obtain that f ( u x ′ ◦ γ ′ ) = ( λ ◦ f ◦ ( λ ′ ) − )( u x ′ ◦ γ ′ ) = λ ( f ( x ′ )) = λ ( T a ( ς )( x ′ )) = u T a ( ς )( x ′ ) ◦ γ . We also have that, for every a ∈ A , u T a ( ς )( x ′ ) ( γ ( a )) =1 ⇔ T a ( ς )( x ′ ) ≤ γ ( a ) ⇔ x ′ ≤ ς ( γ ( a )) ⇔ u x ′ ( ς ( γ ( a ))) = 1 (we applied Lemma 6.8here). Therefore, S ( γ ) ◦ S ( ς ) = S ( α ) ◦ S ( γ ′ ) and, thus, ( α, ς ) ∈ BMO ( γ, γ ′ ). Wewill now show that Θ(( α, ς )) = ( α, f ), i.e. that f ( u x ′ ◦ γ ′ ) = u T a ( ς )( x ′ ) ◦ γ for any x ′ ∈ At( B ′ ). Since the validity of this equation was already demonstrated, we obtainthat Θ is a full functor.Finally, we prove that Θ is essentially surjective on objects. Let (( A, C ) , X ) ∈| C ′′ | . Set Y df = S ( A, C ). Then X ⊆ DeV (( A, C ) , ( , ρ s )) = Y , X is dense in Y and if β : X ֒ → Y is the inclusion map, then β is the Stone- ˇCech compactification of X (wedo not regard it here up to equivalence) (see Remark 6.4). Let γ : (RC( Y ) , ρ Y ) −→ ( P ( X ) , ρ s ) be defined by γ ( G ) df = X ∩ int Y ( G ), for every G ∈ RC( Y ). Then Proposi-tions 6.11 and 6.10 imply that γ ∈ | BMO | . We will prove that Θ( γ ) is C ′′ -isomorphicto (( A, C ) , X ). We have that Θ( γ ) = ((RC( Y ) , ρ Y ) , X γ ), where X γ = { u x ◦ γ | x ∈ X } (since At( P ( X )) = {{ x } | x ∈ X } and writing u x instead of u { x } ). As we alreadynoted, the map λ : X −→ X γ , x u x ◦ γ, is a bijection. We will show that( υ ′ ( A,C ) , λ − ) : (( A, C ) , X ) −→ Θ( γ ) is a C ′′ -isomorphism. Set α df = υ ′ ( A,C ) . We firsthave to prove that ( α, λ − ) is a C ′′ -morphism, i.e. that λ − = S ( α ) | X γ . We have thatfor every x ∈ X , S ( α )( u x ◦ γ ) = ( u x ◦ γ ) ⋄ α and λ − ( u x ◦ γ ) = x . So that, we need toshow that x = ( u x ◦ γ ) ⋄ α . For every a ∈ A , we have that (( u x ◦ γ ) ⋄ α )( a ) = W { (( u x ◦ γ ) ◦ α )( b ) | b ≪ a } = W { u x ( γ ( α ( b ))) | b ≪ a } = W { u x ( X ∩ int Y ( α ( b ))) | b ≪ a } .Recall that int Y ( α ( b )) = { ϕ ∈ DeV (( A, C ) , ( , ρ s )) | ϕ ( b ) = 1 } . Now, we have that u x ( X ∩ int Y ( α ( b ))) = 1 ⇔ x ∈ X ∩ int Y ( α ( b )) ⇔ x ∈ int Y ( α ( b )) ⇔ x ( b ) = 1. Hence,(( u x ◦ γ ) ⋄ α )( a ) = 1 ⇔ ( ∃ b ∈ A such that b ≪ a and x ( b ) = 1) ⇔ x ( a ) = 1. Therefore, x = ( u x ◦ γ ) ⋄ α . Hence, ( α, λ − ) is a C ′′ -morphism. Since α is a DeV -isomorphism and λ − is a Set -isomorphism, we obtain that ( α, λ − ) is a C ′′ -isomorphism. Therefore,Θ is essentially surjective on objects.This completes the proof that Θ is an equivalence. Corollary 6.14. ([9])
There is a dual equivalence between the categories
BMO and
Tych .Proof.
Setting S ′′ df = S ′ ◦ Θ, we obtain, using Theorems 6.6 and 6.13, that S ′′ : BMO −→ Tych is a dual equivalence.
Theorem 6.15.
There exists a full embedding I ′′′ : DeV −→ BMO such that J ◦ S = S ′′ ◦ I ′′′ . Hence, we can say that the dual equivalence S ′′ : BMO −→ Tych extendsde Vries’ dual equivalence Ψ a : DeV −→ CHaus . roof. We define a functor I ′′′ : DeV −→ BMO . Let (
A, C ) ∈ | DeV | and set Y df = S ( A, C ). Then id Y : Y −→ Y is a compactification of Y . Set, for short, i df = id Y .Then, by Proposition 6.11, the map γ i : RC( Y ) −→ P ( Y ) , G int Y ( G ) , is a de Vriesextension. Set γ df = γ i ⋄ υ ′ ( A,C ) . Then it is easy to see that γ is a de Vries extension.We set I ′′′ (( A, C )) df = γ, and for every α ∈ DeV (( A, C ) , ( A ′ , C ′ )), I ′′′ ( α ) df = ( α, T s ( S ( α ))) . Clearly, I ′′′ is well-defined on objects. We show that I ′′′ is well-defined on morphisms,as well. Using the above notation, we have only to show that T s ( S ( α )) ◦ γ = γ ′ ⋄ α ,where γ ′ = I ′′′ (( A ′ , C ′ )). Set, for short, Y ′ df = S ( A ′ , C ′ ), ς df = T s ( S ( α )), υ df = υ ′ ( A,C ) and υ ′ df = υ ′ ( A ′ ,C ′ ) . We have that for every a ∈ A , γ ( a ) = ( γ i ⋄ υ )( a ) = S { γ i ( υ ( b )) | b ≪ a } = S { int Y ( υ ( b )) | b ≪ a } = int Y ( υ ( a )) = { y ∈ Y | y ( a ) = 1 } . Hence, for every a ∈ A , ς ( γ ( a )) = ( S ( α )) − ( { y ∈ Y | y ( a ) = 1 } ) = { y ′ ∈ Y ′ | ( S ( α )( y ′ ))( a ) =1 } = { y ′ ∈ Y ′ | ( y ′ ⋄ α )( a ) = 1 } = { y ′ ∈ Y ′ | W { y ′ ( α ( b )) | b ≪ a } = 1 } = { y ′ ∈ Y ′ | ∃ b ≪ a such that y ′ ( α ( b )) = 1 } . Further, as we have shown above, γ ′ ( a ′ ) = { y ′ ∈ Y ′ | y ′ ( a ′ ) = 1 } for every a ′ ∈ A ′ . Hence, for every a ∈ A ,( γ ′ ⋄ α )( a ) = S { γ ′ ( α ( b )) | b ≪ a } = S {{ y ′ ∈ Y ′ | y ′ ( α ( b )) = 1 } | b ≪ a } = { y ′ ∈ Y ′ | ∃ b ≪ a such that y ′ ( α ( b )) = 1 } = ς ( γ ( a )). Thus, T s ( S ( α )) ◦ γ = γ ′ ⋄ α .Therefore, I ′′′ is well-defined on morphisms.Now, it is easy to see that I ′′′ is a functor. As it follows from Theorem 6.6 andits proof, for showing that J ◦ S = S ′′ ◦ I ′′′ , it suffices to prove that I ′′ = Θ ◦ I ′′′ .Let ( A, C ) ∈ | DeV | . Then I ′′ (( A, C )) = ((
A, C ) , Y ) and Θ( I ′′′ (( A, C ))) = Θ( γ ) =(( A, C ) , X γ ), where X γ = { u y ◦ γ | y ∈ Y } (since At( P ( Y )) = Y and writing u y insteadof u { y } ). Hence, for showing that I ′′ (( A, C )) = (Θ ◦ I ′′′ )(( A, C )), it is enough to provethat y = u y ◦ γ , for every y ∈ Y . We have that for every a ∈ A , u y ( γ ( a )) = 1 ⇔ y ∈ γ ( a ) ⇔ y ∈ { z ∈ Y | z ( a ) = 1 } ⇔ y ( a ) = 1. Thus, y = u y ◦ γ , for every y ∈ Y .Therefore, I ′′ (( A, C )) = (Θ ◦ I ′′′ )(( A, C )). Let now α ∈ DeV (( A, C ) , ( A ′ , C ′ )). Usingthe above notation, we obtain that I ′′ ( α ) = ( α, S ( α )) and Θ( I ′′′ ( α )) = Θ(( α, ς )) =( α, f ς ), where f ς : Y ′ −→ Y, y ′ u T a ( ς )( y ′ ) ◦ γ . So, we have to show that S ( α ) = f ς .Using the Tarski duality, we obtain that η TY ◦ S ( α ) = T a ( T s ( S ( α ))) ◦ η TY ′ . Since η TY ( y ) = { y } for every y ∈ Y , and η TY ′ ( y ′ ) = { y ′ } for every y ′ ∈ Y ′ , we obtain that T a ( ς ) = T a ( T s ( S ( α ))) = S ( α ). Thus, f ς ( y ′ ) = u S ( α )( y ′ ) ◦ γ = S ( α )( y ′ ). Therefore, I ′′ = Θ ◦ I ′′′ . This equality, together with the facts that Θ is an equivalence and I ′′′ is an embedding, imply that I ′′′ is a full embedding.We note that a theorem, analogous to the previous one, was proved in [10], butin a completely different way. References [1]
Ad´amek, J., Herrlich, H. and Strecker, G. E.
Abstract and ConcreteCategories . Online edition, 2004, http://katmat.math.uni-bremen.de/acc.422]
Ad´amek, J., Herrlich, H., Rosick´y, J., Tholen, W.
Injective hulls arenot natural. Algebra Universalis 48 (2002) 379–388.[3]
Aiello, M., Pratt-Hartmann, I. and van Benthem, J. (Eds.)
Handbookof spatial logics . Springer-Verlag, Berlin Heidelberg, 2007.[4]
Alexandroff, P. S.
Outline of Set Theory and General Topology . Nauka,Moskva, 1977 (In Russian).[5]
Alexandroff, P. S. and Ponomarev, V. I.
On bicompact extensions oftopological spaces. Vestn. Mosk. Univ. Ser. Mat. (1959), 93–108. (In Russian).[6]
Arhangel’skii, A. V. and Ponomarev, V. I.
Fundamentals of GeneralTopology: Problems and Exercises . Reidel, Dordrecht, 1984. Originally publishedby Izdatelstvo Nauka, Moscow, 1974.[7]
Bezhanishvili, G.
Stone duality and Gleason covers through de Vries duality.Topology and its Applications 157 (2010), 1064–1080.[8]
Bezhanishvili, G., Bezhanishvili, N., Sourabh, S. and Venema, Y.
Irreducible equivalence relations, Gleason spaces, and de Vries duality. AppliedCategorical Structures 25(3) (2017), 381–401.[9]
Bezhanishvili, G., Morandi, P. J. and Olberding, B.
An extension ofDe Vries duality to completely regular spaces and compactifications. Topologyand its Applications 257 (2019), 85–105.[10]
Bezhanishvili, G., Morandi, P. J. and Olberding, B.
De Vries duality fornormal spaces and locally compact Hausdorff spaces. arXiv:1804.04303 (2018),1–25.[11] ˇCech, E.
Topological Spaces.
Interscience, London, 1966.[12]
Celani, S.
Quasi-modal algebras . Math. Bohem. 126 (2001), 721–736.[13]
Cohn, A. G. and Hazarika, S. M.
Qualitative spatial representation andreasoning: An overview . Fundamenta Informaticae 46 (2001), 1–29.[14]
Comfort, W. and Negrepontis, S.
Chain Conditions in Topology . Cam-bridge Univ. Press (Cambridge, 1982).[15] de Vries, H.
Compact Spaces and Compactifications, an Algebraic Approach .Van Gorcum, The Netherlands, 1962.[16]
Dimov, G.
A de Vries-type duality theorem for the category of locally compactspaces and continuous maps – I. Acta Math. Hungarica 129 (2010), 314–349.[17]
Dimov, G.
Proximity-type Relations on Boolean Algebras and their Connectionswith Topological Spaces.
Doctor of Sciences (= Dr. Habil.) Thesis, Faculty ofMathematics and Informatics, Sofia University “St. Kl. Ohridski”, Sofia, 2013,pp. 1–292.[18]
Dimov, G., Ivanova, E.
Yet another duality theorem for locally compactspaces. Houston Journal of Mathematics 42(2) (2016), 675–700.4319]
Dimov, G., Ivanova-Dimova, E. and Tholen, W.
Extensions of dualitiesand a new approach to the Fedorchuk duality. arXiv:1808.06168v3, 1–35.[20]
Dimov, G. and Vakarelov, D.
Contact Algebras and Region-based Theoryof Space: A Proximity Approach - I. Fundamenta Informaticae 74(2-3) (2006),209-249.[21]
Dimov, G. and Vakarelov, D.
Contact Algebras and Region-based Theoryof Space: A Proximity Approach - II. Fundamenta Informaticae 74(2-3) (2006),251-282.[22]
D¨untsch, I. and Vakarelov, D.
Region-based theory of discrete spaces: Aproximity approach. Annals of Mathematics and Artificial Intelligence 49 (2007),5–14.[23]
Efremoviˇc, V. A.
Infinitesimal spaces. DAN SSSR, 76 (1951), 341–343.[24]
Engelking, R.
General Topology, second ed. Sigma Series in Pure Mathemat-ics, vol. 6, Heldermann Verlag, Berlin, 1989.[25]
Fedorchuk, V. V.
Boolean δ -algebras and quasi-open mappings. Sibirsk. Mat.ˇZ. 14 (5) (1973), 1088–1099; English translation: Siberian Math. J. 14 (1973),759-767 (1974).[26] Gleason, A. M.
Projective topological spaces. Illinois J. Math. 2 (1958),482–489.[27]
Hornsby, K. S., Claramunt, C., Denis, M. and Ligozat, G. (Eds.),
Spatial Information Theory: Proceedings of the 9th International Conference,COSIT 2009 . Lecture Notes in Computer Science 5756, Springer Verlag, 2009.[28]
Johnstone, P. T.
Stone Spaces . Cambridge Univ. Press, Cambridge, 1982.[29]
Koppelberg, S.
Handbook on Boolean Algebras, vol. 1: General Theory ofBoolean Algebras . North Holland, 1989.[30]
Mardeˇsic, S. and Papic, P.
Continuous images of ordered compacta, theSuslin property and dyadic compacta. Glasnik mat.-fis. i astronom. 17 (1962),3–25.[31]
Naimpally, S. and Warrack, B.
Proximity Spaces.
Cambridge, London,1970.[32]
Ponomarev, V. I.
Paracompacta: their projection spectra and continuousmappings. Mat. Sb. (N.S.) 60 (1963), 89–119. (In Russian)[33]
Ponomarev, V. I. and ˇSapiro, L. B.
Absolutes of topological spaces andtheir continuous mappings. Uspekhi Mat. Nauk 31 (1976), 121–136. (In Russian)[34]
Randell, D. A., Cui, Z. and Cohn, A. G.
A spatial logic based on regionsand connection. In: B. Nebel -W. Swartout - C. Rich (Eds.), Proceedings of the3rd International Conference Knowledge Representation and Reasoning, MorganKaufmann, Los Allos, CA, 1992, 165176.[35]
Smirnov, J. M.
On proximity spaces. Mat. Sb. 31 (1952), 543–574.4436]
Stone, M. H.
The theory of representations for Boolean algebras. Trans. Amer.Math. Soc. 40 (1936), 37–111.[37]
Vakarelov, D., Dimov, G., D¨untsch, I. and Bennett, B.
A proximityapproach to some region-based theories of space. J. Applied Non-Classical Logics12 (2002), 527-559.[38]
Walker, R. C.
The Stone- ˇCech Compactification.
Springer-Verlag, 1974.[39]
Wolter, D. and Wallgr¨un, J. O.