Coalgebraic Geometric Logic: Basic Theory

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor on some full subcategory of the category of topological spaces and continuous functions. We investigate derivation systems, soundness and completeness for such geometric modal logics, and we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces, again accompanied by a collection of (open) predicate liftings. Furthermore, we compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category.


Introduction
Propositional geometric logic arose at the interface of (pointfree) topology, logic and theoretical computer science as the logic of finite observations [Abr87,Vic89]. Its language is constructed from a set of proposition letters by applying finite conjunctions and arbitrary disjunctions, these being the propositional operations preserving the property of finite observability. Through an interesting topological connection, formulas of geometric logic can be interpreted in the frame of open sets of a topological space. Central to this connection is the well-known dual adjunction between the category Frm of frames and frame morphisms and the category Top of topological spaces and continuous maps, which restricts to several interesting Stone-type dualities [Joh82].
Coalgebraic logic is a framework in which generalised versions of modal logics are developed parametric in the signature of the language and a functor T : C → C on some base category C. With classical propositional logic as base logic, two natural choices for the base category are Set, the category of sets and functions, and Stone, the category of Stone spaces and continuous functions, i.e. the topological dual to the algebraic category of Boolean algebras. Coalgebraic logic for endofunctors on Set has been well investigated and still is an active area of research, see e.g. [CKP + 08,KP11]. In this setting, modal operators can be defined using the notion of relation lifting [Mos99] or predicate lifting [Pat03]. Coalgebraic logic in the category of Stone coalgebras has been studied in [KKV04,HK04,ES18,BEG20], and there is a fairly extensive literature on the design of a coalgebraic modal logic based on a general Stone-type duality (or dual adjunction), see for instance [BK05,BK06,CJ14,Kli07] and references therein.
In this paper we investigate some links between coalgebraic logic and geometric logic. That is, we use methods from coalgebraic logic to introduce modal operators to the language of geometric logic, with the intention of studying interpretations of these logics in certain topological coalgebras. Note that extensions of geometric logic with the basic modalities and , which are closely related to the topological Vietoris construction, have received much attention in the literature, see [Vic89] for some early history. A first step towards developing coalgebraic geometric logic was taken in [VVV13], where a method is explored to lift a functor on Set to a functor on the category KHaus of compact Hausdorff spaces, and the connection is investigated between the lifted functor and a relation-lifting based "cover" modality.
Our aim here is to develop a framework for the coalgebraic geometric logics that arise if we extend geometric logic with modalities that are induced by appropriate predicate liftings. Guided by the connection between geometric logic and topological spaces, we choose the base category of our framework to be Top itself, or one of its full subcategories such as Sob (sober spaces), KSob (compact sober spaces) or KHaus (compact Hausdorff spaces). On this base category C we then consider an arbitrary endofunctor T which serves as the type of our topological coalgebras. Furthermore, we shall see that if we want our formulas to be interpreted as open sets of the coalgebra carrier, we need the predicate liftings that interpret the modalities of the language to satisfy some natural openness condition. Summarizing, we shall study the coalgebraic geometric logic induced by (1) a functor T : C → C, where C is a full subcategory of Top, and (2) a set Λ of open predicate liftings for T. As running examples we take the combination of the basic modalities for the Vietoris functor, and that of the monotone box and diamond modalities for various topological manifestations of the monotone neighbourhood functor on Set. The structures providing the semantics for our coalgebraic geometric logics are the T-models comprised of a T-coalgebra together with a valuation mapping proposition letters to open sets in the coalgebra carrier.
The main results that we report on here are the following: • Section 4 contains a detailed description of the monotone neighbourhood functor on KHaus, which naturally extends the monotone functor on Stone [HK04] that corresponds to monotone modal logic. • In Section 5 we discuss derivation systems for coalgebraic geometric logic, based on consequence pairs, and derive a general completeness result. • After that, in Section 6 we adapt the method of [KKP04] in order to lift a Set-functor together with a collection of predicate liftings to an endofunctor on Top. We obtain the Vietoris functor and monotone functor on KHaus as restrictions of such lifted functors. • In Section 7, we construct a final object in the category of T-models, where T is an endofunctor on Top which preserves sobriety and admits a Scott-continuous, characteristic geometric modal signature. • Finally, in Section 8 we transfer the notion of Λ-bisimilarity from [GS13,BH17] to our setting, and we compare this to geometric modal equivalence, behavioural equivalence Example 2.2 (Monotone neighbourhood frames). Let D : Set → Set be the functor given on objects by where X is a set. For a morphism f : X → X define Then the category of monotone frames and bounded morphisms is isomorphic to Coalg(D) [Che80,Han03,HK04].
Example 2.3 (Kripke models). Consider for P-models the predicate liftings λ , λ :P → P • P given by Then λ and λ yield the usual Kripke semantics of and .
Example 2.4 (Monotone neighbourhood frames). Monotone neighbourhood models are precisely D-models, where D is the functor defined in Example 2.2. The usual semantics for the box and diamond in this setting can be obtained from the predicate liftings given by λ X (a) = {W ∈ DX | a ∈ W }, λ X (a) = {W ∈ DX | X \ a / ∈ W }. (2.1) We refer to [KP11] for many more examples of coalgebraic logics for Set-functors. 2.4. Geometric logic. Let Φ be a set of proposition letters. The language GL(Φ) of geometric formulas is given by where p ∈ Φ and I is some index set. Coherent formulas are defined in the same way, but without infinitary disjuctions. These are also known as the formulas from positive logic. We describe the logical system as a collection of binary consequence pairs, written as ϕ ψ. A geometric logic is a collection consequence pairs closed under the following rules: identity ϕ ϕ, cut ϕ ψ ψ χ ϕ χ , the conjunction rules ϕ , ϕ ∧ ψ ϕ, ϕ ∧ ψ ψ, ϕ ψ ϕ χ ϕ ψ ∧ χ , the disjunction rules ϕ S (ϕ ∈ S), ϕ ψ (all ϕ ∈ S) S ψ and frame distributivity ϕ ∧ S {ϕ ∧ ψ | ψ ∈ S}.
Note that these are in fact all schemata. We write GL for the minimal geometric logic, i.e. the smallest collection of consequence pairs closed under the axioms and rules given above. We write ϕ GL ψ if the consequence pair ϕ ψ is in GL.
Note that frame distributivity allows us to reduce every formula to a disjunction of finite conjunctions of proposition letters. Therefore, modulo equivalence, the formulas form a set. A collection S of geometric formulas is called directed if for every pair ϕ, ψ ∈ S there exists χ ∈ S such that ϕ χ and ψ χ.
The topological semantics and algebraic semantics of geometric logic are given by topological spaces and frames.
2.5. Frames and spaces. A frame is a complete lattice F in which for all a ∈ F and S ⊆ F the infinite distributive law holds: A frame homomorphism is a function between frames that preserves finite meets and arbitrary joins.
For a, b ∈ F we say that a is well inside b, notation: a b, if there is a c ∈ F such that c ∧ a = ⊥ and c ∨ b = . An element a ∈ F is called regular if a = {b ∈ F | b a} and a frame is called regular if all of its elements are regular. The negation of a ∈ F is defined as ∼a = {b ∈ F | a ∧ b = ⊥}. A frame is said to be compact if S = implies that there is a finite subset S ⊆ S such that S = .
Lemma 2.5. For all elements a, b in a frame F we have a b iff ∼a ∨ b = .
Now suppose a and b are regular elements, then so an arbitrary join of regular elements is regular.
Frames can be presented by generators and relations.
Definition 2.7. A presentation is a pair G, R where G is a set of generators and R is a collection of relations between expressions constructed from the generators using arbitrary joins and finite meets. Let F be a frame and ZF its underlying set. We say that G, R presents F if there is an assignment f : G → ZF of the generators such that (i), (ii) and (iii) hold: (i) The set {f (g) | g ∈ G} generates F , that is, every element of F can be obtained from {f (g) | g ∈ G} using finite meets and arbitrary joins in F .
The assignment f can be extended to an assignement f for any expression x build from the generators in G using ∧ and . We require: (iii) For any frame F and assignment f : G → ZF satisfying property (ii) there exists a frame homomorphism h : F → F such that the diagram The collection of open sets of a topological space X forms a frame, denoted opnX. A continuous map f : X → X induces opnf = f −1 : opnX → opnX and with this definition opn is a contravariant functor Top → Frm. A frame is called spatial if it isomorphic to opnX for some topological space X.
A point of a frame F is a frame homomorphism p : F → 2, with 2 = { , ⊥} the two-element frame. Let ptF be the collection of points of F endowed with the topology { a | a ∈ F }, where a = {p ∈ ptF | p(a) = }. For a frame homomorphism f : F → F define ptf : ptF → ptF by p → p • f . The assignment pt defines a contravariant functor Frm → Top. A topological space that arises as the space of points of a lattice is called sober. The sobrification of a topological space X is pt(opnX).
We denote by Sob and KSob the full subcategories of Top whose objects are sober spaces and compact sober spaces, respectively. Where Frm is the category of frames and frame homomorphisms, SFrm, KSFrm and KRFrm are the full subcategories of Frm whose objects are spatial frames, compact spatial frames and compact regular frames, respectively. The functor Z : Frm → Set is the forgetful functor sending a frame to the underlying set, and restricts to every subcategory of Frm. Note that Ω = Z • opn.
Fact 2.10. The functor pt is dually adjoint to opn. This adjunction restricts to a duality between the category of spatial frames and the category of sober spaces, This duality restricts to the dualities KRFrm ≡ KHaus op .
For a more thorough exposition of frames and spaces, and a proof of the statements in Fact 2.10, we refer to Section C1.2 of [Joh02]. We explicitly mention one isomorphism which is part of this duality, for we will encounter it later on.
Remark 2.11. Let X be a sober space. Then Fact 2.10 entails that there is an isomorphism X → pt(opnX). This isomorphism is given by x → p x , where p x is the point given by for all x ∈ X and a ∈ ΩX.

Logic for topological coalgebras
Although not all of our results can be proved for every full subcategory of Top, we will give the basic definitions in full generality. To this end, we let C be some full subcategory of Top and define coalgebraic logic with C as base category. In particular C = KHaus and C = Sob will be of interest. Throughout this section T is an arbitrary endofunctor on C. Recall that Ω : Top → Set sends a topological spaces to its set of opens, while opn : Top → Frm takes a space to its collection of opens viewed as a frame. Also, recall that Φ is an arbitrary but fixed set of proposition letters. We begin with defining the topological version of a predicate lifting, called an open predicate lifting.
1. An open predicate lifting for T is a natural transformation An open predicate lifting is called monotone in its i-th argument if for every X ∈ C and all a 1 , . . . , a n , b ∈ ΩX we have λ X (a 1 , . . . , a i , . . . , a n ) ⊆ λ X (a 1 , . . . , a i ∪ b, . . . , a n ), and monotone if it is monotone in every argument. It is called Scott-continuous in its i-th argument if for every X ∈ C and every directed set A ⊆ ΩX we have λ X (a 1 , . . . , A, . . . , a n ) = b∈A λ X (a 1 , . . . , b, . . . , a n ) and Scott-continuous if it is Scott-continuous in every argument.
A collection of open predicate liftings for T is called a geometric modal signature for T. A geometric modal signature for a functor T is called monotone if every open predicate lifting in it is monotone, Scott-continuous if every open predicate lifting in it is Scott-continuous, and characteristic if for every topological space X in C the collection {λ X (a 1 , . . . , a n ) | λ ∈ Λ n-ary, a i ∈ ΩX} is a sub-base for the topology on TX.
Remark 3.2. Using the fact that for any two (open) sets a, b the set {a, a ∪ b} is directed, it is easy to see that Scott-continuity implies monotonicity.
Scott-continuity will play a rôle in Section 7, where it is used to show that the collection of formulas modulo (semantic) equivalence is a set, rather than a proper class.
Let S be the Sierpinski space, i.e. the two-element set 2 = {0, 1} topologised by {∅, {1}, 2}. For a topological space X and a ⊆ UX let χ a : X → S be the characteristic map (i.e. χ a (x) = 1 iff x ∈ a). Note that χ a is continuous if and only if a ∈ ΩX. Analogously to predicate liftings for Set-functors [Sch05, Proposition 43], one can classify n-ary predicate liftings as open subsets of TS n . This elucidates the analogy with predicate liftings for Set-functors. Proposition 3.3. Suppose S ∈ C, then there is a bijective correspondence between n-ary open predicate liftings and elements of ΩTS n . This correspondence is given as follows: To an open predicate lifting λ assign the set λ S n (π −1 1 ({1}), . . . , π −1 n ({1})) ∈ ΩTS n , where π i : S n → S is the i-th projection, and conversely, for c ∈ ΩTS n define λ c : Ω n → ΩT by λ c X (a 1 , . . . , a n ) = (T χ a 1 , . . . , χ an ) −1 (c). Furthermore, there is a bijective correspondence between open predicate liftings and continuous functions TS n → S. This is established by identifying elements of ΩTS n with their characteristic map TS n → S. This view on predicate liftings has been investigated in [BKV15, Section 7].
Definition 3.4. The language induced by a geometric modal signature Λ is the collection GML(Φ, Λ) of formulas defined by the grammar where p ranges over the set Φ of proposition letters, I is some index set, and λ ∈ Λ is n-ary. Abbreviate ⊥ := ∅. We call a formula in GML(Φ, Λ) coherent if it does not involve any infinite disjunctions. Definition 3.5. A geometric T-model is a triple X = (X, γ, V ) where (X, γ) is a T-coalgebra and V : Φ → ΩX is a valuation of the proposition letters. A map f : X → X is a geometric T-model morphism from (X, γ, V ) to (X , γ , V ) if f is a coalgebra morphism between the underlying coalgebras and V = f −1 • V . The collection of geometric T-models and geometric T-model morphisms forms a category, which we denote by Mod(T).
Definition 3.6. The semantics of ϕ ∈ GML(Φ, Λ) on a geometric T-model X = (X, γ, V ) is given recursively by We write X, x ϕ iff x ∈ ϕ X . Two states x and x are called modally equivalent if they satisfy the same formulas, notation: x ≡ Λ x . We say that ϕ is a semantic consequence of ψ in Mod(T), notation: ϕ T ψ, if ϕ X ⊆ ψ X for all X ∈ Mod(T).
The following proposition shows that morphisms preserve truth. Its proof is similar to the proof of Theorem 6.17 in [Ven17].
Proposition 3.7. Let Λ be a geometric modal signature for T. Let X = (X, γ, V ) and X = (X , γ , V ) be geometric T-models and let f : X → X be a geometric T-model morphism. Then for all ϕ ∈ GML(Φ, Λ) and x ∈ X we have We state the notion of behavioural equivalence for future reference.
Definition 3.8. Let X = (X, γ, V ) and X = (X , γ , V ) be two geometric T-models and x ∈ X, x ∈ X two states. We say that x and x are behaviourally equivalent in Mod(T), notation: x Mod(T) x , if there exists a geometric T-model Y and T-model morphisms As an immediate consequence of Proposition 3.7 we find that behavioural equivalence implies modal equivalence. We will see in Section 7 that, under mild conditions, the converse is true as well.
Next we have a look at the Vietoris functor on KHaus. Coalgebras for this functor have also been studied in [BBH15], where they are used to interpret the positive modal logic from [Dun95,CJ99]. In Section 4 we study the example of the monotone functor, which gives rise to monotone modal geometric logic. Example 3.10 (Vietoris functor). For a compact Hausdorff space X, let V kh X be the collection of closed subsets of X topologized by the subbase where a ranges over ΩX. For a continuous map f : . If X is compact Hausdorff, then so is V kh X [Mic51, Theorem 4.9], and if f : X → X is a continuous map between compact Hausdorff spaces, then V kh f is well defined and continuous [KKV04, Lemma 3.8], so V kh defines an endofunctor on KHaus. Let X = (X, γ, V ) be a V kh -model. If we set  such that for all X ∈ C and a 1 , . . . , a n ∈ ΩX the set µ X (a 1 , . . . , a n ) is open in TX. Example 3.12. The predicate lifting corresponding to the box modality from Example 3.10 is (monotone) extendable, for it is the restriction of µ : U → U • V kh given by µ X (u) = {b ∈ V kh X | b ⊆ u}. Likewise, all other predicate liftings from Examples 3.9, 3.10 and the monotone functor from Section 4 are extendable as well.
We devote the remainder of this section to investigating strong open predicate liftings.
Recall from Example 3.9 that 2 denotes the two-element set with the trivial topology.
We claim that natural transformations µ : (P • U) n →P • U • T correspond one-to-one with elements ofPUT2, provided 2 ∈ C: To a natural transformation µ associate the set µ 2 (p −1 1 ({1}), . . . , p −1 n ({1})), where p i : 2 n → 2 denotes the i-th projection. Conversely, for c ∈PUT2 define µ c by µ c X (a 1 , . . . , a n ) = (T χ a 1 , . . . , χ an ) −1 (c), where X is a topological  The following proposition gives two sufficient conditions on T for its open predicate liftings to be extendable. For a full subcategory C of Top let preC denote the category of topological spaces in C and (not necessarily continuous) functions.
Proposition 3.14. Let T be an endofunctor on C and suppose 2, S ∈ C.
(1) If T preserves injective functions then every open predicate lifting for T is extendable.
(2) If T extends to preC, then every open predicate lifting for T is extendable.
Proof. For the first item, let c ∈ ΩTS n determine the n-ary open predicate lifting λ c . Since s n is injective, by assumption Ts n is as well, and hence c = (UTs n ) −1 ((UTs n )[c]). Proposition 3.13 now implies that µ (UTs n )[c] is a strong open predicate lifting. It is easy to see that µ (UTs n )[c] extends λ c , hence the latter is extendable.
For the second item we show that, under the assumption, T preserves injective functions. Let f : X → Y be an injective function in C, then there exists a (not necessarily continuous) Monotone open predicate lifting (hence also Scott-continuous ones) for an endofunctor on KHaus are always extendable: Proposition 3.15. Let T be an endofunctor on KHaus and Λ a monotone geometric modal signature for T. Then Λ is monotone extendable.
Proof. Let λ ∈ Λ. We need to show that λ is the restriction of some monotone strong predicate lifting. Define λ X :P n UX →PUTX : (b 1 , . . . , b n ) → {λ X (a 1 , . . . , a n ) | a i ∈ ΩX and a i ⊇ b i }. Monotonicity of λ X ensures λ X (a) = λ X (a) for all a ∈ ΩX and λ is monotone by construction. So we only need to show that λ is indeed a strong open predicate lifting, i.e. a natural transformationP n UX →PUTX. We assume λ to be unary, the general case being similar.
For a continuous map f : X → X between compact Hausdorff spaces we need to show that λ X • f −1 = (Tf ) −1 • λ X . Since, by naturality of λ, the right hand side is equal to . So every element in the intersection of the right hand side is contained in the one on the left hand side and therefore we have ⊆ in (3.1). For the converse, suppose c ∈ ΩX and is one of the elements in the intersection on the left hand side of (3.1).

The monotone neighbourhood functor on KHaus
In this section we define the monotone neighbourhood functor on Frm and show that it (individually) preserves regularity and compactness. This functor is a variation of the Vietoris Locale [Joh85, Section 1]. Subsequently, we give a functor on KHaus which is dual to the restriction of the monotone neighbourhood functor to KRFrm.
Definition 4.1. For a frame F , let MF be the frame generated by a, a, where a ranges over F , subject to the relations where a, b ∈ F and A is a directed subset of F . For a homomorphism f : F → F define Mf : MF → MF on generators by a → f (a) and a → f (a). The assignment M defines a functor on Frm.
The proof of the following proposition closely resembles that of Proposition III4.3 in [Joh82]. In a similar manner one can show that M preserves complete regularity and zero-dimensionaity.
Proposition 4.2. If F is a regular frame, then so is MF .
Proof. We need to show that for all c ∈ MF we have c = {d ∈ MF | d c}. It follows from Lemma 2.6 that it suffices to focus on the generators of MF . Let a ∈ F , then we know {d ∈ MF | d a} ≤ a. Suppose b a in F , then by Lemma 2.5 ∼b ∨ a = and hence ∼b ∨ a = . Also ∼b ∧ b = ⊥ so it follows from (M 2 ) that ∼b ∧ b = ⊥. This proves b a, because the element ∼b is such that ∼b ∨ a = and ∼b ∧ b = ⊥. Since F is regular and {b ∈ F | b a} is directed, it follows that In a similar fashion one may show that a = {d ∈ MF | d a}. This proves the proposition.
We now prove that the functor M preserves compactness. We proceed in a similar manner as [VVV13, Theorem 4.2]. This relies on an auxiliary definition and lemma (Definition 4.3 and Lemma 4.4), in which we give an alternative description of MF . We then prove that this alternative description preserves compactness.
In [Gro18, Corollary 3.42] we proved the same result by first giving a duality result between frames and topological spaces, and then proving preservation of compactness on the topological side. The main difference between that proof and the one we present here is that the current one is constructive.
Write P ω of the finite powerset functor and recall that Z : Frm → Set is the forgetful functor.
Definition 4.3. For a frame F define M F to be the free frame generated by P ω ZF × P ω ZF , qua join-semilattice (that is, the join in M F is given by This results in a frame isomorphic to MF : Clearly these define a bijection. Furthermore it is straightforward to verify that these maps are well defined by checking that the images of generators satisfy relations of the respective frame.
Theorem 4.5. Suppose F is compact. Then MF is compact.
Proof. The frame MD is compact iff there is a preframe homomorphism ϕ : MD → 2 that is right adjoint to the unique frame homomorphism ! : 2 → MD, where 2 = {0, 1} is the two-element frame. By Proposition 4.4 we have MD ∼ = M D, and since all the relations in Definition 4.3 are join-stable, we can use the preframe coverage theorem [JV06, Theorem 5.1] to find that M F viewed as a preframe is the preframe generated by P ω ZF × P ω ZF qua poset, subject to the the relations from Definition 4.3. Define First we check that ϕ is indeed a pre-frame homomorphism. Since ϕ is defined on generators, it suffices to show that it preservers the relations (M 1 ) to (M 6 ), because if it does it can be lifted in a unique way to a preframe homomorphism M F → 2. It is clear that ϕ is a monotone morphism (hence preserves the poset structure of the generators). We check that ϕ preserves the relations one by one.
Then either there is some c ∈ γ such that c ∨ δ = F , which implies ϕ(γ, δ) = 1, or a ∨ δ = F . In the latter case, note that we also have some The first equality holds because a ∧ b = ⊥ F . Again we find ϕ(γ, δ) = 1.
We now know that M restricts to an endofunctor on KRFrm. We write M kr for this restriction.
Remark 4.6. The category Loc of locales and locale morphisms is the opposite of Frm. Therefore, we can also view M as an endofunctor on locales and MA as the monotone neighbourhood locale, where A is a locale.

4.2.
Monotone neighbourhood functor on KHaus. We now describe the topological manifestation of the monotone neighbourhood functor.
Definition 4.7. Let X = (X, τ ) be a compact Hausdorff space. Let D kh X be the collection of sets W ⊆ PX such that u ∈ W iff there exists a closed c ⊆ u such that every open superset of c is in W . Endow D kh X with the topology generated by the subbase where a ranges over ΩX. For continuous functions f : .
For the time being, we regard D kh as a functor KHaus → Top, because we have no evidence yet that it restricts to an endofunctor on KHaus. We aim to prove that D kh is dual to the restriction of M to KRFrm. As a corollary, we then obtain that D kh indeed restricts to KHaus. Theorem 4.9. If X is a compact Hausdorff space then pt(M(opnX)) ∼ = D kh X.
We temporarily fix a compact Hausdorff space X and define the two maps constituting a homeomorphism. We have W p ∈ D kh X because it is the up-set of a collection of closed sets; indeed, for each b ∈ W p there exists a closed subset X \ a ⊆ b with p( a) = ⊥ and by definition all open supersets of X \ a are in W p . Therefore ζ is well defined. In the converse direction we define: Definition 4.11. For a compact Hausdorff space X, define where p W is given on generators by The direction from right to left follows from the fact that W is upwards closed. Conversely, suppose ↑ A ∈ W , then there is a closed set k ⊆ ↑ A with k ∈ W . The elements of A now cover the closed therefore compact set k, so there is a finite The following lemma is key for proving that ζ and θ are continuous and each other's inverses.
and the fact that a = ↑ {a | a a} (this is true because X is assumed to be compact Hausdorff so opnX is compact regular) to find It follows that a ∈ W p iff p( a) = .
We have now aquired sufficient knowledge to prove Theorem 4.9.
Proof of Theorem 4.9. We claim that the maps ζ and θ define a homeomorphism between D kh X and pt(M(opnX)). First we prove that they are each other's inverses, by showing that for all p ∈ pt(M(opnX)) and W ∈ D kh X we have p Wp = p and W p W = W . In order to prove that (the frame homomorphisms) p and p Wp coincide, it suffices to show that they coincide on the generators of M(opnX). By Definition 4.11 and Lemma 4.13 we have for all open sets a, because elements of D kh X are uniquely determined by the closed sets they contain. This follows immediately from the definitions and Lemma 4.13, as We complete the proof by showing that ζ and θ are continuous. The opens of pt(M(opnX)) are generated by a = {p | p( a) = } and a = {p | p( a) = }, for a ∈ ΩX. We have and similarly θ −1 ( a) = a. Continuity of θ follows from the fact that a and a are open in D kh X. Conversely, the opens of D kh X are generated by a and a, where a ranges over ΩX. It is routine to see that ζ −1 ( a) = a and ζ −1 ( a) = a. This proves continuity of ζ. We showed that θ is a continuous function with continuous inverse ζ, hence a homeomorphism. This completes the proof of the theorem.
Corollary 4.14. The assignment D kh defines an endofunctor on KHaus.
Theorem 4.9 yields a map M kr (opnX) → opn(D kh X) for a compact Hausdorff space X given by Unravelling the definitions shows that, on generators, it is given by a → a and a → a.
Definition 4.15. For every compact Hausdorff space X define η X : M kr (opnX) → opn(D kh X) on generators by η X ( a) = a and η X ( a) = a. By the preceding discussion η X is a well-defined frame isomorphism.
It turns out that the maps η X constitute a natural isomorphism.
Proof. It follows from Theorem 4.9 that each of the η X is an isomorphism, so we only need to show naturality. That is, for any morphism f : X → X in KHaus, the following diagram commutes, (Since opn is a contravariant functor, the horizontal arrows are reversed.) For this, suppose a is a generator of M kr (opnX ). Then and by analogous reasoning ΩD kh f • η X ( a) = η X • M kr (opnf )( a). This proves that the diagram commutes.
As an immediate corollary of Lemma 4.16 we obtain: It is easy to see that these are monotone extendable. Write and for the corresponding modal operators and let (X, γ, V ) be a D kh -model.
and similarly x ϕ if X \ ϕ / ∈ γ(x). This is the same as neighbourhood semantics for monotone modal logic over a classical base [Che80,Han03].
Remark 4.18. We will see in Example 6.6 that the functor D kh on KHaus can be generalised to an endofunctor of Top which restricts to Sob.

Axioms, Soundness and completeness
We define the notion of one-step axioms (similar to [KKP04, Definition 3.8]) and one-step rules for a collection of predicate liftings. These give rise to axioms and rules for the language GML(Φ, Λ) from Definition 3.4, and to an endofunctor L on the category of frames. As in Section 3 we let C be some full subcategory of Top, T an endofunctor on C, and we view opn as a contravariant functor C → Frm and Ω as a contravariant functor C → Set.
At the end of Subsection 5.2, in order to derive a general completeness result, we restrict our attention to a language without proposition letters. This need not be problematic: proposition letters can be introduced via (nullary) predicate liftings. In particular, this means that the category of T-models is the same as the category of T-coalgebras.
Ultimately, using the duality proved in Lemma 4.16, we derive that monotone modal geometric logic without proposition letters is sound and complete with respect to D khcoalgebras.
We define the notions of a one-step axiom and a one-step rule for GML(Φ, Λ): Definition 5.1. A one-step axiom for GML(Φ, Λ) is a consequence pair α β, where α, β are one-step formulas. A one-step rule is an expression of the form where I is some index-class, a i , b i are zero-step formulas for i ∈ I and α, β are one-step formulas. First let us investigate how one-step axioms and rules give rise to an (equationally defined) endofunctor L (Λ,Ax) on Frm-when no confusion is likely we drop the subscript and simply write L. Given a frame F , the frame LF can be presented as follows. As generators we take the collection The idea is now to instantiate the (meta)variables of the axiomatisation Ax with the elements of F . Zero-step formulas then naturally evaluate to elements of F . Consequently, an axiom α β gives rise to a relation α ≤ β, and a rule as in (5.1) yields the relation α ≤ β conditionally, that is, we only consider the relation in those cases where a i ≤ b i for all i.
Definition 5.2. For a frame F , define L (Λ,Ax) F = LF to be the frame where R is the collection of relations that arises from substituting the metavariables from the schemata in Ax with elements from F . For a morphism f : F → F define Lf on generators by Lf (λ(a 1 , . . . , a n )) = λ(f (a 1 ), . . . , f (a n )).
Example 5.4. Suppose T = D kh , the monotone neighbourhood functor on KHaus, and λ , λ are given as in Subsection 4.3. The following collection of axioms and rules is sound: where A is a directed set of GL(Φ)-formulas (cf. Subsection 2.4). To be somewhat more precise, we can view both (m 3 ) and (m 6 ) to be the consequence of a rule, the premises of which are given by a set of consequence pairs witnessing the directedness of A. To see, for example, that (m 3 ) is valid in a D kh -coalgebra (X, γ), we need to show that in D kh X, where A is a directed set of open subsets of X. So suppose W ∈ ↑ A, then ↑ A ∈ W . By definition there must be a closed c ⊆ ↑ A such that c ∈ W . Then ↑ A is an open cover of c and since c is closed, hence compact, there must be a finite subcover. But then there must be a single a ∈ A such that c ⊆ a, because A is directed, and as W is up-closed under inclusions we have a ∈ W . This implies W ∈ a, i.e. W is in the right hand side of (5.2).
The functor M from Definition 4.1 is obtained from the procedure of Definition 5.2. Example 5.5. In a similar manner, one can find a collection of sound axioms and rules for the Vietoris functor on KHaus such that the procedure from Definition 5.2 yields the Vietoris locale from [Joh85, Section 1].
For the remainder of this section we work in the following setting: Assumption 5.6. Throughout the remainder of this section, we assume given a collection Λ of predicate liftings for an endofunctor T on C, and a set Ax of axioms and rules for GML(Φ, Λ) which is sound in the sense of Definition 5.3. We write L for the endofunctor on Frm given by the procedure in Definition 5.2 and ρ is the associated natural transformation.
We write ϕ GML(Φ,Λ,Ax) ψ if the consequence pair ϕ ψ is in GML(Φ, Λ, Ax). If no confusion is likely, we omit the subscript from the turnstyle and simply write ϕ ψ. The initial L-algebra L gives rise to an interpretation of GML(Φ, Λ) in every L-algebra: The interpretation of a formula ϕ in A = (A, α) ∈ Alg(L) is given by The interpretation is related to the semantics via the complex algebra: Definition 5.9. The complex L-algebra of a T-coalgebra X = (X, γ) is X + = (opnX, opnγ • ρ X ), where ρ is the natural transformation from Definition 5.3.
We can view the interpretation of a formula ϕ in a T-coalgebra as an element of its complex algebra. Examination of the definitions shows that That is, if ϕ T ψ then (ϕ, ψ) ∈ GML(Λ, Ax). By Lemma 5.8 it suffices to show that [ϕ] ≤ [ψ] in the initial L-algebra L = (L, ) whenever ϕ T ψ.
If ϕ T ψ, then we know that ϕ X + ≤ ψ X + in every complex algebra X + for X ∈ Mod(T). However, there is no guarantee that L should be the complex algebra of some T-model, so we do not automatically get completeness.
The next proposition shows that in order to prove completeness it suffices to find a T-coalgebra X and an L-algebra morphism h : X + → L.
Proposition 5.10. If there exists a T-coalgebra X and an L-algebra morphism h : X + → L, then ϕ T ψ implies ϕ ψ.
Proof. Write i for the unique L-algebra morphism L → X + . Then initiality of L forces h • i = id L . Suppose ϕ T ψ, then we have ϕ X ⊆ ψ X , which in turn implies ϕ X + ≤ ψ X + by (5.3). It follows from monotonicity of h that This proves the proposition.
Ideally, one would use a duality of functors to establish that such an X as in Proposition 5.10 exists. For example, this is how one can prove completeness for (classical) normal modal logic: The Vietoris functor on Stone is the Stone dual of the endofunctor on BA which determines the logic. However, since we do not start with a dual equivalence (like Stone duality) but rather with a dual adjunction, endofunctors on both categories cannot be dual.
To remedy this, we will make use of the fact that the dual adjunction between Top and Frm restricts to several dual equivalences (see Fact 2.10). Note that this does not yet guarantee that the initial L algebra from Lemma 5.7 is the complex algebra of some T-coalgebra! Indeed, its underlying frame need not be in the restricted dual equivalence. We will see that some of the dual equivalences are good enough to "imitate" frames that fall outside it.
We now investigate under which conditions we can use Proposition 5.10. As announced, we shall restrict our attention to the case where Φ = ∅, i.e. we work in a language without proposition letters. This means that there is no need for having valuations, hence Mod(T) is simply (isomorphic to) Coalg(T). If Φ = ∅, we shall write GML(Λ) instead of GML(Φ, Λ) and GML(Λ, Ax) for GML(Φ, Λ, Ax). The absence of proposition letters need not pose a big deficiency: proposition letters can simply be introduced via predicate liftings.
Specifically, we look for a subcategory A of Alg(L) such that: (1) Every L-algebra is the codomain of an L-algebra morphism whose domain is in A; (2) Every algebra in A is the complex algebra corresponding to some T-coalgebra. Clearly, if this is the case we can employ Proposition 5.10.
For the first item, it turns out useful to consider coreflective subcategories of Frm. These are full subcategories F of Frm such that the inclusion functor F → Frm has a right adjoint. We recall an alternative definition from [AHS90] (which is in turn equivalent to the one in [ML71, Section IV.3]). We can now formulate simple conditions that guarantee item (1) to hold. Now suppose that all objects in F are spatial and write T for the subcategory of Top which is dually equivalent to F. Furthermore, assume that T is a subcategory of C (for otherwise T is not defined for every space in T).
If the restriction L of L is dual to the restriction T of T to T, then we know that Coalg(T ) ≡ op Alg(L ). In particular, this means that every element of Alg(L ) is the complex algebra of some T -coalgebra (hence of some T-coalgebra), i.e. item (2) is satisfied. Summarising: Theorem 5.13. Suppose there exists a coreflective subcategory F of Frm such that • The dual T of F is a subcategory of C; • L restricts to an endofunctor L on F; • T restricts to an endofunctor T on F op ; • L is dual to T ; Then GML(Λ, Ax) is complete with respect to Coalg(T), in the sense that for for every consequence pair ϕ ψ of closed GML-formulas we have that ϕ T ψ implies ϕ ψ ∈ GML(Λ, Ax).
Let us apply this theorem to normal and monotone modal geometric logic.
Example 5.14. We denote by M the smallest collection of consequence pairs closed under the axioms and rules from geometric logic (see Subsection 2.4) and the ones presented in Example 5.4. (Note that the congruence rules follow from the monotonicity rules.) It follows from the duality between D kh and M kr , the fact that KRFrm is a coreflective subcategory of Frm, and Theorem 5.13 that M is (sound and) complete with respect to Coalg(D kh ).
Example 5.15. Similar to Example 5.14, one can prove that normal geometric modal logic N is sound and complete with respect to Coalg(V kh ). In this case, the axioms and rules of N are the ones from geometric logic, those introduced for positive modal logic in [Dun95, Section 2], and Scott-continuity. We leave the details to the reader.

Lifting logics from Set to Top
In [KKP05, Section 4] the authors give a method to lift a Set-functor T : Set → Set, together with a collection of predicate liftings Λ for T, to an endofunctor on Stone. We adapt their approach to obtain an endofunctor T Λ on Top, and a collection of Scott-continuous open predicate liftings Λ for T Λ . In this section the notation ↑ is used for directed joins, i.e. joins over directed sets.
6.1. Lifting functors from Set to Top. Let T be an endofunctor on Set and Λ a collection of predicate liftings for T. To define the action of T Λ on a topological space X we take the following steps: Step 1. Construct a frameḞ Λ X of the images of predicate liftings applied to the open sets of X (viewed simply as subsets of T(UX)); Step 2. QuotientḞ Λ X with a suitable relation that ensures ↑ b∈B λ(b) = λ( ↑ B) whenever λ is monotone; Step 3. Employ the functor pt : Frm → Top to obtain a (sober) topological space. This is the content of Definitions 6.1, 6.3 and 6.5. Recall that U : Top → Set is the forgetful functor and that Q is the contravariant functor sending a set to its Boolean powerset algebra.
That is, we close this set under finite intersections and arbitrary unions in Q(T(UX)). For a continuous map f : X → X letḞ Λ f :Ḟ Λ X →Ḟ Λ X be the restriction of Q(T(Uf )) toḞ Λ X .
By continuity of f we have f −1 (a i ) ∈ ΩX so the latter is indeed inḞ Λ X. Functoriality ofḞ Λ follows from functoriality of Q • T • U. Definition 6.3. Let Λ be a collection of predicate liftings for a Set-functor T. For X ∈ Top, let F Λ X be the quotient ofḞ Λ X with respect to the congruence ∼ generated by ↑ b∈B λ(a 1 , . . . , a i−1 , b, a i+1 , . . . , a n ) ∼ λ(a 1 , . . . , a i−1 , ↑ B, a i+1 , . . . , a n ) for all a i ∈ ΩX, B ⊆ ΩX directed, and λ ∈ Λ monotone in its i-th argument. Write q X :Ḟ Λ X → F Λ X for the quotient map and [x] for the equivalence class in F Λ X of an element x ∈Ḟ Λ X. For a continuous function f : (a 1 , . . . , a n ))].
Quotienting by the congruence from Definition 6.3 ensures that the lifted versions of monotone predicate liftings are Scott-continuous, see Proposition 6.11 below. This is useful when constructing a final model, because Scott-continuity ensures that the collection of formulas modulo so-called semantic equivalence is set-sized (see Lemma 7.3 below), and consequently the final model aids in comparing several equivalence notions in Section 8.
Lemma 6.4. The assignment F Λ defines a contravariant functor.
Proof. We need to prove functoriality of F Λ and that F Λ f is well defined for every continuous map f : X → X . In order to show that F Λ is well defined, it suffices to show thatḞ Λ f is invariant under the congruence ∼. If f : X → X is a continuous, then a 1 , . . . , a i−1 , b , a i+1 , . . . , a n )) a 1 , . . . , a i−1 , ↑ B, a i+1 , . . . , a n )) soḞ Λ f is invariant under the congruence. In the ∼-step we use the fact that {f −1 (b ) | b ∈ B} is directed in ΩX. Functoriality of F Λ f follows from functoriality ofḞ Λ .
We are now ready to define the topological Kupke-Kurz-Pattinson lift of a functor on Set together with a collection of predicate liftings, to a functor on Top.
Definition 6.5. Define the topological Kupke-Kurz-Pattinson lift (KKP lift for short) of T with respect to Λ to be the functor This is a functor Top → Top and since pt lands in Sob it restricts to an endofunctor on Sob.
Let us put our theory to action. As stated in Section 4 we can generalise the monotone functor D kh on KHaus from Definition 4.7 to an endofunctor on Top. We will show that lifting the monotone Set-functor D with respect to the predicate liftings for box and diamond from Example 2.4 gives rise to a functor on Top which restricts to D kh . where X ∈ Set. Furthermore recall from Definition 4.7 that for a compact Hausdorff space X the space D kh X is the subset of D(UX) of collections of sets W satisfying for all u ⊆ UX that u ∈ W iff there exists a closed c ⊆ u such that every open superset of c is in W . In particular this means U(D kh X) ⊆ D(UX). The set D kh X is topologised by the subbase By Theorem 4.9 the functor M : Frm → Frm from Definition 4.1 is such that M(opnX) ∼ = opn(D kh X) whenever X is a compact Hausdorff space. We claim that So we only need to show injectivity of ϕ. Our strategy to prove this is to define a map ψ : F {λ ,λ } X → opn(D kh X) and show that it is inverse to ϕ on the level of sets. Since a set-theoretic inverse suffices we do not need to prove that ψ is a homomorphism; we just want it to be well defined. Instead of defining ψ : F {λ ,λ } X → opn(D kh X) directly, we will give a well-defined map ψ :Ḟ {λ ,λ } X → opn(D kh X) whose kernel contains the kernel of the quotient map q X :Ḟ {λ ,λ } X → F {λ ,λ } X. This in turn yields the map ψ we require. In a diagram:Ḟ Define ψ :Ḟ {λ ,λ } X → M(opnX) on generators by λ (a) → a and λ (a) → a. In order to show that this assignments yields a well-defined map (hence extends to a frame homomorphism by Remark 2.8) we need to show that the presentation of an element iṅ F {λ ,λ } X does not affect its image under ψ . That is, if where J i , J i , L k and L k are finite index sets, then i∈I j∈J (a k, )) . and similarly ψ (λ (a)) = λ (a) ∩ U(D kh X). Suppose the identity in (6.3) holds, then we have i∈I j∈J

As stated we have
Since these pairs generate the congruence of Definition 6.3, we have ∼ = ker(q X ) ⊆ ker(ψ ) and hence there exists a map ψ : F {λ ,λ } X → opn( TX) such that the diagram in (6.2) commutes. Therefore ψ is (well) defined on generators by [λ (a)] → a and [λ (a)] → a. One can easily check that ψ • ϕ = id and ϕ • ψ = id by looking at the action on the generators. It follows that ϕ is injective.
This entails that for compact Hausdorff spaces X, Furthermore, it can be seen that for continuous maps f : topological space X we have FX = 2, which is not a T 0 space, hence not a sober space.
Therefore F does not preserve sobriety, while every lifted functor automatically preserves sobriety. Thus F is not the lift of any Set-functor.
6.2. Lifting predicate liftings. We describe how to lift a predicate lifting to an open predicate lifting. Recall that Z : Frm → Set is the forgetful functor which sends a frame to its underlying set.
Definition 6.9. Let Λ be a collection of predicate liftings for a functor T : Set → Set. A predicate lifting λ :P n →P • T in Λ induces an open predicate lifting λ : By λ UX we actually mean the restriction of λ UX to Ω n X ⊆P(UX). The map k FX is the frame homomorphism given by a → {p ∈ pt(F Λ X) | p(a) = 1}. Then Λ := { λ | λ ∈ Λ} is a geometric modal signature for T Λ .
So λ is an open predicate lifting.
The nature of the definitions of T Λ and Λ yields the following desirable results.
(1) Let T : Set → Set be a functor and Λ a collection of predicate liftings for T. Then Λ is characteristic for T Λ .
Proof. Let X be a topological space. For the first item, we need to show that the collection { λ(a 1 , . . . , a n ) | λ ∈ Λ n-ary, a i ∈ ΩX} (6.4) forms a subbase for the topology on T Λ X. An arbitrary nonempty open set of T Λ X is of the form x = {p ∈ pt( F Λ X) | p(x) = 1}, for x ∈ F Λ X. An arbitrary element of F Λ X is the equivalence class of an arbitrary union of finite intersections of elements of the form λ UX (a 1 , . . . , a n ), for λ ∈ Λ and a 1 , . . . , a n ∈ ΩX. So we may write x = 10:28 N. Bezhanishvili, J. de Groot, and Y. Venema Vol. 18:4 i∈I ( j∈J i [λ i,j UX (a i,j 1 , . . . , a i,j n i,j )]) for some index set I, finite index sets J i , λ i,j ∈ Λ and open sets a i,j k ∈ ΩX. We get The second equality follows from Definition 6.9. This shows that the open sets in (6.4) indeed form a subbase for the open sets of T Λ X.
The second item follows immediately from the definitions.
Example 6.12. Let λ be the box-predicate lifting for the monotone functor D on Set from Example 2.4. Then the procedure from Definition 6.9 sends an open a in a compact Hausdorff space X to [λ UX (a)] in F Λ X. We know from Example 6.6 that F Λ X is isomorphic to M(opnX) (provided X is compact Hausdorff) and the element [λ UX (a)] corresponds to a ∈ M(opnX). Therefore We know from Theorem 4.9 that pt • M • opn ∼ = D kh and under this isomorphism a ∈ Ω • pt • M • opnX is sent to a ∈ Ω(D kh X). Therefore Let T be an endofunctor on Set and Λ a set of monotone predicate liftings for T. Moreover, suppose that we have a collection Ax of one-step axioms and rules for the (classical) modal language ML(Λ). Then Ax yields axioms and rules for the language GML( Λ) (by simply replacing every occurrence of λ with λ). We write Ax for the collection of these axioms, together with the Scott-continuity axioms for all open predicate liftings in Λ. Then by construction Ax is sound for T Λ -coalgebras, and the pair ( Λ, Ax ) gives rise to an endofunctor L on Frm via the procedure from Definition 5.2.
It is now natural to ask whether we can apply Theorem 5.13 to this setting to obtain a completeness result. However, while soundness of Ax implies the existence of a natural transformation L • opn → opn • T Λ , there seems to be no guarantee that this yields a natural isomorphism. A potential duality of restrictions of the functors to a suitable subcategory is furthermore frustrated by the fact that we have no generic preservation results, so that T Λ need not even restrict to KSob or KHaus. We highlight this question as an interesting direction for further research.

A final model construction
We construct a final model in Mod(T) for a functor T where either T is an endofunctor on Sob, or T is an endofunctor on Top which preserves sobriety. This assumption need not be problematic: If a functor on Top does not preserve sobriety we can look at its sobrification. There are various ways of obtaining final coalgebras [KKV04, AMV05, MV04, MV06]. Our strategy for obtaining a final coalgebra is similar to that by Moss and Viglizzo [MV04,MV06] in the sense that we use a language (in our case induced by the geometric similarity type) without a logical system. Specifically, given an endofunctor T on Top, we use a notion of semantic equivalence to obtain an equivalence relation on GML(Φ, Λ). We prove that GML(Φ, Λ) modulo this equivalence relation forms a frame, denoted by E. Subsequently, we set L = opn • T • pt : Frm → Frm and endow E with an L-algebra structure δ. The topological space ptE is the state space for the final model, and the map δ gives rise to a T-algebra structure on ptE. Finally, we show that with the canonical valuation this gives rise to a final model in Mod(T).
We hasten to say that the functor L used in this construction need not coincide with the one discussed at the end of Section 6. From a more abstract point of view [BK05,BK06,Kli07], the functor L defined by L = opn • T • pt gives rise to a logic in its own right. We discuss the implications of the final model construction on L in more detail in the conclusion.
It is well known that coalgebras for endofunctors on Set do not in general have final coalgebras. In particular, there exists no final P-coalgebra, as this would imply the existence of a set X such that X and PX have the same cardinality. One way to remedy this is by using topology to restrict the collection of admissible morphisms. We show that in presence of a Scott-continuous characteristic geometric modal signature we can construct a final coalgebra. Intuitively, Scott-continuity gives us a finitary handle on the final coalgebra we construct. (For comparison, observe that the set of predicate liftings Λ = {λ , λ } for the powerset functor P : Set → Set is not Scott-continuous.) Assumption 7.1. Throughout this section, fix an endofunctor T on Top which preserves sobriety, and a Scott-continuous characteristic geometric modal signature Λ for T. Recall that Φ is a set of proposition letters.
Suppose that the functor T : Top → Top with geometric modal signature Λ arises from an endofunctor T on Set with predicate liftings Λ via the procedure in Section 6. Then T preserves sobriety by definition and Λ is characteristic as a consequence of Proposition 6.11(1). If moreover all the predicate liftings in Λ are monotone, then Proposition 6.11(2) tells us that the open predicate liftings in Λ are Scott-continuous.
Recall that a coherent formula is one which does not involve arbitrary disjunctions. Proof. The proof proceeds by induction on the complexity of the formula. Suppose ϕ = ϕ 1 ∨ ϕ 2 . By induction we may assume that ϕ 1 ≡ T,Λ i∈I ψ i and ϕ 2 ≡ T,Λ j∈J ψ j , where all the ψ i and ψ j are coherent, and we have ϕ ≡ T,Λ i∈I∪J ψ i , as desired. If ϕ = ϕ 1 ∧ ϕ 2 , then ϕ ≡ T,Λ ( i∈I ψ i ) ∧ ( j∈J ψ j ) ≡ T,Λ (i,j)∈I×J ψ i ∧ ψ j . Lastly, suppose ϕ = ♥ λ ( i∈I ψ i ), where all the ψ i are coherent. Then we have i∈I ψ i = { i∈I ψ i | I ⊆ I finite} and by construction the set { i∈I ϕ i X | I ⊆ I, I finite} is directed for every T-model X = (X, γ, V ). Hence by Scott-continuity of λ we obtain Therefore ϕ ≡ T,Λ {♥ λ ( i∈I ψ i ) | I ⊆ I finite}, i.e. ϕ is equivalent to an arbitrary disjunction of coherent formulas. The case for n-ary modalities is similar.
Corollary 7.4. The collection E from Definition 7.2 is a set.
Proof. This follows immediately from Lemma 7.3 and the fact that the collection of coherent formulas is a set. It is easy to check that E now forms a frame. The theory of a point x in a geometric T-model X is the collection of formulas that are true at x. The theory of x defines a point of E. This motivates the next definition.
Definition 7.6. Let Z = ptE. For every geometric T-model X = (X, γ, V ) define the theory map by The space Z will turn out to be the state space of a final model in Mod(T) and we will see in Proposition 7.13 that the theory maps are T-model morphisms. We need to show that δ is well defined. For this purpose it suffices to show that the images of the generators of E satisfy the same relations that they satisfy in LE. Recall Z = ptE, and therefore LE = opn(TZ).
Lemma 7.8. If where the J i and L k are finite index sets and I and K are index sets of arbitrary size.
Proof. We will see that this follows from naturality of λ. Our strategy is to show that the truth sets of the left hand side and right hand side of (7.2) coincide in every geometric T-model X = (X, γ, V ).
Observe that the map th X : X → Z, which sends a point to its theory, is continuous because which is open in X for all formulas ϕ. Compute The steps with ( ) hold because inverse images of functions preserve all unions and intersections. This entails that for all geometric T-models and all states x in X we have and hence (7.2) holds. Therefore δ is well defined.
The algebra structure on E entails a coalgebra structure on Z.
Here k T(ptE) : T(ptE) → pt(opn(T(ptE))) is the isomorphism given in Remark 2.11. Since Z = ptE this indeed defines a map Z → TZ.
We can now endow the T-coalgebra (Z, ζ) with a valuation. Thereafter we will show that (Z, ζ) together with this valuation is final in Mod(T).
Definition 7.11. Let V Z : Φ → ΩZ be the valuation p → p.
The triple Z = (Z, ζ, V Z ) is a geometric T-model, simply because it is a T-coalgebra with a valuation. We can prove a truth lemma for Z: Proof. Use induction on the complexity of the formula. The propositional case follows immediately from the definition of V Z . The cases ϕ = ϕ 1 ∧ ϕ 2 and ϕ = i∈i ϕ i are routine. So suppose ϕ = ♥ λ (ϕ 1 , . . . , ϕ n ). We have This proves the lemma.
Proposition 7.13. For every geometric T-model X = (X, γ, V ) the map th X : X → Z is a T-model morphism.
Proof. We need to show that th X is a T-coalgebra morphism and that th −1 X •V Z = V . The latter follows from the fact that for every proposition letter p we have In order to show that th X is a T-coalgebra morphism, we have to show that the following diagram commutes: Since TZ is sober, hence T 0 , it suffices to show that T th X (γ(x)) and ζ(th X (x)) are in precisely the same opens of TZ. Moreover, we know that the open sets of TZ are generated by the sets λ Z ( ϕ 1 , . . . , ϕ n ), so it suffices to show that for all λ ∈ Λ and ϕ i ∈ GML(Φ, Λ) we have T th X (γ(x)) ∈ λ Z ( ϕ 1 , . . . , ϕ n ) iff ζ(th X (x)) ∈ λ Z ( ϕ 1 , . . . , ϕ n ).
This follows from the following computation, This proves the proposition.
The developed theory results in the following theorem.
Theorem 7.14. Let T be an endofunctor on Top which preserves sobriety, and Λ a Scottcontinuous characteristic geometric modal signature for T. Then the geometric T-model Proof. Proposition 7.13 states that for every geometric T-model X = (X, γ, V ) there exists a T-coalgebra morphism th X : X → Z, so we only need to show that this morphism is unique. Let f : X → Z be any coalgebra morphism. We know from Proposition 3.7 that coalgebra morphisms preserve truth, so for all Therefore we must have f (x) = th X (x).
As an immediate corollary we obtain the following theorem. Recall from Definition 3.8 that two states x and x are behaviourally equivalent in Mod(T) if there are T-model morphisms f and f with f (x) = f (x ).
Proof. If x and x are behaviourally equivalent, then they are modally equivalent by Proposition 3.7. Conversely, if they are modally equivalent, then th X (x) = th X (x ) by construction, so they are behaviourally equivalent.
Remark 7.16. If T is an endofunctor on Sob instead of Top, then the same procedure yields a final model in Mod(T). In particular, T need not be the restriction of a Topendofunctor. However, if T is an endofunctor on KSob or KHaus the procedure above does not guarantee a final coalgebra in Mod(T); indeed the state space Z of the final coalgebra Z we just constructed need not be compact sober nor compact Hausdorff.
Of course, there may be a different way to attain similar results for KSob or KHaus. We leave this as an interesting open question. In Theorem 8.9 we prove an analogue of Theorem 7.15 for endofunctors on KSob.

Bisimulations
This section is devoted to bisimulations and bisimilarity between coalgebraic geometric models. Bisimulations are important tools in the study of modal logics. They provide a structural notion of semantic equivalence: bisimilar worlds satisfy precisely the same logical formulas. If the converse is also true then the logical language is powerful enough to distinguish non-bisimilar states. This is called the Hennessy-Milner property [HM85]. We compare two notions of bisimilarity, modal equivalence (from Definition 3.6) and behavioural equivalence (Definition 3.8). Again, where C is a full subcategory of Top and T an endofunctor on C, we give definitions and propositions in this generality where possible. When necessary, we will restrict our scope to particular instances of C.
Definition 8.1. Let X = (X, γ, V ) and X = (X , γ , V ) be two geometric T-models. Let B be an object in C such that B ⊆ X × X , with projections π : B → X and π : B → X . Then B is called an Aczel-Mendler bisimulation between X and X if for all (x, x ) ∈ B we have x ∈ V (p) iff x ∈ V (p) and there exists a transition map β : B → TB that makes π and π coalgebra morphisms. That is, β is such that the following diagram commutes:

Tπ Tπ
Two states x ∈ UX, x ∈ UX are called bisimilar, notation x x , if they are linked by an Aczel-Mendler bisimulation.
Note that defines a relation between the sets underlying the models X and X . So while Aczel-Mendler bisimulations are defined as topological spaces, the resulting notion of bisimilarity is simply a relation.
It follows from Proposition 3.7 that bisimilar states satisfy the same formulas. Furthermore, if C has pushouts then it follows from taking pushouts that Aczel-Mendler bisimilarity implies behavioural equivalence. If moreover T preserves weak pullbacks, the converse holds as well [Rut00].
However, we do not wish to make this assumption on topological spaces, since few functors seem to preserve weak pullbacks. For example, the Vietoris functor does not preserve weak pullbacks [BFV10, Corollary 4.3] and neither does the monotone functor from Definition 4.7. (To see the latter statement, consider the example given in Section 4 of [HK04] and equip the sets in use with the discrete topology.) Therefore we define Λ-bisimulations for Top-coalgebras as an alternative to Aczel-Mendler bisimulations. This notion is an adaptation of ideas in [BH17,GS13]. Under some conditions on Λ, Λ-bisimilarity coincides with behavioural equivalence.
In the next definition we need the concept of coherent pairs: If X and X are two sets and B ⊆ X × X is a relation, then a pair (a, a ) ∈ PX × PX is called B-coherent if B[a] ⊆ a and B −1 [a ] ⊆ a. For details and properties see Section 2 in [HKP09].
Definition 8.2. Let T be an endofunctor on C, Λ a geometric modal signature for T and X = (X, γ, V ) and X = (X , γ , V ) two geometric T-models. A Λ-bisimulation between X and X is a relation B ⊆ UX × UX such that for all (x, x ) ∈ B and p ∈ Φ we have and for all λ ∈ Λ and all tuples of B-coherent pairs of opens (a i , a i ) ∈ ΩX × ΩX : γ(x) ∈ λ X (a 1 , . . . , a n ) iff γ (x ) ∈ λ X (a 1 , . . . , a n ). (8.1) Two states are called Λ-bisimilar if there is a Λ-bisimulation linking them, notation: x Λ x . We give an alternative characterisation of (8.1) to elucidate the connection with [BH17,GHK20].
Remark 8.3. In the abstract setting of [GHK20], bisimulations are taken to be spans in the base category satisfying certain conditions. By contrast, in Definition 8.2 above we define bisimulations using relations between topological spaces, rather than spans in Top. In order to explain the connection between our approach and that in [BH17,GHK20], we equip a relation between topological spaces with the subspace topology.
As desired, Λ-bisimilar states always satisfy the same formulas.
Proposition 8.4. Let T be an endofunctor on C and Λ a geometric modal signature for T.
Proposition 8.5. Let T be an endofunctor on C and Λ a geometric modal signature for T.
The collection of Λ-bisimulations between two models enjoys the following interesting property.
Proposition 8.6. Let Λ be a geometric modal signature of a functor T : Top → Top and let X = (X, γ, V ) and X = (X , γ , V ) be two geometric T-models. The collection of Λ-bisimulations between X and X forms a complete lattice.
Proof. It is obvious that the collection of Λ-bisimulations is a poset. We will show that this collection is closed under taking arbitrary unions; the result then follows from the fact that any complete semilattice is also a complete lattice, see e.g. [BS81, Theorem 4.2].
Let J be some index set and for all j ∈ J let B j be Λ-bisimulations between X and X and set B = j∈J B j . We claim that B is a Λ-bisimulation.
Let (a i , a i ) be B-coherent pairs of opens. Suppose xBx and γ(x) ∈ λ X (a 1 , . . . , a n ). Then there is j ∈ J with xB j x hence x ∈ V (p) iff x ∈ V (p). As B j [a i ] ⊆ B[a i ] ⊆ a i and B −1 j [a i ] ⊆ B −1 [a i ] ⊆ a i , all B-coherent pairs (a i , a i ) are also B j -coherent. Since B j is a Λ-bisimulation we get γ (x ) ∈ λ X (a 1 , . . . , a n ). The converse direction is proven symmetrically.
We know by now that Λ-bisimilarity implies modal equivalence. Furthermore, we have seen in Theorem 7.15 that modal equivalence coincides with behavioural equivalence whenever T is an endofunctor on Top which preserves sobriety and Λ is a Scott-continuous characteristic geometric modal signature. In order to prove a converse, i.e. that behavioural equivalence implies Λ-bisimilarity, we need to assume that the geometric modal signature is extendable.
Recall that two elements x, x in two models are behaviourally equivalent in Mod(T), notation: x Mod(T) x , if there exist morphisms f, f in Mod(T) such that f (x) = f (x ).
Proposition 8.7. Let T be an endofunctor on C and Λ a monotone extendable geometric modal signature for T. Let X = (X, γ, V ) and X = (X , γ , V ) be two geometric T-models. is a Λ-bisimulation linking x and x . Clearly xBx . It follows from Proposition 3.7 that u and u satisfy precisely the same formulas whenever (u, u ) ∈ B. Suppose λ ∈ Λ is n-ary and for 1 ≤ i ≤ n let (a i , a i ) be a B-coherent pair of opens. Suppose uBu and γ(u) ∈ λ X (a 1 , . . . , a n ). We will show that γ (u ) ∈ λ X (a 1 , . . . , a n ). The converse direction is similar.
(Coherence of (a i , a i )) This proves the proposition.
Remark 8.8. If C = KHaus in the proposition above, then Proposition 3.15 allows us to drop the assumption that Λ be extendable.
Let T be an endofunctor on Top and let Λ be a geometric modal signature for T. The following diagram summarises the results from Propositions 8.4 and 8.7 and Theorem 7.15. The arrows indicate that one form of equivalence implies the other. Here (1) holds if T preserves weak pullbacks, (2) is true when Λ is Scott-continuous and characteristic and T preserves sobriety (cf. Theorem 7.15), and (3) holds when Λ is monotone extendable. Note that the converse of (2) always holds, because morphisms preserve truth (Proposition 3.7). The implications in the diagram hold for endofunctors on Sob as well (use Remark 7.16). Moreover, with some extra effort it can be made to work for endofunctors on KSob as well. In order to achieve this, we have to redo the proof for the bi-implication between modal equivalence and behavioural equivalence. This is the content of the following theorem.
Proof. If x and x are behaviourally equivalent then they are modally equivalent by Proposition 3.7. The converse direction can be proved using similar reasoning as in Section 7. The major difference is the following: We define an equivalence relation ≡ 2 on GML(Φ, Λ) by ϕ ≡ 2 ψ iff ϕ X = ψ X and ϕ X = ψ X . (Note that X and X are now fixed.) That is, ϕ ≡ 2 ψ iff ϕ and ψ are satisfied by precisely the same states in X and X (compare Definition 7.2). The frame E 2 := GML(Φ, Λ)/≡ 2 can then be shown to be a compact frame and hence Z 2 := ptE 2 is a compact sober space. The remainder of the proof is analogous to the proof of Theorem 7.15. A detailed proof can be found in [Gro18, Theorem 3.34].
We summarise the results for Top and two of its full subcategories: Theorem 8.10. Let T be a sobriety-preserving endofunctor on Top or an endofunctor on Sob or KSob, and Λ a monotone extendable Scott-continuous characteristic geometric modal signature for T. If x and x are two states in two geometric T-models, then x Λ x iff x ≡ Λ x iff x Mod(T) x .

Conclusion
We have started building a framework for coalgebraic geometric logic and we have investigated examples of concrete functors. There are still many unanswered and interesting questions. We outline possible directions for further research.
Modal equivalence versus behavioural equivalence: From Theorem 8.10 we know that modal equivalence and behavioural equivalence coincide in Mod(T) if T is an endofunctor on KSob, Sob or an endofunctor on Top which preserves sobriety. A natural question is whether the same holds when T is an endofunctor on KHaus. When does a lifted functor restrict to KHaus?: We know of two examples, namely the powerset functor with the box and diamond lifting, and the monotone functor on Set with the box and diamond lifting, where the lifted functor on Top restricts to KHaus. It would be interesting to investigate whether there are explicit conditions guaranteeing that the KKP lift of a functor restricts to KHaus. These conditions could be either for the Set-functor one starts with, or the collection of predicate liftings for this functor, or both. Modalities and finite observations: Geometric logic is generally introduced as the logic of finite observations, and this explains the choice of connectives (∧, and, in the first-order version, ∃). We would like to understand to which degree modalities can safely be added to the base language, without violating the (semantic) intuition of finite observability. Clearly there is a connection with the requirement of Scott-continuity (preservation of directed joins), and we would like to make this connection precise, specifically in the topological setting. Connection to geometric predicate logic: The extension of geometric logic with predicates is discussed in e.g. [Joh02, Chapter D1] and [Vic07]. In this setting toposes replace the rôle of frames. This raises the question of how modal geometric logic and geometric predicate logics relate. For example, can we find an analogue of the Van Benthem characterisation theorem, characterising modal geometric logic as a bisimulation-invariant fragment of geometric predicate logic? Order-enriched category theory: The fact that all examples in this paper use open predicate liftings that are monotone suggests that we could also work in the setting of order-enriched category theory. The richer structure of order-enriched category theory was successfully used in the study of positive coalgebraic logics [KKV12,BKV15].