Coalgebras for the powerset functor and Thomason duality
aa r X i v : . [ m a t h . L O ] A ug COALGEBRAS FOR THE POWERSET FUNCTORAND THOMASON DUALITY
G. BEZHANISHVILI, L. CARAI, P. J. MORANDI
Abstract.
We describe the endofunctor H on the category CABA of complete and atomicboolean algebras and complete boolean homomorphisms such that the category
Alg ( H ) ofalgebras for H is dually equivalent to the category Coalg ( P ) of coalgebras for the powersetendofunctor P on Set . As a consequence, we derive Thomason duality from Tarski duality. Introduction
It is a classic result in modal logic that the category MA of modal algebras is duallyequivalent to the category DFr of descriptive frames. Descriptive frames can be thought ofas pairs (
X, R ) where X is a Stone space and R is a binary relation on X that is continuous,meaning that the associated map ρ R : X → V ( X ) to its Vietoris space, given by ρ R ( x ) = R [ x ] := { y | xRy } , is a well-defined continuous map. This result can be traced back to the work of J´onsson-Tarski [13], Halmos [12], and Kripke [14]. In the form of a duality theorem between MA and DFr it was stated by Esakia [6] and Goldblatt [10] .Let Stone be the category of Stone spaces. Then
DFr is isomorphic to the category
Coalg ( V )of coalgebras for the Vietoris endofunctor V : Stone → Stone . Let BA be the category ofboolean algebras and SL the category of meet-semilattices with top. Then the forgetfulfunctor U : BA → SL has a left adjoint H : SL → BA . As was observed by Abramsky [1] andKupke, Kurz, and Venema [15], MA is isomorphic to the category Alg ( H ) of algebras for H ,and Stone duality between BA and Stone is lifted to a dual equivalence between
Alg ( H ) and Coalg ( V ). This yields an alternate proof of Esakia-Goldblatt duality between MA and DFr .In [19], Thomason proved a discrete version of Esakia-Goldblatt duality. Let
CAMA be thecategory whose objects are complete and atomic modal algebras where the modal operator (cid:3) is completely multiplicative, and whose morphisms are complete modal algebra homo-morphisms. Let also
KFr be the category of Kripke frames and p-morphisms. Thomasonduality establishes that
CAMA is dually equivalent to
KFr . This generalizes Tarski dualitybetween the category
CABA of complete and atomic boolean algebras and complete booleanhomomorphisms and the category
Set of sets and functions.
Mathematics Subject Classification.
Key words and phrases.
Modal logic, coalgebra, Tarski duality, Thomason duality. We point out that Esakia phrased it for the subcategory of
DFr where the relation R is reflexive andtransitive. Consequently, he worked with the subcategory of MA consisting of closure algebras of McKinseyand Tarski [18]. It is natural to try to obtain Thomason duality in the same vein as Esakia-Goldblattduality was obtained in [15] by lifting Stone duality. For this it is natural to replace theVietoris endofunctor on
Stone with the powerset endofunctor P on Set . It is known (see,e.g., [20]) that
KFr is isomorphic to
Coalg ( P ). In this short note we describe an analogueof the endofunctor H : BA → BA for CABA and prove that the category of algebras forthis endofunctor is dually equivalent to
Coalg ( P ). Thomason duality then follows as aconsequence. To the best of our knowledge, this approach has not been considered in thepast. 2. Tarski and Thomason dualities
It is an old result of Tarski (see, e.g., [9, p. 121]) that a boolean algebra A is isomorphic toa powerset algebra iff A is complete and atomic. This result extends to a dual equivalencebetween the category Set of sets and functions and the category
CABA of complete and atomicboolean algebras and complete boolean homomorphisms. Following [4, Thm. 4.3], we referto this duality as Tarski duality. The dual equivalence is established by the contravariantfunctors ℘ : Set → CABA and at : CABA → Set . The functor ℘ assigns to each set X thepowerset ℘ ( X ) and to each function f : X → Y its inverse image f − : ℘ ( Y ) → ℘ ( X ).The functor at assigns to each A ∈ CABA its set of atoms. If α : A → B is a completeboolean homomorphism, it has a left adjoint α ∗ : B → A , which sends atoms to atoms, andthe functor at assigns to α the function α ∗ : at ( B ) → at ( A ). One unit ε : 1 Set → at ◦ ℘ of this dual equivalence is given by ε X ( x ) = { x } for each x ∈ X ∈ Set , and the other unit ϑ : 1 CABA → ℘ ◦ at by ϑ A ( a ) = ↓ a ∩ at ( A ) for each a ∈ A ∈ CABA .Tarski duality was extended to modal algebras by Thomason [19]. We recall that a modalalgebra is a pair A = ( A, (cid:3) ) where A is a boolean algebra and (cid:3) is a unary function on A preserving all finite meets. If A is complete, then we say that (cid:3) is completely multiplicative if it preserves arbitrary meets. A modal algebra homomorphism between two modal algebras A and B is a boolean homomorphism α : A → B satisfying α ( (cid:3) a ) = (cid:3) α ( a ) for each a ∈ A . Definition 2.1.
Let
CAMA be the category whose objects are complete and atomic modalalgebras with completely multiplicative (cid:3) and whose morphisms are complete modal algebrahomomorphisms.A
Kripke frame is a pair F = ( X, R ) where X is a set and R is a binary relation on X . For x ∈ X , let R [ x ] = { y ∈ X | xRy } be the R -image of x . A p-morphism between two Kripkeframes F and G is a function f : X → Y satisfying f [ R [ x ]] = R [ f ( x )] for each x ∈ X . Definition 2.2.
Let
KFr be the category of Kripke frames and p-morphisms.
Theorem 2.3 (Thomason duality) . CAMA is dually equivalent to
KFr . The contravariant functors establishing Thomason duality extend the contravariant func-tors of Tarski duality. More precisely, the functor ℘ : KFr → CAMA associates to each(
X, R ) ∈ KFr the algebra ( ℘ ( X ) , (cid:3) R ) ∈ CAMA where (cid:3) R is defined by (cid:3) R ( S ) = { x ∈ X | R [ x ] ⊆ S } . Also, it associates to each KFr -morphism f : X → Y the CAMA -morphism f − : ℘ ( Y ) → ℘ ( X ). The functor at : CAMA → KFr associates to each ( A, (cid:3) ) ∈ CAMA
OALGEBRAS FOR THE POWERSET FUNCTOR AND THOMASON DUALITY 3 the Kripke frame ( at ( A ) , R (cid:3) ) where xR (cid:3) y iff x ∧ (cid:3) ¬ y = 0. Also, it associates to each CAMA -morphism α : A → B the KFr -morphism α ∗ : at ( B ) → at ( A ).3. Kripke frames and coalgebras for the powerset functor
The following definition is well known (see, e.g., [2, Def. 5.37]).
Definition 3.1.
Let C be a category and T : C → C an endofunctor on C .(1) An algebra for T is a pair ( A, f ) where A is an object of C and f : T ( A ) → A is a C -morphism.(2) Let ( A , f ) and ( A , f ) be two algebras for T . A morphism between ( A , f ) and( A , f ) is a C -morphism α : A → A such that the following square is commutative. T ( A ) T ( A ) A A f T ( α ) f α (3) Let Alg ( T ) be the category whose objects are algebras for T and whose morphismsare morphisms of algebras.The notion of coalgebras for T is dual to that of algebras for T . Definition 3.2. (1) A coalgebra for an endofunctor T : C → C is a pair ( B, g ) where B is an object of C and g : B → T ( B ) is a C -morphism.(2) A morphism between two coalgebras ( B , g ) and ( B , g ) for T is a C -morphism α : B → B such that the following square is commutative. B B T ( B ) T ( B ) g α g T ( α ) (3) Let Coalg ( T ) be the category whose objects are coalgebras for T and whose mor-phisms are morphisms of coalgebras.We next view P : Set → Set as an endofunctor on
Set associating to each set X itspowerset P ( X ) and to each function f : X → Y the function P ( f ) : P ( X ) → P ( Y ) thatmaps each subset S ⊆ X to its direct image f [ S ]. Remark 3.3.
There are two powerset endofunctors on
Set , one covariant which we denoteby P , and one contravariant. The contravariant one is the composition of ℘ : Set → CABA and the forgetful functor U : CABA → Set .The following result is well known (see, e.g., [20]).
Theorem 3.4.
KFr is isomorphic to
Coalg ( P ) . G. BEZHANISHVILI, L. CARAI, P. J. MORANDI
Proof. (Sketch). To each Kripke frame F = ( X, R ) we associate the coalgebra ρ R : X →P ( X ) defined by ρ R ( x ) = R [ x ]. If f : X → X is a p-morphism between Kripke frames( X , R ) and ( X , R ), then f is also a morphism between the coalgebras ( X , ρ R ) and( X , ρ R ). This defines a covariant functor C : KFr → Coalg ( P ). To each coalgebra ( X, ρ )for P , we associate the Kripke frame ( X, R ρ ) where xR ρ y iff y ∈ ρ ( x ). If f is a morphismbetween two coalgebras ( X , ρ ) and ( X , ρ ), then f is also a p-morphism between the Kripkeframes ( X , R ρ ) and ( X , R ρ ). This defines a covariant functor K : Coalg ( P ) → KFr . It isstraightforward to see that R = R ρ R for each ( X, R ) ∈ KFr and ρ = ρ R ρ for each ( X, ρ ) ∈ Coalg ( P ). Thus, the functors C and K yield an isomorphism of KFr and
Coalg ( P ). (cid:3) Free objects in
CABA and the functor H It is well known (see Markowski [17] and Dwinger [5, Thm. 4.2]) that free objects on anyset exist in the category of complete and completely distributive lattices. Since a completeboolean algebra is completely distributive iff it is atomic, it follows that free objects on any setalso exist in
CABA . This is in contrast to the well-known fact [7, 11] that free objects do notexist in the category of complete boolean algebras and complete boolean homomorphisms. Itfollows from [3] that the free object in
CABA on a set X can be constructed as the canonicalextension of the free boolean algebra on X . To keep the paper self-contained, we give adirect proof of this result, and then use it to build the endofunctor H : CABA → CABA .We recall [13, 8] that a canonical extension of a boolean algebra A is a complete booleanalgebra A σ together with a boolean embedding e : A → A σ such that each element x ∈ A σ isa join of meets (and hence also a meet of joins) of e [ A ] and for S, T ⊆ A , from V e [ S ] ≤ W e [ T ]it follows that V S ≤ W T for some finite S ⊆ S and T ⊆ T . It is well known that A σ isisomorphic to ℘ uf ( A ) where uf ( A ) is the set of ultrafilters of A . Theorem 4.1.
Let X be a set. The canonical extension of the free boolean algebra over X is the free object in CABA on X .Proof. Let F be the free boolean algebra over X , f : X → F the associated map, and e : F → F σ the boolean embedding into the canonical extension. We show that ( F σ , e ◦ f )has the universal mapping property in CABA . Let A ∈ CABA and g : X → A be a function.Since A is a boolean algebra, there is a unique boolean homomorphism ϕ : F → A with ϕ ◦ f = g . This induces a map uf ( ϕ ) : uf ( A ) → uf ( F ) on the sets of ultrafilters given by uf ( ϕ )( y ) = ϕ − ( y ). Define ϕ + : at ( A ) → uf ( F ) by ϕ + ( x ) = ϕ − ( ↑ x ). If we identify atomswith the principal ultrafilters, we can think of ϕ + as the restriction of uf ( ϕ ) to at ( A ).We identify F σ with ℘ uf ( F ). Then e : F → F σ becomes the Stone map e ( a ) = { y ∈ uf ( F ) | a ∈ y } . The map ϕ + : at ( A ) → uf ( F ) yields a CABA -morphism ℘ ( ϕ + ) : F σ → ℘ at ( A ). Since A ∈ CABA , the map ϑ A : A → ℘ at ( A ) is an isomorphism. We set ψ = ϑ − A ◦ ℘ ( ϕ + ). Clearly OALGEBRAS FOR THE POWERSET FUNCTOR AND THOMASON DUALITY 5 ψ : F σ → A is a CABA -morphism. We show that ϑ A ◦ ϕ = ℘ ( ϕ + ) ◦ e . F F σ X A ℘ at ( A ) eϕ ℘ ( ϕ + ) ψf g ϑ A Let a ∈ F . Since ϑ A ϕ ( a ) = { x ∈ at ( A ) | x ≤ ϕ ( a ) } and e ( a ) = { y ∈ uf ( F ) | a ∈ y } , we have( ℘ ( ϕ + ) ◦ e )( a ) = ϕ − e ( a ) = { x ∈ at ( A ) | ϕ + ( x ) ∈ e ( a ) } = { x ∈ at ( A ) | a ∈ ϕ + ( x ) } = { x ∈ at ( A ) | a ∈ ϕ − ( ↑ x ) } = { x ∈ at ( A ) | x ≤ ϕ ( a ) } = ϑ A ϕ ( a ) . This shows that ϑ A ◦ ϕ = ℘ ( ϕ + ) ◦ e , so ψ ◦ ( e ◦ f ) = ϕ ◦ f = g .It is left to show uniqueness. Suppose that µ : F σ → A is a CABA -morphism satisfying µ ◦ ( e ◦ f ) = g . Then ( µ ◦ e ) ◦ f = ( ψ ◦ e ) ◦ f = ϕ ◦ f . By uniqueness of ϕ , we have µ ◦ e = ϕ = ψ ◦ e . Therefore, µ and ψ agree on e [ F ]. Since e [ F ] is join-meet dense in F σ and µ, ψ are CABA -morphisms, we conclude that µ = ψ . (cid:3) Let
CSL be the category whose objects are complete meet-semilattices and whose mor-phisms preserve arbitrary meets. We clearly have the forgetful functor U : CABA → CSL .We show that it has a left adjoint H : CSL → CABA . If L ∈ CSL , let F ( L ) be the free objectin CABA on L viewed as a set, and let f L : L → F ( L ) be the associated map. Define a L = _ n f L (cid:16)^ T (cid:17) △ ^ { f L ( t ) | t ∈ T } | T ⊆ L o and the principal ideal I L = ↓ a L , where △ is the symmetric difference in F ( L ). We then set H ( L ) to be the quotient F ( L ) /I L . Since F ( L ) ∈ CABA and I L is principal, H ( L ) ∈ CABA .For a ∈ L , let (cid:3) a = [ f L ( a )] ∈ H ( L ). Let α L : L → H ( L ) be the composition of the quotientmap π : F ( L ) → H ( L ) and f L . Then α L ( a ) = (cid:3) a for each a ∈ L . L F ( L ) H ( L ) f L α L π By definition of I L we see that (cid:3) V T = V { (cid:3) t | t ∈ T } in H ( L ) for each T ⊆ L . Thus, α L isa CSL -morphism.
Theorem 4.2.
The correspondence L
7→ H ( L ) defines a functor H : CSL → CABA that isleft adjoint to the forgetful functor U : CABA → CSL .Proof.
By [16, p. 89] it is enough to show that for each L ∈ CSL , A ∈ CABA , and a
CSL -morphism g : L → A there is a unique CABA -morphism τ : H ( L ) → A such that τ ◦ α L = g .By Theorem 4.1, there is a unique CABA -morphism ϕ : F ( L ) → A with ϕ ◦ f L = g . To seethat ϕ factors through I L , let T ⊆ L . Since g is a CSL -morphism, g ( V T ) = V { g ( t ) | t ∈ T } .Therefore, ϕf L (cid:16)^ T (cid:17) = g (cid:16)^ T (cid:17) = ^ g [ T ] G. BEZHANISHVILI, L. CARAI, P. J. MORANDI and ϕ (cid:16)^ { f L ( t ) | t ∈ T } (cid:17) = ^ { ϕf L ( t ) | t ∈ T } = ^ { g ( t ) | t ∈ T } = ^ g [ T ] . From these two equations we see that ϕ ( a L ) = 0. Therefore, ϕ ( I L ) ⊆ ker( ϕ ), and hence ϕ induces a CABA -morphism τ : H ( L ) → A with τ ◦ α L = g . Since H ( L ) is generated by α L [ L ]and τ is a CABA -morphism, τ is uniquely determined by the equation τ ◦ α L = g . L F ( L ) H ( L ) A f L g α L πϕ τ (cid:3) Remark 4.3.
To describe how H acts on morphisms, let α : L → M be a CSL -morphism.Then f M ◦ α : L → H ( M ) is a CSL -morphism, so there is a unique
CABA -morphism H ( α ) : H ( L ) → H ( M ) such that H ( α ) ◦ f L = f M ◦ α . Therefore, if a ∈ L , then H ( α )( (cid:3) a ) = H ( α ) f L ( a ) = f M α ( a ) = (cid:3) α ( a ) . L M H ( L ) H ( M ) αf L f M H ( α ) We conclude the section by showing that
CAMA is isomorphic to
Alg ( H ). Theorem 4.4.
CAMA is isomorphic to
Alg ( H ) .Proof. Let ( A, (cid:3) ) ∈ CAMA . Since (cid:3) : A → A is a CSL -morphism, by Theorem 4.2, there is aunique
CABA -morphism τ (cid:3) : H ( A ) → A such that τ (cid:3) ( (cid:3) a ) = (cid:3) a for each a ∈ A . Therefore,( A, τ (cid:3) ) ∈ Alg ( H ). Let α : A → B be a CAMA -morphism and a ∈ A . Since α commutes with (cid:3) , by Remark 4.3, τ (cid:3) H ( α )( (cid:3) a ) = τ (cid:3) ( (cid:3) α ( a ) ) = (cid:3) α ( a ) = α ( (cid:3) a ) = ατ (cid:3) ( (cid:3) a ) . Since H ( A ) is generated by { (cid:3) a | a ∈ A } , we obtain τ (cid:3) ◦ H ( α ) = α ◦ τ (cid:3) . Therefore, α is alsoa morphism in Alg ( H ). This defines a covariant functor A : CAMA → Alg ( H ).Conversely, let A ∈ CABA and τ : H ( A ) → A be a CABA -morphism. If we define (cid:3) τ on A by (cid:3) τ a = τ ( (cid:3) a ), then (cid:3) τ is completely multiplicative, so ( A, (cid:3) τ ) ∈ CAMA . Let α : A → B be a morphism in Alg ( H ) and a ∈ A . By Remark 4.3, (cid:3) α ( a ) = τ (cid:3) ( (cid:3) α ( a ) ) = τ (cid:3) H ( α )( (cid:3) a ) = ατ (cid:3) ( (cid:3) a ) = α ( (cid:3) a ) . Therefore, α is also a CAMA -morphism. This defines a covariant functor M : Alg ( H ) → CAMA .Let ( A, (cid:3) ) ∈ CAMA . For a ∈ A , we have (cid:3) τ (cid:3) a = τ (cid:3) ( (cid:3) a ) = (cid:3) a . Therefore, (cid:3) τ (cid:3) = (cid:3) .Next, let ( A, τ ) ∈ Alg ( H ). For a ∈ A , we have τ (cid:3) τ ( (cid:3) a ) = (cid:3) τ a = τ ( (cid:3) a ). Since H ( A ) isgenerated by { (cid:3) a | a ∈ A } , we obtain that τ (cid:3) τ = τ . Thus, the functors A and M yield anisomorphism of CAMA and
Alg ( H ). (cid:3) OALGEBRAS FOR THE POWERSET FUNCTOR AND THOMASON DUALITY 7 Duality between
Alg ( H ) and Coalg ( P ) and Thomason duality As follows from the previous section, we have the following diagram.
CABA SetCABA Set at H P ℘ at ℘ Figure 1.
We emphasize again that P : Set → Set is covariant, while ℘ : Set → CABA is contravari-ant. In this section we show that this diagram commutes up to natural isomorphism, fromwhich we derive our main result that
Alg ( H ) is dually equivalent to Coalg ( P ). Thomasonduality then follows as an easy corollary. Definition 5.1.
Let A ∈ CABA . With each S ⊆ at ( A ) we associate an element x S of H ( A )by setting x S = (cid:3) W S ∧ ¬ _ { (cid:3) W U | U ( S } . Note that x ∅ = (cid:3) . Lemma 5.2.
Let A ∈ CABA , S, T ⊆ at ( A ) , and a ∈ A . (1) x S = 0 . (2) If S = T , then x S = x T . (3) If W S ≤ a , then x S ≤ (cid:3) a . (4) If W S (cid:2) a , then x S ∧ (cid:3) a = 0 . (5) x S ∈ at H ( A ) .Proof. (1). Define g : A → ℘ P at ( A ) by g ( a ) = { U ⊆ at ( A ) | W U ≤ a } . To see that g is a CSL -morphism, let X ⊆ A . Then g (cid:16)^ X (cid:17) = n U ⊆ at ( A ) | _ U ≤ ^ X o = n U ⊆ at ( A ) | _ U ≤ a for each a ∈ X o = \ nn U ⊆ at ( A ) | _ U ≤ a o | a ∈ X o = ^ { g ( a ) | a ∈ X } . By Theorem 4.2, g extends to a CABA -morphism τ : H ( A ) → ℘ P at ( A ) such that τ ( (cid:3) a ) = g ( a ). We have τ ( x S ) = τ ( (cid:3) W S ) \ [ (cid:8) τ ( (cid:3) W U ) | U ( S (cid:9) = g (cid:16)_ S (cid:17) \ [ n g (cid:16)_ U (cid:17) | U ( S o = n V | _ V ≤ _ S o \ [ nn W | _ W ≤ _ U o | U ( S o = { V | V ⊆ S } \ [ {{ W | W ⊆ U } | U ( S } = { V | V ⊆ S } \ { U | U ( S } = { S } . Since τ ( x S ) = { S } 6 = ∅ , we conclude that x S = 0 in H ( A ). G. BEZHANISHVILI, L. CARAI, P. J. MORANDI (2). If S = T , then τ ( x S ) = { S } 6 = { T } = τ ( x T ) by the proof of (1), so x S = x T .(3). If W S ≤ a , then x S ≤ (cid:3) W S ≤ (cid:3) a .(4). Let W S (cid:2) a and set U = S ∩ ↓ a ( S . Then W U = ( W S ) ∧ a . Therefore, x S ∧ (cid:3) a ≤ ( (cid:3) W S ∧ ¬ (cid:3) W U ) ∧ (cid:3) a = ( (cid:3) W S ∧ (cid:3) a ) ∧ ¬ (cid:3) W U = (cid:3) ( W S ) ∧ a ∧ ¬ (cid:3) W U = (cid:3) W U ∧ ¬ (cid:3) W U = 0 . Thus, x S ∧ (cid:3) a = 0.(5). Since H ( A ) is generated by { (cid:3) a | a ∈ A } and is complete and atomic, hence com-pletely distributive, each element is a join of meets of elements of the form (cid:3) b and ¬ (cid:3) c (with b, c ∈ A ). Therefore, since x S = 0 by (1), it follows from (3) and (4) that x S is an atom. (cid:3) Theorem 5.3.
For A ∈ CABA define ζ A : P at ( A ) → at H ( A ) by ζ A ( S ) = x S for each S ⊆ at ( A ) . This yields a natural isomorphism ζ : P ◦ at → at ◦ H .Proof. Let A ∈ CABA . By Lemma 5.2(5), ζ A is well defined, and by Lemma 5.2(2), it isone-to-one. To see that it is onto, let x be an atom of H ( A ), b = V { a ∈ A | x ≤ (cid:3) a } , and S = at ( A ) ∩ ↓ b . We show that x = x S . We have W S = b and x ≤ V { (cid:3) a | x ≤ (cid:3) a } = (cid:3) b .Therefore, b ≤ a iff x ≤ (cid:3) a and hence _ S ≤ a iff x ≤ (cid:3) a for each a ∈ A . From this it follows that x ≤ (cid:3) W S , and if U ( S , then x (cid:3) W U . Because x is an atom, we must have x ≤ ¬ (cid:3) W U . Therefore, x ≤ x S by definition of x S . Since x S isan atom by Lemma 5.2(5), we conclude that x = x S . Thus, ζ A is a bijection.To show naturality, let α : A → A be a CABA -morphism. We have the following diagram. P at ( A ) at H ( A ) P at ( A ) at H ( A ) P at ( α ) ζ A at H ( α ) ζ A If T ⊆ at ( A ), then ( at H ( α ) ◦ ζ A )( T ) = at H ( α )( x T ) = H ( α ) ∗ ( x T )and ( ζ A ◦ P at ( α ))( T ) = ζ A P ( α ∗ )( T ) = ζ A ( α ∗ [ T ]) = x α ∗ [ T ] . Let u ∈ H ( A ). Then H ( α ) ∗ ( x T ) ≤ u iff x T ≤ H ( α )( u ). Therefore, if a ∈ A , then H ( α )( (cid:3) a ) = (cid:3) α ( a ) , so H ( α ) ∗ ( x T ) ≤ (cid:3) a iff x T ≤ (cid:3) α ( a ) . By Lemma 5.2, if U ⊆ at ( A i ) and b ∈ A i , then x U ≤ (cid:3) b iff W U ≤ b . Also, since α ∗ is left adjoint to α , it preserves joins.Therefore, x T ≤ (cid:3) α ( a ) ⇐⇒ _ T ≤ α ( a ) ⇐⇒ α ∗ (cid:16)_ T (cid:17) ≤ a ⇐⇒ _ α ∗ [ T ] ≤ a ⇐⇒ x α ∗ [ T ] ≤ (cid:3) a . Thus, H ( α ) ∗ ( x T ) ≤ (cid:3) a iff x α ∗ [ T ] ≤ (cid:3) a , and so H ( α ) ∗ ( x T ) = x α ∗ [ T ] . This yields( at H ( α ) ◦ ζ A )( T ) = H ( α ) ∗ ( x T ) = x α ∗ [ T ] = ( ζ A ◦ P at ( α ))( T ) . Consequently, ζ is natural. (cid:3) OALGEBRAS FOR THE POWERSET FUNCTOR AND THOMASON DUALITY 9
As a consequence of Theorem 5.3 we obtain that Figure 1 is commutative up to naturalisomorphism. In particular, H is naturally isomorphic to ℘ ◦ P ◦ at . Thus, we arrive at thefollowing representation of H ( A ). Theorem 5.4. If A ∈ CABA , then H ( A ) ∼ = ℘ P at ( A ) . Remark 5.5.
Theorem 5.4 is an analogue of the following result [1, 15]: If A ∈ BA and H ( A ) is the free boolean algebra on the underlying meet-semilattice of A , then H ( A ) isisomorphic to the boolean algebra of clopens of the Vietoris space of the Stone dual of A . Remark 5.6.
In the proof of Lemma 5.2(1) we defined a
CABA -morphism τ : H ( A ) → ℘ P at ( A ) and showed that τ ( x S ) = { S } for each S ⊆ at ( A ). To see that τ is in fact a CABA -isomorphism, by Lemma 5.2(5) and the first paragraph of the proof of Theorem 5.3, at H ( A ) = { x S | S ⊆ at ( A ) } . Since the atoms of ℘ P at ( A ) are precisely the { S } for S ⊆ at ( A ),we have that τ is a bijection on atoms, and hence τ is a CABA -isomorphism. Therefore, theisomorphism of Theorem 5.4 can be realized by τ . Remark 5.7.
It follows from the commutativity of Figure 1 that H◦ ℘ is naturally isomorphicto ℘ ◦P . While we do not need an explicit description of the natural isomorphism η : H◦ ℘ → ℘ ◦ P , we point out that η is determined from the formula η X ( (cid:3) S ) = { T | T ⊆ S } for each S ⊆ X ∈ Set .We are ready to prove our main result.
Theorem 5.8.
Tarski duality between
CABA and
Set lifts to a dual equivalence between
Alg ( H ) and Coalg ( P ) .Proof. If (
A, f ) ∈ Alg ( H ), then ζ − A ◦ at ( f ) : at ( A ) → P at ( A ) is an object of Coalg ( P ).Let α : A → A be a morphism in Alg ( H ). By Theorem 5.3, ζ is a natural isomorphism.Therefore, at ( α ) is a morphism of the corresponding coalgebras. This defines a contravariantfunctor Φ : Alg ( H ) → Coalg ( P ). at ( A ) at ( A ) P at ( A ) P at ( A ) at ( α ) ζ − A ◦ at ( f ) ζ − A ◦ at ( f ) P at ( α ) If (
X, g ) ∈ Coalg ( P ), then ℘ ( g ) ◦ η X : H ℘ ( X ) → ℘ ( X ) is an object of Alg ( H ). Let α : X → X be a morphism in Coalg ( P ). Since η is a natural isomorphism, ℘ ( α ) is a morphismof the corresponding algebras. This defines a contravariant functor Ψ : Coalg ( P ) → Alg ( H ). H ℘ ( X ) H ℘ ( X ) ℘ ( X ) ℘ ( X ) H ℘ ( α ) ℘ ( g ) ◦ η X ℘ ( g ) ◦ η X ℘ ( α ) Let (
A, f ) ∈ Alg ( H ). Then Φ( A, f ) = ( at ( A ) , ζ − A ◦ at ( f )). Therefore, ΨΦ( A, f ) =( ℘ at ( A ) , ℘ ( ζ − A ◦ at ( f )) ◦ η ℘ at ( A ) ). Thus, it follows from the naturality of ζ and η that the following square is commutative. H ( A ) H ℘ at ( A ) A ℘ at ( A ) f H ( ϑ A ) ℘ ( ζ − A ◦ at ( f )) ◦ η ℘ at ( A ) ϑ A Thus, ϑ : 1 Alg ( H ) → Ψ ◦ Φ is a natural isomorphism. Similary, ε : 1 Coalg ( P ) → Φ ◦ Ψ is anatural isomorphism. Therefore,
Alg ( H ) and Coalg ( P ) are dually equivalent. (cid:3) Putting Theorems 3.4, 4.4, and 5.8 together yields Thomason duality. The contravariantfunctor ℘ : KFr → CAMA in Thomason duality is the composition
KFr Coalg ( P ) Alg ( H ) CAMA C Ψ M and the contravariant functor at : CAMA → KFr is the composition
CAMA Alg ( H ) Coalg ( P ) KFr . A Φ K Acknowledgements
We would like to thank Phil Scott for a careful reading of the manuscript and for hisuseful suggestions. We also thank Nick Bezhanishvili, Sebastian Enqvist, Jim de Groot, andClemens Kupke for their comments.
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