Duality for powerset coalgebras

Let CABA be the category of complete and atomic boolean algebras and complete boolean homomorphisms, and let CSL be the category of complete meet-semilattices and complete meet-homomorphisms. We show that the forgetful functor from CABA to CSL has a left adjoint. This allows us to describe an endofunctor H on CABA such that the category Alg(H) of algebras for H is dually equivalent to the category Coalg(P) of coalgebras for the powerset endofunctor P on Set. As a consequence, we derive Thomason duality from Tarski duality, thus paralleling how J\'onsson-Tarski duality is derived from Stone duality.


Introduction
It is a classic result in modal logic, known as Jónsson-Tarski duality, that the category MA of modal algebras is dually equivalent to the category DFr of descriptive frames. This result can be traced back to the work of Jónsson-Tarski [JT51], Halmos [Hal56], and Kripke [Kri63]. In the modern form it was proved by Esakia [Esa74] and Goldblatt [Gol76]. 1 Jónsson-Tarski duality is a generalization of the celebrated Stone duality between the category BA of boolean algebras and the category Stone of Stone spaces. It was observed by Abramsky [Abr88] and Kupke, Kurz, and Venema [KKV04] that Jónsson-Tarski duality can be proved by lifting Stone duality using algebra/coalgebra methods. This can be done by utilizing the classic Vietoris construction (see, e.g., [Joh82,Ch. III.4]). Indeed, associating with each Stone space X its Vietoris space V(X) gives rise to an endofunctor V : Stone → Stone such that DFr is isomorphic to the category Coalg(V) of coalgebras for V. Let SL be the category of meet-semilattices with top. Then the forgetful functor U : BA → SL has a left adjoint L : SL → BA. Letting K = LU gives an endofunctor K : BA → BA such that MA is isomorphic to the category Alg(K) of algebras for K. Moreover, the following diagram commutes up to natural isomorphism, yielding that Stone duality lifts to a dual objects in CABA do exist. This can be seen by observing that the Eilenberg-Moore algebras of the double contravariant powerset monad are exactly the objects of CABA [Tay02], and that categories of algebras for monads have free objects [AHS06,Prop. 20.7(2)]. A more concrete construction of free objects in CABA can be given by utilizing the theory of canonical extensions of Jónsson and Tarski [JT51]. Indeed, in Theorem 3.4 we will prove that the free object in CABA over a set X is the canonical extension F σ of the free boolean algebra F over X. We then quotient F σ by the complete congruence generated by the relations defining a completely multiplicative modal operator, yielding the desired L : CSL → CABA (see Theorem 3.7).
An alternate construction of L : CSL → CABA that parallels the alternate construction of L : SL → BA can be given by taking the powerset of hom CSL (M, 2) for each M ∈ CSL. Since hom CSL (M, 2) is isomorphic to the order-dual of M , this amounts to taking the powerset of M . 2 Therefore, we again obtain that the left adjoint can be constructed either purely algebraically, utilizing that free objects exist in CABA, or else using the powerset construction. Thus, we arrive at the following diagram, which parallels the constructions of the left adjoints L : SL → BA and L : CSL → CABA: In Section 4 we define the endofunctor H : CABA → CABA as the composition H = LU. In Theorem 4.3 we prove that the following diagram commutes (up to natural isomorphism).

CABA Set
Tarski duality H P

Tarski duality
This paves the way towards proving that the category Alg(H) of algebras for H is dually equivalent to the category Coalg(P) of coalgebras for P (see Theorem 4.10). It is well known that Jónsson-Tarski and Thomason dualities are connected through the canonical extension and forgetful functors (−) σ : MA → CAMA and U : DFr → KFr, making the following diagram commutative.

Thomason duality
We conclude the paper by Remark 4.13 in which we show that there are analogous canonical extension and forgetful functors (−) σ : Alg(K) → Alg(H) and U : Coalg(V) → Coalg(P) that make a similar diagram commutative. 2 We thank one of the referees for suggesting this approach.

Coalgebraic approach to Jónsson-Tarski duality
In this section we give a brief account of Jónsson-Tarski duality, and provide a construction of the left adjoint L : SL → BA of the forgetful functor U : BA → SL which is alternative to [KKV04,Prop. 3.12]. This we do by utilizing Pontryagin duality for semilattices [HMS74].
We start by recalling the definition of a modal algebra.
Definition 2.1. A modal algebra is a pair (B, ) where B is a boolean algebra and is a unary function on B preserving finite meets. A modal algebra homomorphism between modal algebras (B 1 , 1 ) and (B 2 , 2 ) is a boolean homomorphism α : B 1 → B 2 such that α( 1 a) = 2 α(a) for each a ∈ B 1 . Let MA be the category of modal algebras and modal algebra homomorphisms.
A subset of a topological space X is clopen if it is both closed and open, and X is zero-dimensional if X has a basis of clopen sets. A Stone space is a zero-dimensional compact Hausdorff space.
For a binary relation R on X, we write Definition 2.2. A descriptive frame is a pair (X, R) where X is a Stone space and R is a binary relation on X such that R[x] is closed for each x ∈ X and R −1 [U ] is clopen for each clopen U ⊆ X.
Such relations are often called continuous relations for the following reason. Let V(X) be the Vietoris space of X. We recall (see, e.g., [Joh82, Sec. III.4]) that V(X) is the set of closed subsets of X topologized by the subbasis Then R is continuous iff the associated map ρ R : X → V(X), given by ρ R (x) = R[x], is a well-defined continuous map (that ρ R is well defined follows from (i), and that it is continuous from (ii)).
Let DFr be the category of descriptive frames and continuous p-morphisms, where a p-morphism between (X 1 , R 1 ) and (X 2 , R 2 ) is a map f : One unit β : 1 BA → clop • uf of this dual equivalence is given by the Stone maps β A : A → clop(uf(A)) for A ∈ BA, and the other unit η : 1 Stone → uf • clop by the homeomorphisms η X : X → uf(clop(X)) for X ∈ Stone, which are given by These functors naturally generalize to yield Jónsson-Tarski duality. As we pointed out in the introduction, an alternative approach to Jónsson-Tarski duality is by lifting Stone duality using algebra/coalgebra methods. We recall that the Vietoris construction extends to an endofunctor V : Stone → Stone by sending a continuous map f : for each G ∈ V(X). We next consider the category Coalg(V) of coalgebras for V. For this we recall the notion of a coalgebra for an endofunctor (see, e.g., [Ven07,Def. 9.1]).

Definition 2.4.
(1) A coalgebra for an endofunctor T : C → C is a pair (A, f ) where A is an object of the category C and f : A → T (A) is a C-morphism.
(2) A morphism between two coalgebras (A 1 , f 1 ) and (A 2 , f 2 ) for T is a C-morphism α : A 1 → A 2 such that the following square is commutative.
(3) Let Coalg(T ) be the category whose objects are coalgebras for T and whose morphisms are morphisms of coalgebras.
The dual endofunctor K : BA → BA of the Vietoris endofunctor V : Stone → Stone was described in [KKV04]. Let SL be the category of meet-semilattices with top and meethomomorphisms preserving top. Then K is the composition LU, where L : SL → BA is the left adjoint of the forgetful functor U : BA → SL. In [KKV04,Prop. 3.12] the left adjoint is constructed algebraically, by taking the free boolean algebra F over the underlying set of M ∈ SL and then taking the quotient of F by the relations remembering that M is a meet-semilattice with top. We give an alternative description of L, which utilizes Pontryagin duality for semilattices [HMS74], which we briefly recall next.
Let StoneSL be the category whose objects are topological meet-semilattices, where the topology is a Stone topology, and whose morphisms are continuous meet-homomorphisms. Pontryagin duality for SL establishes a dual equivalence between SL and StoneSL. The contravariant functor (−) * : SL → op StoneSL sends M to its dual M * := hom SL (M, 2), where 2 = {0, 1} is the two-element chain and meet on M * is pointwise meet. If 2 is given the discrete topology and 2 M the product topology, then M * is easily seen to be a closed subspace of 2 M , and so the subspace topology is a Stone topology. Moreover, pointwise meet is continuous, and hence M * ∈ StoneSL. On morphisms, if σ : M → N is an SL-morphism, then σ * : N * → M * is defined by σ * (γ) = γ • σ. The contravariant functor in the other direction sends A ∈ StoneSL to A * := hom StoneSL (A, 2) and σ : A → B to σ * : B * → A * , defined in the same way as the previous functor. Finally, one natural isomorphism sends each M ∈ SL to its double dual M * * by sending m to the map σ → σ(m) for each m ∈ M and σ ∈ M * . The other natural isomorphism sends each A ∈ StoneSL to its double dual A * * and is given by the same formula.
Theorem 2.5. Associating with each M ∈ SL the boolean algebra clop(M * ) of clopen subsets of its dual M * yields an alternative description of the functor L : SL → BA that is left adjoint to the forgetful functor U : BA → SL.  Let Alg(K) be the category of algebras for K (see [AHS06,Def. 5.37]). We recall that for an endofunctor T , algebras for T are defined by reversing the arrows in the definition of coalgebras for T . Since K is dual to V, we have that Alg(K) is dually equivalent to Coalg(V). Because Alg(K) is isomorphic to MA and Coalg(V) is isomorphic to DFr, this gives an alternate proof of Jónsson-Tarski duality (see [KKV04]). 3. Two constructions of the left adjoint L : CSL → CABA Let KFr be the category of Kripke frames and p-morphisms. Forgetting the topology of a descriptive frame yields the forgetful functor U : DFr → KFr. To describe the modal algebras corresponding to Kripke frames, we recall the notion of a completely multiplicative modal operator.
Let CAMA be the category whose objects are complete atomic modal algebras with completely multiplicative , and whose morphisms are complete modal algebra homomorphisms.
Thomason duality generalizes Tarski duality between CABA and Set the same way Jónsson-Tarski duality generalizes Stone duality. We recall that CABA is the category of complete atomic boolean algebras and complete boolean homomorphisms and Set is the category of sets and functions. The contravariant functors of Tarski duality are ℘ : Set → op CABA and at : CABA → op Set. The functor ℘ assigns to each set X the powerset ℘(X) and to each function f : . The functor at assigns to each A ∈ CABA its set of atoms. If α : A → B is a complete boolean homomorphism, it has a left adjoint α * : B → A, which sends atoms to atoms, and the 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 : To derive Thomason duality from Tarski duality the same way Jónsson-Tarski duality was derived from Stone duality, we need to replace the Vietoris endofunctor V on Stone with the powerset endofunctor P on Set. We recall that the endofunctor P : Set → Set associates to each set X its powerset P(X) and to each function f : X → Y the function P(f ) : P(X) → P(Y ) that maps each subset S ⊆ X to its direct image f [S]. We also need to replace the endofunctor K : BA → BA with an appropriate endofunctor H : CABA → CABA.
To describe H, we need to construct the left adjoint to U : CABA → CSL, where CSL is the category of complete meet-semilattices and complete meet-homomorphisms. As in the previous section, this can be done purely algebraically or using duality. To construct H algebraically, we need that free objects exist in CABA. Care is needed here since it is a well-known result of Gaifman [Gai64] and Hales [Hal64] that free objects do not exist in the category of complete boolean algebras and complete boolean homomorphisms. On the other hand, free objects do exist in CABA, and this can be seen by observing that the Eilenberg-Moore algebras of the double contravariant powerset monad are exactly the objects of CABA [Tay02], and that categories of algebras for monads have free objects [AHS06, Prop. 20.7(2)].
A more concrete construction of free objects in CABA can be given utilizing the theory of canonical extensions. It is well known that free objects over any set exist in the category of complete and completely distributive lattices (see Markowski [Mar79] and Dwinger [Dwi81, Thm. 4.2]). By [BHJ21, Cor. 2.3], the free complete and completely distributive lattice over a set X is the canonical extension of the free bounded distributive lattice over X. We show that the same is true in CABA. For this we need to recall the definition of a canonical extension of a boolean algebra. Theorem 3.4. Let X be a set. The canonical extension of the free boolean algebra over X is the free object in CABA over 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.
We identify F σ with ℘(uf(F )). Then e : F → F σ becomes the Stone map β F . 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 −1 We next show that the forgetful functor U : CABA → CSL has a left adjoint L : CSL → CABA. Let A ∈ CABA. We recall that a boolean congruence ≡ on A is a complete congruence

This shows that ϑ
It is well known that the quotient algebra A/≡ is also an object in CABA. As usual, for a ∈ A we write [a] for the equivalence class of a. Then the quotient map π : A → A/≡, given by a → [a], is a CABA-morphism.
Remark 3.5. There is a well-known one-to-one correspondence between congruences and ideals of a boolean algebra A, which associates to each boolean congruence ≡ on A the equivalence class of 0. If A ∈ CABA, this correspondence restricts to a one-to-one correspondence between complete congruences and principal ideals. In this case, the equivalence class of 0 is generated by the element x = {a b | a ≡ b}, where denotes symmetric difference in A.
Let γ * be the left adjoint of γ, and consider its restriction γ * : at(A) → M . Also recall that Then τ is the composition of two CABA-morphisms, so is a CABA-morphism. Moreover, for S ⊆ M , we have Thus, for each S ⊆ M , we have   Theorem 3.7). The resulting two functors are naturally isomorphic. In this section we will always assume that H(A) is the powerset of A, but will indicate how the corresponding result can be proved if we think of H(A) as the quotient of the free object in CABA over A.
We show that the diagram in Figure 1 is commutative up to natural isomorphism. The horizontal arrows in the diagram represent the contravariant functors at : Set → op CABA and ℘ : Set → op CABA of Tarski duality, whereas the vertical arrows represent the endofunctors H and P on CABA and Set, respectively. This together with standard algebra/coalgebra machinery then allows us to prove that Alg(H) is dually equivalent to Coalg(P), thus yielding an alternate proof of Thomason duality.  (1) H • ℘ = ℘ • P.
(2) at • H is naturally isomorphic to P • at. Proof.
(1) If X ∈ Set, then ℘P(X) and H℘(X) are both objects in CABA obtained by taking the double powerset of X ordered by inclusion. We show that the two compositions also agree on morphisms. Let f : X → Y be a map. It is sufficient to show that ℘P(f )(↓S) = H℘(f )(↓S) for each S ⊆ Y . By Remark 3.11, we have On the other hand, Thus, ℘P(f )(↓S) = H℘(f )(↓S), completing the proof.
(2) follows from (1) since the horizontal arrows in the diagram in Figure 1 form a dual equivalence.
Remark 4.4. If we use the alternative description of H, then Theorem 4.3(1) should be phrased as H • ℘ is naturally isomorphic to ℘ • P. The natural isomorphism ξ : H • ℘ → ℘ • P is given on the generators of H℘(X) by ξ X ( S ) = ↓S for each S ⊆ X ∈ Set. In the next remark we give an explicit description of the natural isomorphism ζ : at•H → P • at. This will be used in Remark 4.11.
for each a ∈ A. Since H • ℘ = ℘ • P and ε, ϑ are natural isomorphisms of Tarski duality, we have that the composition atH(ϑ A ) • ε Pat(A) is a bijection.
We show that for each a ∈ A we have It follows from Remark 3.10 that In particular, {{x ∈ at(A) | x ≤ a}} ≤ H(ϑ A )({a}), and so because atH(ϑ A ) is left adjoint to H(ϑ A ). Therefore, atH(ϑ A )({{x ∈ at(A) | x ≤ a}}) = {a} since both sides of the last inequality are atoms. Thus, ζ A = (atH(ϑ A ) • ε Pat(A) ) −1 , and hence ζ is a natural isomorphism.
We next utilize Theorem 4.3 and standard algebra/coalgebra machinery to show that Tarski duality lifts to a dual equivalence between Alg(H) and Coalg(P). We start with the following well-known result (see, e.g., [Ven07,Sec. 9]). Since we will be using the functors establishing the isomorphism of Theorem 4.6 in Remark 4.13, we sketch the proof.
Remark 4.8. If we use the alternative description of H, then the previous theorem can be proved using Theorem 3.7 and Remark 3.8. The advantage of using this description of H lies in the suggestive definitions τ ( a ) = a and τ a = τ ( a ).
We are ready to lift Tarski duality to a dual equivalence between Alg(H) and Coalg(P). For this we utilize [Jac17, Thm. 2.5.9] which states that, under certain conditions, adjunctions lift to adjunctions between categories of algebras. For our purposes, we require the following reformulation of [Jac17, Thm. 2.5.9] for dual equivalences.
Remark 4.12. Putting Theorems 4.6, 4.7, and 4.10 together yields an alternate proof of Thomason duality. We recall that the contravariant functors establishing Thomason duality extend the contravariant functors of Tarski duality. Namely, the functor ℘ : KFr → op CAMA associates to each (X, R) ∈ KFr the algebra (℘(X), R ) ∈ CAMA where R is defined by  For S ⊆ X, we have   For x, y ∈ at(A), we have xR ζ A •at(τ ) y iff y ∈ ζ A at(τ )(x) iff y ∈ ζ A ((τ ) * (x)).
By Remark 3.10, for a ∈ A, we have τ ({a}) = a ∧ ¬ { b | b < a}. Therefore, for each x ∈ at(A), we have x ≤ τ ({a}) iff x ≤ a and x b for each b < a.
Thus, xR ζ A •at(τ ) y iff xR y, so F at A(A, ) = at(A, ), and hence F at A = at.
Remark 4.13. We conclude the paper by connecting the coalgebraic approaches to Jónsson-Tarski and Thomason dualities. As follows from [KKV04], Alg(K) is dually equivalent to Coalg(V), from which Jónsson-Tarski duality follows. By Theorem 4.10, Alg(H) is dually equivalent to Coalg(P), from which Thomason duality follows. Let U : Stone → Set be the forgetful functor. For each X ∈ Stone, viewing the underlying set of the Vietoris space V(X) as a subset of P(X), we have an inclusion map i : UV(X) → PU(X). We extend U to a forgetful functor on the level of coalgebras. Let (X, g) ∈ Coalg(V), so g : X → V(X) is a continuous map. Set U(X, g) := (U(X), g ) where g : U(X) → PU(X) is given by g = i • U(g).