Presenting de Groot duality of stably compact spaces
aa r X i v : . [ c s . L O ] M a r Presenting de Groot duality of stably compactspaces
Tatsuji Kawai
Japan Advanced Institute of Science and Technology [email protected]
Abstract
We give a constructive account of the de Groot duality of stably com-pact spaces in the setting of strong proximity lattice, a point-free repre-sentation of a stably compact space. To this end, we introduce a notion ofstrong continuous entailment relation, which can be thought of as a pre-sentation of a strong proximity lattice by generators and relations. Thenew notion allows us to identify de Groot duals of stably compact spacesby analysing the duals of their presentations. We carry out a numberof constructions on strong proximity lattices using strong continuous en-tailment relations and study their de Groot duals. The examples includevarious powerlocales, patch topology, and the space of valuations. Theseexamples illustrate the simplicity of our approach by which we can reasonabout the de Groot duality of stably compact spaces.
Keywords: stably compact space; de Groot duality; strong proximity lat-tice; entailment relation; locale
MSC2010:
De Groot duality of stably compact spaces induces a family of dualities on var-ious powerdomain constructions. In the point-free setting, Vickers [24] showedthat the de Groot dual of the upper powerlocale of a stably compact locale isthe lower powerlocale of its dual. In the point-set setting, Goubault-Larrecq [8]showed that the dual of the Plotkin powerdomain of a stably compact space isthe Plotkin powerdomain of its dual; the same holds for the probabilistic pow-erdomain.In this paper, we give an alternative account of these results in the settingof strong proximity lattice [16], the Karoubi envelop of the category of spectrallocales and locale maps. Strong proximity lattices have a structural dualitywhich reflects the de Groot duality of stably compact spaces in a simple way(see Section 6). Moreover, a strong proximity lattice is just a distributive latticewith an extra structure, so it does not require infinitary joins inherent in the Stably compact locales are also known as stably locally compact locales [10, Chapter VII,Section 4.6] or arithmetic lattices [16]. The upper and lower powerlocales of a locale correspondto the Smyth and Hoare powerdomains of the corresponding space, respectively. strong continuous entailment relation , which can bethought of as a presentation of a strong proximity lattice by generators andrelations by Scott’s entailment relations [18]. The notion is a variant of thatof an entailment relation with the interpolation property due to Coquand andZhang [4]. Here, the structure due to Coquand and Zhang is strengthened sothat it has an intrinsic duality which reflects the de Groot duality of stablycompact spaces. The resulting structure, strong continuous entailment rela-tion, allows us to identify de Groot duals of stably compact locales presentedby generators and relations by analysing the duals of their presentations. Weillustrate the ease with which we can reason about de Groot duality by carryingout a number of constructions on strong proximity lattices using strong con-tinuous entailment relations. The examples include various powerlocales, patchtopology, and the space of valuations.Throughout this paper, we work in the point-free setting, identifying sta-bly compact spaces with their point-free counterparts, stably compact locales.This allows us to work constructively in the predicative sense as manifested inAczel’s constructive set theory [1]. However, the point of this work is not theconstructively but the simplicity of our approach by which we can analyse deGroot duals of various constructions on stably compact spaces.
Related works
Besides the work of Coquand and Zhang [4] and that of Jung and S¨underhauf[16] mentioned above, many authors studied stably compact spaces from thepoint-free perspective (see Escard´o [6]; Jung, Kegelmann, and Moshier [15];Vickers [24]). Among them, the notion of entailment system by Vickers [24],which develops the idea of Jung et al. [15], is particularly related to the notionof strong continuous entailment relation. These structures are equipped withstructural dualities which reflects the de Groot duality of stably compact spaces.The essential difference between our approach and that of Vickers is thefollowing: the theory of strong continuous entailment relation is built on the factthat stably compact locales are the retracts of spectral locales and locale maps.Hence, the theory of strong continuous entailment relation essentially deals withthe objects of the Karoubi envelop of the latter category. On the other hand,the theory of entailment system deals with the objects of the Karoubi envelop ofthe category of spectral locales and preframe homomorphisms (see Section 4). In this view, the former theory treats stably compact locales as locales while thelatter theory treats them as preframes; this has to do with the simplicity of thetreatment of joins in the geometric presentation of a locale represented by theformer structure (see Section 5). Thus, if one is interested in the localic structureof stably compact locales rather than that of preframe, it would be more naturalto work with strong continuous entailment relations. In particular, this couldpotentially facilitate some of the localic constructions on stably compact localesinvolving finite joins, such as patch topology and the Vietoris powerlocale, in thesetting of strong continuous entailment relations, although proper comparison More specifically, it suffices to consider only free frames rather than spectral locales.
2s needed.Apart from the point-free approaches mentioned above, we are motivated bythe corresponding results for stably compact spaces due to Goubault-Larrecq [8].To derive these results, he used the notion of A -valuation due to Heckmann [9].It would be interesting to know if there is any connection between our approachand the A -valuation approach. However, since we prefer to work constructivelyin the point-free setting, we do not compare the two approaches in this paper. Organisation
In Section 2, we fix some basic notions on locales. In Section 3, we introduce thenotion of proximity lattice as the Karoubi envelop of the category of spectrallocales and preframe homomorphisms. In Section 4, we give an alternative rep-resentation for proximity lattices, called continuous entailment relation, basedon the notion of entailment system. In Section 5, we strengthen the notion ofproximity lattice to strong proximity lattice by looking into the Karoubi en-velop of the category of spectral locales and locale maps. We also introduce thecorresponding notion of strong continuous entailment relation. In Section 6, weformulate the duality of proximity lattices and continuous entailment relations,and show that these dualities reflect the de Groot duality of stably compactlocales. In Section 7, we study the de Groot duals of various constructions onstably compact locales by exploiting the correspondence between strong prox-imity lattices and strong continuous entailment relations. A frame is a poset ( X, ∧ , W ) with finite meets ∧ and joins W for all subsets of X where finite meets distribute over all joins. A homomorphism from a frame X to a frame Y is a function f : X → Y which preserves finite meets and alljoins. The category of locales is the opposite of the category of frames and framehomomorphisms. We write Ω( X ) for the frame corresponding to a locale X , butwe often regard a frame as a locale and vice versa without change of notation.Given a set S of generators , a geometric theory over S is a set of axioms ofthe form V A ⊢ W i ∈ I V B i , where A is a finite subset of S and ( B i ) i ∈ I is a set-indexed family of finite subsets of S . Single conjunctions and single disjunctionsare identified with elements of S . We use the following abbreviations: ⊤ ≡ V ∅ , ⊥ ≡ _ ∅ , W B ≡ _ b ∈ B { b } , _ i ∈ I b i ≡ _ i ∈ I { b i } . An interpretation of a geometric theory T (over S ) in a locale X is a function f : S → Ω( X ) such that V a ∈ A f ( a ) ≤ X W i ∈ I V b ∈ B i f ( b ) for each axiom V A ⊢ W i ∈ I V B i of T . There is a locale Sp( T ) with a universal interpretation i T : S → Ω(Sp( T )): for any interpretation f : S → Ω( X ) of T , there exists a unique framehomomorphism f : Ω(Sp( T )) → Ω( X ) such that f ◦ i T = f . In this case, Sp( T )is called the locale (or frame) presented by T . A model of a geometric theory T over S is a subset α ⊆ S such that A ⊆ α → ∃ i ∈ I ( B i ⊆ α ) for each axiom Here, finite means finitely enumerable . A set A is finitely enumerable if there is a surjectivefunction f : { , . . . , n − } → A for some natural number n . Finitely enumerable sets are alsoknown as Kuratowski finite sets; see e.g., Johnstone [12, D5.4]. A ⊢ W i ∈ I V B i of T . If the models of T form a distinguished class of objects,we call Sp( T ) the locale whose models are members of that class . We recall the construction of Karoubi envelop (cf. [7, Chapter 2, Exercise B]).
Definition 3.1. An idempotent in a category C is a morphism f : A → A such that f ◦ f = f . The Karoubi envelop (or splitting of idempotents ) of C is a category Split ( C ) where objects are idempotents in C and morphisms h : ( f : A → A ) → ( g : B → B ) are morphisms h : A → B in C such that g ◦ h = h = h ◦ f .One can show that if C is a full subcategory of D where every idempotentsplits in D and every object in D is a retract of an object of C , then D isequivalent to Split ( C ).It is well known that stably compact locales are exactly the retracts of spec-tral locales, whose frames are the ideal completions of distributive lattices [10,Chapter VII, Theorem 4.6]. Less well known is the fact that stably compact lo-cales are exactly the preframe retracts of spectral locales [24, Section 3] so thatthe category of stably compact locales and preframe homomorphisms can becharacterised as the Karoubi envelop of the category of spectral locales and pre-frame homomorphisms. Here, a preframe is a poset with directed joins (joinsof directed subsets) and finite meets which distribute over directed joins. A preframe homomorphism between preframes is a function which preserves fi-nite meets and directed joins. The latter fact leads to the notion of proximitylattice [16] by considering a finitary description of the dual of the category ofspectral locales and preframe homomorphisms. Definition 3.2.
Let S and S ′ be distributive lattices. A proximity relation from S to S ′ is a relation r ⊆ S × S ′ such that(ProxI) r − b def = { a ∈ S | a r b } is an ideal of S for all b ∈ S ′ ,(ProxF) ra def = { b ∈ S ′ | a r b } is a filter of S ′ for all a ∈ S .Here, an ideal is a downward closed subset of S closed under finite joints. A filter is an upward closed subset of S closed under finite meets.Let DLat
Prox be the category of distributive lattices and proximity relations:the identity on a distributive lattice S is the order ≤ on S ; the composition ofproximity relations is the relational composition. We differ from the standard distinction between “interpretation” and “model” whereinthe former interprets the language (here generators) and the latter in addition satisfies theaxioms of the theory over the language. In this paper, in contrast, both “interpretation”and “model” mean an interpretation which satisfies the axioms: the latter keeps the usualmeaning of a model in the lattice of truth values, the powerset of a singleton {∗} , whereas theformer means an axiom preserving interpretation in a frame more general than that of thetruth values. In locale theory, this can be expressed as the distinction between generalisedpoints and global points (see Vickers [23]). To be precise, the results of Vickers [24] are stronger; the stably compact locales arepreframe retracts of free frames (cf. footnote 2). ideal completion of a distributive lattice S , denoted by Idl( S ), is theframe of ideals of S : the directed join of ideals is their union; finite joins andfinite meets are defined by0 def = { } , I ∨ J def = [ a ∈ I,b ∈ J ↓ ( a ∨ b ) , def = S, I ∧ J def = { a ∧ b | a ∈ I, b ∈ J } , where ↓ a def = { b ∈ S | b ≤ a } , the principal ideal generated by a . Every ideal I is a directed join of principal ideals: I = _ a ∈ I ↓ a. (3.1) Proposition 3.3.
For any proximity relation r : S → S ′ , there exists a uniquepreframe homomorphism f : Idl( S ′ ) → Idl( S ) such that f ( ↓ b ) = r − b for all b ∈ S ′ . Moreover, this bijection preserves identities and compositions of proximityrelations.Proof. It is easy to see that a proximity relation r : S → S ′ uniquely extends toa meet-semilattice homomorphism f r : S ′ → Idl( S ) defined by f r ( b ) def = r − b. (3.2)By Vickers [20, Theorem 9.1.5 (i) (iv)], the function f r extends uniquely to apreframe homomorphism f : Idl( S ′ ) → Idl( S ) by f ( I ) def = _ b ∈ I f r ( b ) = r − I. (3.3)The second statement follows from the first and (3.1).Let Spectral
Pre be the category of spectral locales and preframe homomor-phisms: objects of
Spectral
Pre are distributive lattices and morphisms are pre-frame homomorphisms between the ideal completions.
Theorem 3.4.
The category
DLat
Prox is dually equivalent to
Spectral
Pre .Proof.
Immediate from Proposition 3.3.Since
Split ( Spectral
Pre ) is equivalent to the category of stably compact lo-cales and preframe homomorphisms,
Split ( DLat
Prox ) is dually equivalent to thelatter category. The objects and morphisms of
Split ( DLat
Prox ) are called prox-imity lattices and proximity relations respectively (cf. Jung and S¨underhauf [16]and de Gool [19]). In what follows, we write
ProxLat for
Split ( DLat
Prox ). Notation . We write ( S, ≺ ) for a proximity lattice, where S is a distributivelattice and ≺ is an idempotent proximity relation on S . We write r : ( S, ≺ ) → ( S ′ , ≺ ′ ) for a proximity relation from ( S, ≺ ) to ( S ′ , ≺ ′ ), i.e., a proximity relation r : S → S ′ between the underlying distributive lattices such that ≺ ′ ◦ r = r ◦≺ = r . 5ach proximity lattice ( S, ≺ ) represents a stably compact locale whose frameis the collection RIdl( S ) of rounded ideals of S ordered by inclusion [16, Theo-rem 11]. Here, an ideal I ⊆ S is rounded if a ∈ I ↔ ∃ b ≻ a ( b ∈ I ). Directedjoins and finite meets in RIdl( S ) are calculated as in Idl( S ); on the other hand,finite joins in RIdl( S ) are defined by0 def = ↓ ≺ , I ∨ J def = [ a ∈ I,b ∈ J ↓ ≺ ( a ∨ b ) , where ↓ ≺ a def = { b ∈ S | b ≺ a } . Every rounded ideal I is a directed join of itsmembers: I = W a ∈ I ↓ ≺ a . Let Spec( S ) denote the locale whose frame is RIdl( S ). We give an alternative presentation of a proximity lattice in terms of Vickers’sentailment system [24]. We need some constructions on finite subsets. For anyset S , let Fin ( S ) denote the set of finite subsets of S . For each U ∈
Fin ( Fin ( S )),define U ∗ ∈ Fin ( Fin ( S )) inductively by ∅ ∗ def = {∅} , ( U ∪ { A } ) ∗ def = (cid:8) B ∪ C | B ∈ U ∗ & C ∈ Fin + ( A ) (cid:9) , where Fin + ( A ) denotes the set of inhabited finite subsets of A . Writing DL( S )for the free distributive lattice over S , the mapping U 7→ U ∗ transforms adisjunction of conjunctions of generators in DL( S ) to the conjunction of dis-junctions of generators, or the other way around (cf. Vickers [23, Theorem 8.7]). Definition 4.1.
Let
S, S ′ , S ′′ be sets and r, s be relations r ⊆ Fin ( S ) × Fin ( S ′ )and s ⊆ Fin ( S ′ ) × Fin ( S ′′ ).1. The relation r is said to be upper if A r B → A ∪ A ′ r B ∪ B ′ for all A, A ′ ∈ Fin ( S ) and B, B ′ ∈ Fin ( S ′ ).2. The cut composition s · r ⊆ Fin ( S ) × Fin ( S ′′ ) is defined by A s · r C def ⇐⇒ ∃V ∈ Fin ( Fin ( S ′ )) [ ∀ B ′ ∈ V ∗ ( A r B ′ ) & ∀ B ∈ V ( B s C )] . Definition 4.2 (Vickers [24, Section 6]) . An entailment system is a pair ( S, ≪ )where S is a set and ≪ is an upper relation on Fin ( S ) such that ≪ · ≪ = ≪ .A Karoubi morphism r : ( S, ≪ ) → ( S ′ , ≪ ′ ) of entailment systems is an upperrelation r ⊆ Fin ( S ) × Fin ( S ′ ) such that ≪ ′ · r = r = r · ≪ . Let
Entsys be the category of entailment systems and Karoubi morphismsbetween them: the identity on ( S, ≪ ) is ≪ and the composition of morphismsis the cut composition; see Vickers [24, Section 5 and Section 6] for the details. The notion of rounded ideal makes sense in the more general setting of information sys-tems [21]. In the notation of Vickers [24, Section 4], the set U ∗ is equal to { Im γ | γ ∈ Ch( U ) } , where Ch( U ) is the set of choices of U and Im γ is the image of a choice γ ; see Definition 12and Definition 13, and the proof of Proposition 14 in [24]. In [24], the cut composition of r and s is denoted by r † s using the forward notation forthe relational composition. See in particular [24, Lemma 30].
6s shown in [24],
Entsys is dually equivalent to the category of stably compactlocales and preframe homomorphisms. An entailment system ( S, ≪ ) representsa stably compact locale (or frame) which is a preframe retract of the free frameover S , or equivalently, the spectral locale determined by the free distributivelattice over S . The relation ≪ then represents the retraction.We now focus on the full subcategory of Entsys consisting of reflexive entail-ment systems [24, Section 6.1], also known as entailment relations [2, 18].
Definition 4.3. An entailment relation on a set S is a relation ⊢ on Fin ( S )such that a ⊢ a (R) A ⊢ BA ′ , A ⊢ B, B ′ (M) A ⊢ B, a a, A ⊢ BA ⊢ B (T)for all a ∈ S and A, A ′ , B, B ′ ∈ Fin ( S ), where “ a ” denotes { a } and “ , ” denotesthe union.Each entailment relation ( S, ⊢ ) is an entailment system; we write EntRel forthe full subcategory of
Entsys consisting of entailment relations. As noted byVickers [24, Section 6.1], the category
EntRel is dually equivalent to
Spectral
Pre .Hence
EntRel is equivalent to
DLat
Prox , which we now elaborate.
Definition 4.4.
Let
S, S ′ be sets and r be a relation r ⊆ Fin ( S ) × Fin ( S ′ ).Define a relation e r ⊆ Fin ( Fin ( S )) × Fin ( Fin ( S ′ )) by U e r V def ⇐⇒ ∀ A ∈ U∀ B ∈ V ∗ ( A r B ) . Lemma 4.5.
Let
S, S ′ , S ′′ be sets and r, s be upper relations r ⊆ Fin ( S ) × Fin ( S ′ ) and s ⊆ Fin ( S ′ ) × Fin ( S ′′ ) . Then g s · r = e s ◦ e r . Proof.
See Vickers [24, Proposition 22 and Proposition 31].An entailment relation ( S, ⊢ ) determines a distributive lattice DL( S, ⊢ ) whoseunderlying set is Fin ( Fin ( S )) equipped with an equality defined by U = ⊢ V def ⇐⇒ U e ⊢ V & V e ⊢ U . The lattice structure is defined as follows:0 def = ∅ , U ∨ V def = U ∪ V , def = {∅} , U ∧ V def = { A ∪ B | A ∈ U & B ∈ V} . It is easy to check that joins and meets are well-defined with respect to = ⊢ andthat the order determined by DL( S, ⊢ ) is e ⊢ .If r : ( S, ⊢ ) → ( S ′ , ⊢ ′ ) is a Karoubi morphism, then e ⊢ ′ ◦ e r = ] ⊢ ′ · r = e r = g r · ⊢ = e r ◦ e ⊢ by Lemma 4.5. Then, it easy to see that e r is a proximity relation from DL( S, ⊢ )to DL( S ′ , ⊢ ′ ). In Vickers’s notation [24, Proposition 25], we have U e r V ⇐⇒ U r V ∗ . roposition 4.6. The assignment r e r determines a functor E : EntRel → DLat
Prox , which establishes equivalence of
EntRel and
DLat
Prox .Proof.
Functoriality of E follows from the construction of DL( S, ⊢ ) and Lemma 4.5.Moreover, E is faithful because A r B ↔ { A } e r { B } ∗ . To see that E is full, forany proximity relation r : DL( S, ⊢ ) → DL( S ′ , ⊢ ′ ), define ˆ r ⊆ Fin ( S ) × Fin ( S ′ ) by A ˆ r B def ⇐⇒ { A } r { B } ∗ . Clearly, we have U e ˆ r V ↔ U r V . Moreover A ( ⊢ ′ · ˆ r ) B ⇐⇒ { A } ( ] ⊢ ′ · ˆ r ) { B } ∗ ⇐⇒ { A } ( e ⊢ ′ ◦ e ˆ r ) { B } ∗ ⇐⇒ { A } ( e ⊢ ′ ◦ r ) { B } ∗ ⇐⇒ { A } r { B } ∗ ⇐⇒ A ˆ r B. Similarly ˆ r · ⊢ = ˆ r , so ˆ r is a Karoubi morphism. Thus E is full. To see that E is essentially surjective, for any distributive lattice ( S, , ∨ , , ∧ ), define anentailment relation ( S, ⊢ ) by A ⊢ B def ⇐⇒ V A ≤ W B. (4.1)Then, define relations r ⊆ S × Fin ( Fin ( S )) and s ⊆ Fin ( Fin ( S )) × S by a r U def ⇐⇒ a ≤ _ A ∈U V A, U s a def ⇐⇒ _ A ∈U V A ≤ a. It is straightforward to show that r and s are proximity relations between S and DL( S, ⊢ ) and inverse to each other. Remark . By the standard construction (Mac Lane [17, Chapter IV, Sec-tion 4, Theorem 1]), we can define a quasi-inverse D : DLat
Prox → EntRel of E : EntRel → DLat
Prox by D ( S, , ∨ , , ∧ ) def = ( S, ⊢ ) , where ⊢ is defined by (4.1). The functor D sends a proximity relation r : S → S ′ to a Karoubi morphism D ( r ) : D ( S ) → D ( S ′ ) defined by A D ( r ) B def ⇐⇒ V A r W B. By Proposition 4.6, the category
Split ( EntRel ) is equivalent to
ProxLat , andhence dually equivalent to the category of stably compact locales and preframehomomorphisms. Unfolding the definition of
Split ( EntRel ), we have the follow-ing characterisations of its objects and morphisms (Definition 4.8 and 4.9).
Definition 4.8. A continuous entailment relation is an entailment relation( S, ⊢ ) equipped with an idempotent Karoubi endomorphism ≪ ⊆ Fin ( S ) × Fin ( S )on ( S, ⊢ ). We write ( S, ⊢ , ≪ ) for a continuous entailment relation.8ote that each continuous entailment relation ( S, ⊢ , ≪ ) has an associatedentailment system ( S, ≪ ). Definition 4.9.
Let ( S, ⊢ , ≪ ) and ( S ′ , ⊢ ′ , ≪ ′ ) be continuous entailment rela-tions. A proximity map from ( S, ⊢ , ≪ ) to ( S ′ , ⊢ ′ , ≪ ′ ) is a Karoubi morphismbetween entailment systems ( S, ≪ ) and ( S ′ , ≪ ′ ).In the following, we write ContEnt for
Split ( EntRel ). Since an entailmentsystem ( S, ≪ ) can be identified with a continuous entailment relation ( S, ≬ , ≪ )where A ≬ B def ⇐⇒ ( ∃ a ∈ S ) a ∈ A ∩ B, the category Entsys can be regarded as a full subcategory of
ContEnt . Onthe other hand, each continuous entailment relation ( S, ⊢ , ≪ ) is isomorphic to( S, ≬ , ≪ ) with ≪ being the isomorphism. Thus, we have the following. Proposition 4.10.
The categories
ContEnt and
Entsys are equivalent.
Proximity lattices and continuous entailment relations have nice structural du-ality, which will be elaborated in Section 6. However, stably compact localesrepresented by these structures do not seem to admit simple geometric presen-tations. In the case of a proximity lattice ( S, ≺ ), for example, Spec( S ) can bepresented by a geometric theory over S with the following axioms: a ⊢ ⊥ (if a ≺ , c ⊢ a ∨ b (if c ≺ a ′ ∨ b ′ , a ′ ≺ a, b ′ ≺ b ) , ⊤ ⊢ , a ∧ b ⊢ ( a ∧ b ) , a ⊢ b (if a ≤ b ) ,a ⊢ b (if a ≺ b ) , a ⊢ _ b ≺ a b. (5.1)In the case of continuous entailment relations (or entailment systems), the pre-sentations of the locales represented by these structures are more elaborate;see Vickers [24, Corollary 44]. To obtain a simpler presentation of Spec( S ), westrengthen the notion of proximity lattice to strong proximity lattice [16], whichcan be obtained from the well-known fact that stably compact locales are theretracts of the spectral locales. We start from a finitary description of locale maps between spectral locales.
Definition 5.1.
Let S and S ′ be distributive lattices. A proximity relation r : S → S ′ is said to be join-preserving if(Prox0) a r ′ → a = 0,(Prox ∨ ) a r ( b ∨ ′ c ) → ∃ b ′ , c ′ ∈ S ( a ≤ b ′ ∨ c ′ & b ′ r b & c ′ r c ).Proposition 3.3 restricts to join-preserving proximity relations and framehomomorphisms. 9 roposition 5.2. For any join-preserving proximity relation r : S → S ′ betweendistributive lattices, there exists a unique frame homomorphism f : Idl( S ′ ) → Idl( S ) such that f ( ↓ b ) = r − b for all b ∈ S ′ .Proof. The proof is similar to Proposition 3.3. One only has to note that ajoin-preserving proximity relation r : S → S ′ uniquely extends to a lattice ho-momorphism f r : S ′ → Idl( S ) defined as (3.2). The desired conclusion thenfollows from Vickers [20, Theorem 9.1.5 (iii) (iv)].Let DLat
JProx be the subcategory of
DLat
Prox where morphisms are join-preserving proximity relations. Let
Spectral be the category of spectral localesand locale maps. The following is immediate from Proposition 5.2.
Theorem 5.3.
The category
DLat
JProx is equivalent to
Spectral . Since
Split ( Spectral ) is equivalent to the category of stably compact localesand locale maps,
Split ( DLat
JProx ) is equivalent to the latter category. Theobjects and morphisms of
Split ( DLat
JProx ) are called ∨ -strong proximity lattices and joint-preserving proximity relations respectively (cf. van Gool [19, Definition1.2 and Definition 1.9]). We write JSProxLat for
Split ( DLat
JProx ) and adopt thesimilar notation as in Notation 3.5 for ∨ -strong proximity lattices and join-preserving proximity relations between them.As in the case of proximity lattices, each ∨ -strong proximity lattice ( S, ≺ )represents a stably compact locale by the collection RIdl( S ) of rounded ideals.In this case, the locale Spec( S ) can be presented by a simpler geometric theorythan (5.1), as we now show.Let X and Y be locales. A function f : Ω( X ) → Ω( Y ) is Scott continu-ous if it preserves directed joins. A Scott continuous function is a suplatticehomomorphism if it preserves finite joins (and hence all joins).
Definition 5.4.
Let ( S, ≺ ) be a ∨ -strong proximity lattice and X be a locale. A dcpo interpretation of ( S, ≺ ) in X is an order preserving function f : S → Ω( X )such that f ( a ) = W b ≺ a f ( b ) . A dcpo interpretation f : S → Ω( X ) is called a suplattice (preframe) interpretation if it preserves finite joins (resp. finite meets); f is called an interpretation if it preserves both finite joins and finite meets.Any ∨ -strong proximity lattice ( S, ≺ ) admits an interpretation i S : S → RIdl( S ) defined by i S ( a ) def = ↓ ≺ a. (5.2) Proposition 5.5.
Let ( S, ≺ ) be a ∨ -strong proximity lattice and X be a lo-cale. For any (dcpo, suplattice, preframe) interpretation f : S → Ω( X ) , thereexists a unique frame homomorphism (resp. Scott continuous function, suplat-tice homomorphism, preframe homomorphism) f : RIdl( S ) → Ω( X ) such that f ◦ i S = f .Proof. See Vickers [20, Theorem 9.1.5] where an analogous fact for spectrallocales is presented. The unique extension of f is defined by f ( I ) def = W a ∈ I f ( a ).Since f is a dcpo interpretation, we have f ◦ i S = f . Remark . For dcpo and preframe interpretations, Proposition 5.5 holds forproximity lattices as well. In this case, however, the function i S : S → RIdl( S )does not necessarily preserve finite joins, so it is only a preframe interpretation.10 orollary 5.7. For any ∨ -strong proximity lattice ( S, ≺ ) , the locale Spec( S ) ispresented by a geometric theory over S with the following axioms: ⊢ ⊥ , ( a ∨ b ) ⊢ a ∨ b, ⊤ ⊢ , a ∧ b ⊢ ( a ∧ b ) , a ⊢ b ( if a ≤ b ) ,a ⊢ b ( if a ≺ b ) , a ⊢ _ b ≺ a b. Proof.
Immediate from the frame version of Proposition 5.5.Note that models of the geometric theory in Corollary 5.7 are rounded primefilters of ( S, ≺ ), i.e., those prime filters F on S such that a ∈ F ↔ ∃ b ≺ a ( b ∈ F ).Let JSProxLat
Prox be the full subcategory of
ProxLat consisting of ∨ -strongproximity lattices. Theorem 5.8.
The category
JSProxLat
Prox is equivalent to
ProxLat .Proof.
Given a proximity lattice ( S, ≺ ), define a preorder ≤ ∨ on Fin ( S ) by A ≤ ∨ B def ⇐⇒ ∀ C ≺ L A ∃ D ≺ L B ( W C ≺ W D ) , where A ≺ L B def ⇐⇒ ∀ a ∈ A ∃ b ∈ B ( a ≺ b ) . (5.3)Let S ∨ be the set Fin ( S ) equipped with the equality = ∨ determined by ≤ ∨ , i.e.,= ∨ def = ≤ ∨ ∩ ≥ ∨ , and define a lattice structure ( S ∨ , ∨ , ∨ ∨ , ∨ , ∧ ∨ ) by0 ∨ def = ∅ , A ∨ ∨ B def = A ∪ B, ∨ def = { } , A ∧ ∨ B def = { a ∧ b | a ∈ A, b ∈ B } . (5.4)It is straightforward to check that the above operations respect = ∨ and thatthe lattice is distributive. Next, define a relation ≺ ∨ on Fin ( S ) by A ≺ ∨ B def ⇐⇒ ∃ C ≺ L B ( A ≤ ∨ C ) . Again, it is straightforward to check that ≺ ∨ respects = ∨ . Since ≺ L is idem-potent and ≺ is a proximity relation, ≺ ∨ is an idempotent relation. We claimthat ( S ∨ , ≺ ∨ ) def = ( S ∨ , ∨ , ∨ ∨ , ∨ , ∧ ∨ , ≺ ∨ ) is a ∨ -strong proximity lattice. Forexample, to see that ( S ∨ , ≺ ∨ ) is ∨ -strong, suppose that A ≺ ∨ B ∪ C . Then,there exist B ′ ≺ L B and C ′ ≺ L C such that A ≤ ∨ B ′ ∨ ∨ C ′ . Since ≺ L ⊆ ≺ ∨ ,we have B ′ ≺ ∨ B and C ≺ ∨ C . Moreover, A ≺ ∨ ∨ clearly implies A ≤ ∨ ∨ .Define relations r ⊆ S × Fin ( S ) and s ⊆ Fin ( S ) × S by a r A def ⇐⇒ ∃ B ≺ L A ( a ≺ W B ) , A s a def ⇐⇒ A ≺ ∨ { a } . Then, r and s clearly respect = ∨ , so they are relations between S and S ∨ . Itis straightforward to show that r and s are proximity relations between ( S, ≺ )and ( S ∨ , ≺ ∨ ) and are inverse to each other. The left part of axiom ( a ∨ b ) ⊢ a ∨ b denotes the generator a ∨ b , while the right part isthe disjunction of two generators a and b . The similar remark applies to a ∧ b ⊢ ( a ∧ b ).
11y introducing ∨ -strong proximity lattices, we have obtained a simpler ge-ometric theory for Spec( S ) (cf. Corollary 5.7). This, however, comes at thecost of the structural duality of proximity lattices. Nevertheless, the notion of ∨ -strong proximity lattice is categorically equivalent to the one with a strongerself-dual structure than that of a proximity lattice. Definition 5.9 (Jung and S¨underhauf [16, Definition 18]) . A strong proximitylattice is a ∨ -strong proximity lattice ( S, ≺ ) satisfying(Prox1) 1 ≺ a → a = 1,(Prox ∧ ) b ∧ c ≺ a → ∃ b ′ ≻ b ∃ c ′ ≻ c ( b ′ ∧ c ′ ≤ a ).Let SProxLat and
SProxLat
Prox be the full subcategories of
JSProxLat and
JSProxLat
Prox , respectively, consisting of strong proximity lattices. In Sec-tion 5.3, we show that
SProxLat and
SProxLat
Prox are equivalent to the largercategories.
We characterise a full subcategory of
ContEnt , whose objects correspond tostrong proximity lattices. The notion introduced below is a modification of thatof entailment relation with the interpolation property by Coquand and Zhang [4],which satisfies only one direction of (5.5).
Definition 5.10. A strong continuous entailment relation is an entailment re-lation ( S, ⊢ ) equipped with an idempotent relation ≺ ⊆ S × S satisfying ∃ A ′ ∈ Fin ( S ) ( A ≺ U A ′ ⊢ B ) ⇐⇒ ∃ B ′ ∈ Fin ( S ) ( A ⊢ B ′ ≺ L B ) (5.5)for all A, B ∈ Fin ( S ) where ≺ L is defined as (5.3) and ≺ U is defined by A ≺ U B def ⇐⇒ ∀ b ∈ B ∃ a ∈ A ( a ≺ b ) . We write ( S, ⊢ , ≺ ) for a strong continuous entailment relation.Each strong continuous entailment relation ( S, ⊢ , ≺ ) represents a stably com-pact locale by a geometric theory T ( S, ⊢ , ≺ ) over S with the following axioms: V A ⊢ W B (if A ⊢ B ) , a ⊢ b (if a ≺ b ) , a ⊢ _ b ≺ a b. (5.6) Theorem 5.11 (Coquand and Zhang [4]) . For any strong continuous entail-ment relation ( S, ⊢ , ≺ ) , the locale presented by T ( S, ⊢ , ≺ ) is stably compact.Moreover, any stably compact locale can be presented in this way.Proof. See Coquand and Zhang [4, Theorem 1].In the following, we often identify a strong continuous entailment relation( S, ⊢ , ≺ ) with the theory T ( S, ⊢ , ≺ ).We relate strong continuous entailment relations to continuous entailmentrelations. Lemma 5.12.
Let
S, S ′ , S ′′ , S ′′′ be sets and r, s, t be relations r ⊆ Fin ( S ) × Fin ( S ′ ) , s ⊆ Fin ( S ′ ) × Fin ( S ′′ ) , and t ⊆ Fin ( S ′′ ) × Fin ( S ′′′ ) . . If s is upper and rA = { B ∈ Fin ( S ′ ) | A r B } is closed under finite joinsfor each A ∈ Fin ( S ) , then ( t · s ) ◦ r = t · ( s ◦ r ) .
2. If s is upper and t − D = { C ∈ Fin ( S ′′ ) | C t D } is closed under finite joinsfor each D ∈ Fin ( S ′′′ ) , then ( t ◦ s ) · r = t ◦ ( s · r ) . Proof.
1. Suppose that
A t · ( s ◦ r ) D . Then there exists V ∈
Fin ( Fin ( S ′′ )) suchthat ∀ C ′ ∈ V ∗ ( A s ◦ r C ′ ) and ∀ C ∈ V ( C t D ), so for each C ′ ∈ V ∗ there exists B C ′ ∈ Fin ( S ′ ) such that A r B C ′ and B C ′ s C ′ . Put B = S C ′ ∈V ∗ B C ′ . Then A r B and ∀ C ′ ∈ V ∗ ( B s C ′ ), and hence A ( t · s ) ◦ r D . The converse is easy.2. The proof is dual of 1.For each strong continuous entailment relation ( S, ⊢ , ≺ ), define a relation ≪ ⊢ on Fin ( S ) by ≪ ⊢ def = ⊢ ◦ ≺ U = ≺ L ◦ ⊢ . Proposition 5.13.
The structure ( S, ⊢ , ≪ ⊢ ) is a continuous entailment rela-tion.Proof. By item 1 of Lemma 5.12, we have ⊢ · ≪ ⊢ = ⊢ · ( ⊢ ◦ ≺ U ) = ( ⊢ · ⊢ ) ◦ ≺ U = ⊢ ◦ ≺ U = ≪ ⊢ . Similarly ≪ ⊢ · ⊢ = ≪ ⊢ by item 2 of Lemma 5.12. Then ≪ ⊢ · ≪ ⊢ = ( ≺ L ◦ ⊢ ) · ≪ ⊢ = ≺ L ◦ ( ⊢ · ≪ ⊢ ) = ≺ L ◦ ≪ ⊢ = ≪ ⊢ . Hence ≪ ⊢ is an idempotent Karoubi endomorphism on ( S, ⊢ ).Let SContEnt
Prox be a category where objects are strong continuous entail-ment relations and morphisms are proximity maps between the underlying con-tinuous entailment relations. By the assignment ( S, ⊢ , ≺ ) ( S, ⊢ , ≪ ⊢ ), we canidentify SContEnt
Prox with a full subcategory of
ContEnt .In what follows, we show that
SContEnt
Prox and
SProxLat
Prox are equivalent.First, note that the functor E : EntRel → DLat
Prox (cf. Proposition 4.6) inducesa functor F : ContEnt → ProxLat , which establishes equivalence of ContEnt and
ProxLat . The functor F sends eachcontinuous entailment relation ( S, ⊢ , ≪ ) to a proximity lattice (DL( S, ⊢ ) , f ≪ )and each proximity map r to a proximity relation e r . By Remark 4.7, F has aquasi-inverse G : ProxLat → ContEnt which sends each proximity lattice ( S, ≺ )to a continuous entailment relation ( S, ⊢ , ≪ ), where ⊢ is given by (4.1) and ≪ is defined by A ≪ B def ⇐⇒ V A ≺ W B. (5.7) Lemma 5.14.
For each strong continuous entailment relation ( S, ⊢ , ≺ ) , thestructure F ( S, ⊢ , ≪ ⊢ ) = (DL( S, ⊢ ) , g ≪ ⊢ ) is a strong proximity lattice.Proof. We must show that (DL( S, ⊢ ) , g ≪ ⊢ ) satisfies (Prox0), (Prox ∨ ), (Prox1),and (Prox ∧ ). As a demonstration, we show (Prox ∨ ). Suppose that U g ≪ ⊢ V ∨W .For each B ∈ V ∗ and C ∈ W ∗ , there exist B ′ ≺ L B and C ′ ≺ L C such that A ⊢ B ′ ∪ C ′ for all A ∈ U . Put V ′ = { B ′ | B ∈ V ∗ } ∗ and W ′ = { C ′ | C ∈ W ∗ } ∗ .Then, we have U e ⊢ V ′ ∨ W ′ , V ′ g ≪ ⊢ V and W ′ g ≪ ⊢ W .13 emma 5.15. For each strong proximity lattice ( S, ≺ ) , the structure ( S, ⊢ , ≺ ) ,where ⊢ is defined by (4.1) , is a strong continuous entailment relation. Moreoverthe relation ≪ ⊢ determined by ( S, ⊢ , ≺ ) is characterised by (5.7) .Proof. Straightforward.By Lemma 5.14 and Lemma 5.15, we have the following.
Theorem 5.16.
The functor F : ContEnt → ProxLat restricts to
SContEnt
Prox and
SProxLat
Prox , which establishes equivalence of the latter two categories.
The following corresponds to the notion of join-preserving proximity relation.
Definition 5.17.
Let ( S, ⊢ , ≺ ) and ( S ′ , ⊢ ′ , ≺ ′ ) be strong continuous entailmentrelations. A proximity map r : ( S, ⊢ , ≺ ) → ( S ′ , ⊢ ′ , ≺ ′ ) is join-preserving if(JP) A r B → ∃ U ∈
Fin ( Fin ( S )) (cid:16) { A } e ⊢ U & ∀ A ′ ∈ U∃ b ∈ B ( A ′ r { b } ) (cid:17) .Since join-preserving proximity maps are closed under composition (see theremark below Proposition 5.20), strong continuous entailment relations and join-preserving proximity maps form a subcategory SContEnt of SContEnt
Prox .Let = ( ∅ , ≬ , =) be a terminal object in SContEnt . The notion of join-preserving proximity map is consistent with the theory T ( S, ⊢ , ≺ ) in (5.6). Proposition 5.18.
For any strong continuous entailment relation ( S, ⊢ , ≺ ) ,there exists a bijective correspondence between the models of T ( S, ⊢ , ≺ ) and thejoin-preserving proximity maps from to S .Proof. A model α of T ( S, ⊢ , ≺ ) corresponds to a join-preserving proximity map r α : → S defined by ∅ r α A def ⇐⇒ α ≬ A. Conversely, a join-preserving proximity map r : → S corresponds to a model α r of T ( S, ⊢ , ≺ ) defined by α r def = { a ∈ S | ∅ r { a }} . It is straightforward to check that the above correspondence is bijective.We now restrict Theorem 5.16 to
SContEnt and
SProxLat . The followingshould be compared with Vickers [24, Theorem 42].
Lemma 5.19.
The condition (JP) is equivalent to the following: (JP0)
A r ∅ → A ⊢ ∅ , (JP ∨ ) A r B ∪ C → ∃ U , V ∈
Fin ( Fin ( S )) (cid:16) { A } e ⊢ U ∪ V & U e r { B } ∗ & V e r { C } ∗ (cid:17) .Proof. Assume (JP). For (JP0), if
A r ∅ , then we must have { A } e ⊢ ∅ so A ⊢ ∅ .For (JP ∨ ), suppose that A r B ∪ C . By (JP), there exist U , V ∈
Fin ( Fin ( S )) suchthat { A } e ⊢ U ∪ V , and ∀ B ′ ∈ U∃ b ∈ B ( B ′ r { b } ) and ∀ C ′ ∈ V∃ c ∈ C ( C ′ r { c } ).Then, ∀ B ′ ∈ U ( B ′ r B ) and ∀ C ′ ∈ V ( C ′ r C ).Conversely, assume (JP0) and (JP ∨ ). We show (JP) by induction on the sizeof B . The base case B = ∅ follows from (JP0). For the inductive case, suppose14hat A r B ∪ { b } . By (JP ∨ ), there exist U , V ∈
Fin ( Fin ( S )) such that A e ⊢ U ∪ V , U e r { B } ∗ , and V e r ∗ . By induction hypothesis, for each C ∈ U there exists U C ∈ Fin ( Fin ( S )) such that C e ⊢ U C and ∀ B ′ ∈ U C ∃ b ′ ∈ B ( B ′ r { b ′ } ). Then S C ∈U U C ∪ V witnesses (JP) for B ∪ { b } . Proposition 5.20.
A proximity map r : ( S, ⊢ , ≺ ) → ( S ′ , ⊢ ′ , ≺ ′ ) is join-preservingif and only if e r : (DL( S, ⊢ ) , g ≪ ⊢ ) → (DL( S ′ , ⊢ ′ ) , g ≪ ⊢ ′ ) is join-preserving.Proof. Suppose that r is join-preserving.(Prox0) Suppose U e r ∅ . By (JP0), we have A ⊢ ∅ for all A ∈ U . Thus U e ⊢ ∅ .(Prox ∨ ) Suppose U e r V ∨ W . Since (
V ∨ W ) ∗ = ⊢ ′ V ∗ ∧ W ∗ in DL( S ′ , ⊢ ′ ), foreach A ∈ U , B ∈ V ∗ , and C ∈ W ∗ , we have A r B ∪ C . By (JP ∨ ), there exist V A,B,C , W A,B,C ∈ Fin ( Fin ( S )) such that A e ⊢ V A,B,C ∪ W
A,B,C , V A,B,C e r { B } ∗ ,and W A,B,C e r { C } ∗ . Put V ′ = _ A ∈U ^ B ∈V ∗ _ C ∈W ∗ V A,B,C , W ′ = _ A ∈U ^ C ∈W ∗ _ B ∈V ∗ W A,B,C . Then, V ′ e r V and W ′ e r W . Since { A } e ⊢ ^ B ∈V ∗ ^ C ∈W ∗ V A,B,C ∪ W
A,B,C e ⊢ ^ B ∈V ∗ _ C ∈W ∗ V A,B,C ∨ ^ C ∈W ∗ _ B ∈V ∗ W A,B,C for each A ∈ U , we have U e ⊢ V ′ ∪ W ′ .Conversely, suppose that e r is join-preserving. We show (JP0) and (JP ∨ ).(JP0) Suppose A r ∅ . Then { A } e r ∅ . Thus { A } e ⊢ ∅ by (Prox0), and so A ⊢ ∅ .(JP ∨ ) Suppose A r B ∪ C . Then { A } e r { B } ∗ ∨ { C } ∗ . By (Prox ∨ ), there exist U , V ∈
Fin ( Fin ( S )) such that { A } e ⊢ U ∪ V , U e r { B } ∗ , and V e r { C } ∗ .In particular, since join-preserving proximity relations are closed under com-position, so do join-preserving proximity maps. Theorem 5.21.
The categories
SContEnt and
SProxLat are equivalent.Proof.
By Theorem 5.16 and Proposition 5.20, the functor F : ContEnt → ProxLat restricts to a full and faithful functor from
SContEnt to SProxLat . Since ev-ery isomorphic proximity relation between ∨ -strong proximity lattices is join-preserving, F establishes an equivalence of SContEnt and
SProxLat .By an abuse of notation, we write F : SContEnt → SProxLat and G : SProxLat → SContEnt for the restrictions of the functor F : ContEnt → ProxLat and its quasi-inverse G : ProxLat → ContEnt . Remark . Many of the examples in Section 7 start from a strong prox-imity lattice ( S, ≺ ) and specify a strong continuous entailment relation whichrepresents the desired construction on Spec( S ). The functor F : SContEnt → SProxLat then allows us to calculate the corresponding construction on ( S, ≺ ).The presentations of stably compact locales are invariant under the equiva-lence of SContEnt and
SProxLat in the following sense.
Proposition 5.23.
1. For any strong proximity lattice ( S, ≺ ) , the locale Spec( S ) is presented by G ( S, ≺ ) . . For any continuous entailment relation ( S, ⊢ , ≺ ) , the locale Spec( F ( S, ⊢ , ≺ )) is presented by ( S, ⊢ , ≺ ) .Proof.
1. This is clear from the definition of G ( S, ≺ ) and Corollary 5.7.2. First, we define a bijection between interpretations of ( S, ⊢ , ≺ ) in a locale X and interpretations of GF ( S, ⊢ , ≺ ) in X via a mapping a
7→ : S → Fin ( Fin ( S )). Let f : S → Ω( X ) be an interpretation of ( S, ⊢ , ≺ ) in X . Define f : Fin ( Fin ( S )) → Ω( X ) by f ( U ) def = W A ∈U V a ∈ A f ( a ) , which clearly satisfies f ( ) = f ( a ) for all a ∈ S . We show that f preservesthe order on DL( S, ⊢ ), which implies that f respects the equality on DL( S, ⊢ ).Suppose U e ⊢ V . Since f is an interpretation of ( S, ⊢ , ≺ ), we have f ( U ) = W A ∈U V a ∈ A f ( a ) ≤ X V B ′ ∈V ∗ W b ′ ∈ B ′ f ( b ′ ) = W B ∈V V b ∈ B f ( b ) = f ( V ) , where ≤ X is the order on X . Thus, f is a function on DL( S, ⊢ ). Similarly, wehave U g ≪ ⊢ V → f ( U ) ≤ X f ( V ). It is also easy to check that f preserves finitemeets and finite joins. Furthermore, for any A ∈ Fin ( S ), we have V a ∈ A f ( a ) = V a ∈ A _ b ≺ a f ( b ) = _ B ≺ U A V b ∈ B f ( b ) , which implies f ( U ) ≤ X W V g ≪ ⊢ U f ( V ). Thus, f is an interpretation of GF ( S, ⊢ , ≺ ) in X . Since U = ⊢ W A ∈U V a ∈ A for each U ∈
Fin ( Fin ( S )), f is a uniqueinterpretation of GF ( S, ⊢ , ≺ ) in X such that f ( ) = f ( a ) for all a ∈ S .Define j S : S → RIdl( F ( S, ⊢ , ≺ )) by j S ( a ) = ↓ g ≪ ⊢ . Then, it is straight-forward to show that j S is a universal interpretation of ( S, ⊢ , ≺ ). To construct a new entailment relation, one often specifies a set of initial entail-ments from which the entire relation is generated.
Definition 5.24. An axiom on a set S is a pair ( A, B ) ∈ Fin ( S ) × Fin ( S ). Givena set ⊢ of axioms on S , an entailment relation ( S, ⊢ ) is said to be generated by ⊢ if ⊢ is the smallest entailment relation on S that contains ⊢ .We usually write A ⊢ B for ( A, B ) ∈ ⊢ . Lemma 5.25. If ⊢ is a set of axioms on a set S , then the entailment relation ⊢ generated by ⊢ is inductively defined by the following rules: A ≬ BA ⊢ B (R ′ ) A ⊢ C ∀ c ∈ C ( A ′ , c ⊢ B ) A, A ′ ⊢ B (AxL) Proof.
First, we show that the relation ⊢ generated by (R ′ ) and (AxL) is anentailment relation. The proof is by induction on the height of derivations ofthe premises of each condition in Definition 4.3. For example, to see that ⊢ satisfies (T), we show that ⊢ satisfies more general condition: A ⊢ B, a a, A ′ ⊢ B ′ A, A ′ ⊢ B, B ′ (T ′ )16uppose A ⊢ B, a and a, A ′ ⊢ B ′ . Then A ⊢ B, a is derived by either (R ′ ) or(AxL). The former case is easy. In the latter case, A ⊢ B, a is of the form C ′ , C ⊢ B, a for some C ⊢ D such that ∀ d ∈ D ( C ′ , d ⊢ B, a ). By inductionhypothesis, we have A ′ , C ′ , d ⊢ B, B ′ for all d ∈ D . Hence A ′ , C, C ′ ⊢ B, B ′ by(AxL). Next, if ⊢ ′ is another entailment relation on S containing ⊢ , then ⊢ ′ satisfies (R ′ ) and (AxL), so ⊢ ′ must contain ⊢ .Dually, we have the following. Lemma 5.26. If ⊢ is a set of axioms on a set S , the entailment relation ⊢ generated by ⊢ is inductively defined by the following rules: A ≬ BA ⊢ B (R ′ ) ∀ c ∈ C ( A ⊢ B ′ , c ) C ⊢ BA ⊢ B ′ , B (AxR)The following is useful when defining a new strong continuous entailmentrelation using axioms. Lemma 5.27.
Let ⊢ be an entailment relation on a set S generated by a set ⊢ of axioms. If ≺ is an idempotent relation on S such that1. C ≺ U A ⊢ B → ∃ B ′ ∈ Fin ( S ) ( C ⊢ B ′ ≺ L B ) ,2. A ⊢ B ≺ L C → ∃ A ′ ∈ Fin ( S ) ( A ≺ U A ′ ⊢ C ) ,then ( S, ⊢ , ≺ ) is a strong continuous entailment relation.Proof. Let ≺ be an idempotent relation on S satisfying 1 and 2. We show onlyone direction of (5.5), A ⊢ B = ⇒ ∀ C ≺ U A ∃ B ′ ∈ Fin ( S ) ( C ⊢ B ′ ≺ L B ) , by induction on the derivation of A ⊢ B . If A ⊢ B is derived by (R ′ ), then theconclusion is trivial. Suppose that A, A ′ ⊢ B is derived by (AxL). Then, thereexists C ∈ Fin ( S ) such that A ⊢ C and ∀ c ∈ C ( A ′ , c ⊢ B ). Let D ≺ U A ∪ A ′ .Since D ≺ U A , there exists C ′ ≺ L C such that D ⊢ C ′ by 1. By inductionhypothesis, for each c ′ ∈ C ′ , there exists B c ′ ≺ L B such that D, c ′ ⊢ B c ′ . Put B ′ = S c ′ ∈ C ′ B c ′ . Then, by successive applications of (T), we obtain D ⊢ B ′ .The other direction of (5.5) follows from 2 and Lemma 5.26.As an application of generated strong continuous entailment relations, weshow that SProxLat
Prox and
JSProxLat
Prox are equivalent. Recall that the func-tor F : ContEnt → ProxLat restricts to an equivalence of
SContEnt
Prox and
SProxLat
Prox (Theorem 5.16). Composing F with the inclusion SProxLat
Prox ֒ → JSProxLat
Prox , we get a full and faithful functor F ′ : SContEnt
Prox → JSProxLat
Prox . Lemma 5.28.
The functor F ′ is essentially surjective.Proof. Given a ∨ -strong proximity lattice ( S, ≺ ), define an entailment relation ⊢ ∧ on S by specifying its axioms as follows: A ⊢ ∧ B def ⇐⇒ ∃ C ∈ Fin ( S ) ( A ≺ U C & V C ≤ W B ) . Using Lemma 5.27, one can show that ( S, ⊢ ∧ , ≺ ) is a strong continuous entail-ment relation. On the other hand, let G ( S, ≺ ) = ( S, ⊢ , ≪ ) be the continuous17ntailment relation determined by the quasi-inverse G of F (see (4.1) and (5.7)).It suffices to show that ( S, ⊢ ∧ , ≺ ) and ( S, ⊢ , ≪ ) are isomorphic as continuousentailment relations. By induction on ⊢ ∧ , we see that A ≪ ⊢ ∧ B ⇐⇒ ∃ C ∈ Fin ( S ) ( A ≺ U C & V C ≺ W B ) . Then, it is straightforward to show that ≪ · ≪ ⊢ ∧ = ≪ ⊢ ∧ = ≪ ⊢ ∧ · ≪ ⊢ ∧ and ≪ ⊢ ∧ · ≪ = ≪ = ≪ · ≪ . Thus ≪ and ≪ ⊢ ∧ are proximity maps ≪ : ( S, ⊢ , ≪ ) → ( S, ⊢ ∧ , ≪ ⊢ ∧ ) and ≪ ⊢ ∧ : ( S, ⊢ ∧ , ≪ ⊢ ∧ ) → ( S, ⊢ , ≪ ) and inverse to each other. Theorem 5.29.
The categories
SContEnt
Prox , ContEnt , ProxLat , JSProxLat
Prox ,and
SProxLat
Prox are equivalent.Proof.
By Lemma 5.28, Theorem 5.16, and Theorem 5.8.Since every isomorphic proximity relation between ∨ -strong proximity lat-tices are join-preserving, we also have the following by Theorem 5.21. Theorem 5.30.
The categories
SContEnt , JSProxLat , and
SProxLat are equiv-alent.
In point-set topology, the de Groot dual of a stably compact space has the sameset of points equipped with the cocompact topology: the topology generatedby the complements of compact saturated subsets of the original space. ByHofmann–Mislove theorem, compact saturated subsets correspond to Scott openfilters, which are amenable to point-free treatment. Thus, the de Groot dual ofa stably compact locale X is defined to be the locale whose frame is the Scottopen filters on Ω( X ); see Escard´o [5]. We relate the de Groot duality to thestructural dualities of proximity lattices and continuous entailment relations. Definition 6.1.
The dual S ◦ of a distributive lattice S = ( S, , ∨ , , ∧ ) is thedistributive lattice ( S, , ∧ , , ∨ ) with the opposite order. The dual S d of aproximity lattice S = ( S, ≺ ) is the proximity lattice ( S ◦ , ≻ ).Our aim is to give a localic account of [16, Section 4], which shows thatRIdl( S d ) is isomorphic to the frame of Scott open filters on RIdl( S ). Definition 6.2.
Let ( S, ≺ ) be a proximity lattice. Write Σ(Spec( S )) for thelocale whose models are rounded ideals of S , i.e., Σ(Spec( S )) is presented by ageometric theory T Σ over S with the following axioms: ⊤ ⊢ , a ∧ b ⊢ ( a ∨ b ) , a ⊢ b (if b ≤ a ) ,a ⊢ b (if b ≺ a ) , a ⊢ _ b ≻ a b. Let Upper( S ) be the collection of rounded upper subsets of ( S, ≺ ), i.e., thosesubset U ⊆ S such that a ∈ U ↔ ∃ b ≺ a ( b ∈ U ). Clearly, Upper( S ) is closed18nder all joins, which are just unions. Moreover, (ProxI) ensures that Upper( S )has finite meets defined by1 def = S = ↑ ≺ , U ∧ V def = [ a ∈ U,b ∈ V ↑ ≺ ( a ∨ b ) , where ↑ ≺ a def = { b ∈ S | b ≻ a } . These finite meets clearly distribute over alljoins. Hence Upper( S ) is a frame. Lemma 6.3.
The frames
Ω(Σ(Spec( S ))) and Upper( S ) are isomorphic.Proof. It is straightforward to show that a function i Σ : S → Upper( S ) definedby i Σ ( a ) def = ↑ ≺ a is a universal interpretation of T Σ . Proposition 6.4.
The frame
Ω(Σ(Spec( S ))) is the Scott topology on RIdl( S ) .Proof. It is known that Upper( S ) is the Scott topology on RIdl( S ); see Vickers[21, Lemma 2.11] or Jung and S¨underhauf [16, Lemma 14]. Then, the claimfollows from Lemma 6.3.Scott open filters on a locale X are models of the upper powerlocale of X ,which is characterised by the following universal property; see Vickers [22]. Definition 6.5.
The upper powerlocale of a locale X is a locale P U ( X ) togetherwith a preframe homomorphism i U : Ω( X ) → Ω(P U ( X )) such that for any pre-frame homomorphism f : Ω( X ) → Ω( Y ) to a locale Y , there exists a uniqueframe homomorphism f : Ω(P U ( X )) → Ω( Y ) such that f ◦ i U = f . Proposition 6.6.
For any proximity lattice S , Σ(Spec( S d )) is the upper pow-erlocale of Spec( S ) .Proof. By Definition 6.2, the locale Σ(Spec( S d )) is presented by a geometrictheory T over S with the following axioms: ⊤ ⊢ , a ∧ b ⊢ ( a ∧ b ) , a ⊢ b (if a ≤ b ) ,a ⊢ b (if a ≺ b ) , a ⊢ _ b ≺ a b. By the preframe version of Proposition 5.5 (see also Remark 5.6), the universalinterpretation i T : S → Ω(Σ(Spec( S d ))) of T in Σ(Spec( S d )) uniquely extendsto a preframe homomorphism i U : RIdl( S ) → Ω(Σ(Spec( S d ))) via the function i S : S → RIdl( S ) defined by (5.2). Then, it is straightforward to show that i U satisfies the universal property of the upper powerlocale of Spec( S ). Theorem 6.7.
For any proximity lattice S , the frame RIdl( S d ) is isomorphicto the frame of Scott open filters on RIdl( S ) . Thus, Spec( S d ) is the de Grootdual of Spec( S ) .Proof. Since RIdl( S d ) is the collection of models of Σ(Spec( S d )), it is isomorphicto the frame of Scott open filters on RIdl( S ) by Proposition 6.6We extend the duality to morphisms. The following are obvious.19 emma 6.8. If r : S → S ′ is a proximity relation between proximity lattices,then the relational opposite r − is a proximity relation r − : S ′ d → S d . Proposition 6.9.
The assignment r r − determines a dual isomorphism ( · ) − : ProxLat ∼ = −→ ProxLat op . All of the categories we have introduced so far (
ProxLat , ContEnt etc.) areorder-enriched categories, where homsets are ordered by the set-theoretic inclu-sion. Thus, the following notion applies.
Definition 6.10.
Let C be an order-enriched category. For morphisms f : A → B and g : B → A , we say that f is a left adjoint to g and g is a right adjoint to f if f ◦ g ≤ B id B and id A ≤ A g ◦ f , where ≤ A and ≤ B are the orderson Hom ( A, A ) C and Hom ( B, B ) C respectively. In this case, ( f, g ) is called anadjoint pair of morphisms from A to B .Let ProxLat
Perf be the subcategory of
ProxLat where morphisms from S to S ′ are adjoint pairs of proximity relations from S ′ to S . The identity on ( S, ≺ )is ( ≺ , ≺ ), and the composition of adjoint pairs ( s, r ) and ( s ′ , r ′ ) is ( s ◦ s ′ , r ′ ◦ r ). Theorem 6.11.
The assignment ( s, r ) : S → S ′ ( r − , s − ) : S d → S ′ d deter-mines an isomorphism ( · ) d : ProxLat
Perf ∼ = −→ ProxLat
Perf .Proof.
Since the functor ( · ) − : ProxLat → ProxLat op preserves the order on mor-phisms, for any morphism ( s, r ) : S → S ′ in ProxLat
Perf (i.e. an adjoint pair ofproximity relations from S ′ to S ), the pair ( r − , s − ) is an adjoint pair from S ′ d to S d , i.e., a morphism ( r − , s − ) : S d → S ′ d in ProxLat
Perf .A locale map f : X → Y is perfect if the corresponding frame homomor-phism Ω( f ) : Ω( Y ) → Ω( X ) has a Scott continuous right adjoint g : Ω( X ) → Ω( Y ). In this case, g is necessarily a preframe homomorphism. An adjoint pair( s, r ) : S → S ′ of proximity relations in ProxLat
Perf corresponds to a perfect mapfrom Spec( S ) to Spec( S ′ ). Hence, Theorem 6.11 is a manifestation of the deGroot duality of stably compact locales in the setting of proximity lattice. We describe an analogous duality on the category
ContEnt , and relate it to theduality on
ProxLat via the equivalence of the two categories.
Definition 6.12.
The dual ⊢ ◦ of an entailment relation ⊢ on S is the relationalopposite: A ⊢ ◦ B def ⇐⇒ B ⊢ A . The dual S d of a continuous entailment relation S = ( S, ⊢ , ≪ ) is the continuous entailment relation ( S, ⊢ ◦ , ≫ ).If r : S → S ′ is a proximity map between continuous entailment relations,then r − is a proximity map r − : S ′ d → S d . Then, the following is obvious. Proposition 6.13.
The assignment r : S → S ′ r − : S ′ d → S d determines adual isomorphism ( · ) − : ContEnt ∼ = −→ ContEnt op . Theorem 6.14.
The equivalence F : ContEnt → ProxLat commutes with thedual isomorphisms ( · ) − on ContEnt and
ProxLat up to natural isomorphism. roof. For each continuous entailment relation ( S, ⊢ , ≪ ), define a relation r S ⊆ D ( S, ⊢ ) ◦ × D ( S, ⊢ ◦ ) by U r S V def ⇐⇒ U ∗ f ≫ V . Since U f ≪ V ↔ V ∗ f ≫ U ∗ , one can easily show that r S is a proximity relationfrom F ( S, ⊢ , ≪ ) d to F (( S, ⊢ , ≪ ) d ) with an inverse t S defined by V t S U def ⇐⇒ U f ≪ V ∗ . To see that r S is natural in S , for any proximity map r : ( S, ⊢ , ≪ ) → ( S ′ , ⊢ ′ , ≪ ′ )and for any U ∈
Fin ( Fin ( S ′ )) and V ∈
Fin ( Fin ( S )), we have U ( r S ◦ ( e r ) − ) V ⇐⇒ ∃W ∈
Fin ( Fin ( S )) (cid:0) U ( e r ) − W & W r S V (cid:1) ⇐⇒ ∃W ∈ Fin ( Fin ( S )) (cid:0) W e r U & V ∗ f ≪ W (cid:1) ⇐⇒ V ∗ e r U⇐⇒ ∃W ∈
Fin ( Fin ( S ′ )) (cid:16) W ∗ f ≪ ′ U & V ∗ e r W ∗ (cid:17) ⇐⇒ ∃W ∈ Fin ( Fin ( S ′ )) (cid:16) U r S ′ W & W g ( r − ) V (cid:17) ⇐⇒ U ( g ( r − ) ◦ r S ′ ) V . Let
ContEnt
Perf be the category of continuous entailment relations and ad-joint pairs of proximity maps which is defined similarly as
ProxLat
Perf . Thefollowing is analogous to Theorem 6.11.
Proposition 6.15.
The assignment ( s, r ) : S → S ′ ( r − , s − ) : S d → S ′ d determines an isomorphism ( · ) d : ContEnt
Perf ∼ = −→ ContEnt
Perf . Theorem 6.16.
The category
ContEnt
Perf is equivalent to
ProxLat
Perf . Theequivalence commutes with the isomorphisms ( · ) d on ContEnt
Perf and
ProxLat
Perf up to natural isomorphism.Proof.
Since the functor F : ContEnt → ProxLat preserves the order on mor-phisms, it can be restricted to an equivalence between
ContEnt
Perf and
ProxLat
Perf .The second statement follows from Theorem 6.14.We introduce the notion of dual for strong continuous entailment relations.
Definition 6.17.
The dual of a strong continuous entailment relation ( S, ⊢ , ≺ )is a strong continuous entailment relation ( S, ⊢ ◦ , ≻ ).Note that the inclusion SContEnt
Prox ֒ → ContEnt commutes with the dualitieson both categories. Since strong proximity lattices are closed under the dualityin the sense of Definition 6.1, Theorem 6.16 restricts to the full subcategories
SContEnt
Perf and
SProxLat
Perf of ContEnt
Perf and
ProxLat
Perf , respectively, whichconsist of strong continuous entailment relations and strong proximity lattices.
We present a number of constructions on stably compact locales in the setting ofstrong proximity lattices and strong continuous entailment relations and analyse21heir de Groot duals. For the sake of simplicity, we prefer to work with strong proximity lattices rather proximity lattices because the geometric theories ofthe locales represented by the former are simpler and easier to work with.Our main tool is the following observation, together with Lemma 5.27.
Lemma 7.1. If ( S, ⊢ ) is an entailment relation generated by a set ⊢ of axioms,then the dual ⊢ ◦ is generated by ⊢ ◦ def = { ( B, A ) | A ⊢ B } .Proof. Immediate from the structural symmetry of entailment relations.
We deal with the lower, upper, and Vietoris powerlocales and consider theirinteractions with the construction Σ(Spec( S )), the locale whose frame is theScott topology on RIdl( S ). For the localic account of powerlocales, the readeris referred to Vickers [22, 23]. Let ( S, ≺ ) be a strong proximity lattice.1. The locale Σ(Spec( S )) is presented by a strong continuous entailmentrelation Σ( S ) = ( S, ⊢ Σ , ≻ ) where ⊢ Σ is generated by the following axioms: ⊢ Σ a, b ⊢ Σ a ∨ b a ⊢ Σ b ( if b ≤ a )
2. The upper powerlocale of
Spec( S ) is presented by a strong continuousentailment relation P U ( S ) = ( S, ⊢ U , ≺ ) where ⊢ U is generated by thefollowing axioms: ⊢ U a, b ⊢ U a ∧ b a ⊢ U b ( if a ≤ b ) In particular, (the locale whose frame is) the Scott topology and the upper pow-erlocale of a stably compact locale are stably compact.Proof.
It is straightforward to check that Σ( S ) and P U ( S ) satisfy the conditionin Lemma 5.27. Then, item 1 is immediate from Definition 6.2, while item 2follows from Proposition 6.6.Note that A ⊢ Σ B ↔ ∃ b ∈ B ( b ≤ W A ) and A ⊢ U B ↔ ∃ b ∈ B ( V A ≤ b ).The constructions Σ( S ) and P U ( S ) extend to functors Σ : SProxLat op → SContEnt and P U : SProxLat → SContEnt , which send each join-preserving proximity rela-tion r : ( S, ≺ ) → ( S ′ , ≺ ′ ) to join-preserving proximity maps Σ( r ) : Σ( S ′ ) → Σ( S )and P U ( r ) : P U ( S ) → P U ( S ′ ) defined by A Σ( r ) B def ⇐⇒ ∃ b ∈ B ( b r W A ) ,A P U ( r ) B def ⇐⇒ ∃ b ∈ B ( V A r b ) . The notion of lower powerlocale is the dual of that of upper powerlocale.
Definition 7.3.
The lower powerlocale of a locale X is a locale P L ( X ) togetherwith a suplattice homomorphism i L : Ω( X ) → Ω(P L ( X )) such that for anysuplattice homomorphism f : Ω( X ) → Ω( Y ) to a locale Y , there exists a uniqueframe homomorphism f : Ω(P L ( X )) → Ω( Y ) such that f ◦ i L = f .22 emma 7.4. For any strong proximity lattice ( S, ≺ ) , the lower powerlocaleof Spec( S ) is presented by a strong continuous entailment relation P L ( S ) =( S, ⊢ L , ≺ ) where ⊢ L is generated by the following axioms: ⊢ L a ∨ b ⊢ L a, b a ⊢ L b ( if a ≤ b ) In particular, the lower powerlocale of a stably compact locale is stably compact.Proof.
Immediate from the suplattice version of Proposition 5.5.Note that A ⊢ L B ↔ ∃ a ∈ A ( a ≤ W B ) . The construction P L ( S ) ex-tends to a functor P L : SProxLat → SContEnt , which sends each join-preservingproximity relation r : ( S, ≺ ) → ( S ′ , ≺ ′ ) to a join-preserving proximity mapP L ( r ) : P L ( S ) → P L ( S ′ ) defined by A P L ( r ) B def ⇐⇒ ∃ a ∈ A ( a r W B ) . Theorem 7.5.
For any strong proximity lattice S , we have1. P U ( S ) d ∼ = P L ( S d ) and P L ( S ) d ∼ = P U ( S d ) ,2. Σ( S d ) ∼ = P U ( S ) and Σ( S ) d ∼ = P L ( S ) .Proof. Immediate from Lemma 7.1, Lemma 7.2, and Lemma 7.4.Item 1 of Theorem 7.5 is known: Vickers gave a localic proof using entail-ment systems [24, Theorem 54], and Goubault-Larrecq proved the correspondingresult for stably compact spaces [8, Theorem 3.1]. It is notable, however, thatour proof is a simple analysis of axioms of entailment relations.In the following, compositions such as P U (Σ( S )) should be read as P U ( F (Σ( S ))),where F : SContEnt → SProxLat is the functor establishing the equivalence ofthe two categories (see Remark 5.22).
Proposition 7.6.
For any strong proximity lattice S , we have1. P U (Σ( S )) ∼ = Σ(P L ( S )) ,2. P U (P L ( S )) ∼ = Σ(Σ( S )) .Proof. By Theorem 6.14 and item 2 of Theorem 7.5, we haveP U (Σ( S )) ∼ = P U (P L ( S ) d ) ∼ = Σ(P L ( S )) . The proof of item 2 is similar.
The double powerlocale of a locale X is a locale P D ( X ) togetherwith a Scott continuous function i D : Ω( X ) → Ω(P D ( X )) such that for any Scottcontinuous function f : Ω( X ) → Ω( Y ) to a locale Y , there exists a unique framehomomorphism f : Ω(P D ( X )) → Ω( Y ) such that f ◦ i D = f . Lemma 7.8.
For any strong proximity lattice ( S, ≺ ) , the double power locale of Spec( S ) can be presented by a strong continuous entailment relation P D ( S ) =( S, ⊢ D , ≺ ) , where ⊢ D is generated by the following axioms: a ⊢ D b ( a ≤ b ) . roof. By the dcpo version of Proposition 5.5.Note that A ⊢ D B ↔ ∃ a ∈ A ∃ b ∈ B ( a ≤ b ) . The construction ( S, ≺ ) ( S, ⊢ D , ≺ ) extends to a functor P D : SProxLat → SContEnt , which sends eachjoin-preserving proximity relation r : ( S, ≺ ) → ( S ′ , ≺ ′ ) to a join-preserving prox-imity map P D ( r ) : ( S, ⊢ D , ≺ ) → ( S, ⊢ ′ D , ≺ ′ ) defined by A P D ( r ) B def ⇐⇒ ∃ a ∈ A ∃ b ∈ B ( a r b ) . Proposition 7.9.
For any strong proximity lattice S , we have P D ( S ) d ∼ = P D ( S d ) . Proof.
Immediate from Lemma 7.1 and Lemma 7.8.We prove some well-known characterisations of the double powerlocale (seeProposition 7.13 and Proposition 7.15). To this end, we begin with the con-struction of the lower powerlocale of a strong continuous entailment relation.Given a strong continuous entailment relation ( S, ⊢ , ≺ ) define an entailmentrelation ⊢ L on Fin ( S ) by the following axioms: A ⊢ L A , . . . , A n − (if ∀ B ∈ { A i | i < n } ∗ A ⊢ B )Define an idempotent relation ≺ L on Fin ( S ) by A ≺ L B def ⇐⇒ A ≺ U B. Then (
Fin ( S ) , ⊢ L , ≺ L ) is a strong continuous entailment relation by Lemma 5.27. Lemma 7.10.
For any U , V ∈
Fin ( Fin ( S )) , we have U ≪ ⊢ L V ⇐⇒ ∃ A ∈ U∀ B ∈ V ∗ ( A ≪ ⊢ B ) . Proof.
By induction on ⊢ L , one can show that U ⊢ L V def ⇐⇒ ∃ A ∈ U∀ B ∈ V ∗ ( A ⊢ B ) . (7.1)Then, the direction ⇒ is obvious from (7.1). Conversely, suppose that thereexists A ∈ U such that ∀ B ∈ V ∗ ( A ≪ ⊢ B ). Then, for each B ∈ V ∗ , there exists C B such that A ≺ U C B ⊢ B . Put C = S B ∈U ∗ C B . Then A ≺ U C and C ⊢ B for all B ∈ V ∗ . Hence U ( ≺ L ) U { C } ⊢ L V and so U ≪ ⊢ L V . Lemma 7.11.
The strong continuous entailment relation ( Fin ( S ) , ⊢ L , ≺ L ) presentsthe lower powerlocale of Spec( F ( S, ⊢ , ≺ )) .Proof. From the characterisation of ≪ ⊢ L in Lemma 7.10, the entailment system( Fin ( S ) , ≪ ⊢ L ) coincides with the construction of the lower powerlocale of theentailment system ( S, ≪ ⊢ ) in Vickers [24, Theorem 53]. Corollary 7.12.
The upper powerlocale of
Spec( F ( S, ⊢ , ≺ )) can be presentedby a strong continuous entailment relation ( Fin ( S ) , ⊢ U , ≺ U ) defined by U ⊢ U V def ⇐⇒ ∃ B ∈ V∀ A ∈ U ∗ ( A ⊢ B ) ,A ≺ U B def ⇐⇒ A ≺ L B. roof. We have P L ( S d ) d ∼ = P U ( S ) by item 2 of Theorem 7.5. The corollaryfollows by unfolding the definition of P L ( S d ) d using Lemma 7.10.For any strong proximity lattice ( S, ≺ ), define a preorder ≤ ∨ on Fin ( S ) by A ≤ ∨ B def ⇐⇒ W A ≤ W B. Let S ∨ be the set Fin ( S ) equipped with the equality determined by ≤ ∨ . Definea lattice structure S ∨ def = ( S ∨ , ∨ , ∨ ∨ , ∨ , ∧ ∨ ) as in (5.4) and an idempotentrelation ≺ ∨ on S ∨ by A ≺ ∨ B def ⇐⇒ W A ≺ W B. Then, ( S ∨ , ≺ ∨ ) is a strong proximity lattice, which is isomorphic to ( S, ≺ ) viaproximity relations r : ( S, ≺ ) → ( S ∨ , ≺ ∨ ) and s : ( S ∨ , ≺ ∨ ) → ( S, ≺ ) defined by a r A def ⇐⇒ a ≺ W A, A s a def ⇐⇒ W A ≺ a. The following is a special case of Vickers [23], which holds for more generalcontext of locally compact locale. Proposition 7.13.
For any strong proximity lattice ( S, ≺ ) , we have P D ( S ) ∼ = Σ(Σ( S )) . Proof.
By item 2 of Proposition 7.6, it suffices to show that P D ( S ) ∼ = P U (P L ( S )).By Lemma 7.4 and Corollary 7.12, the locale P U (P L ( S )) is presented by a strongcontinuous entailment relation ( Fin ( S ) , ⊢ UL , ≺ UL ) on Fin ( S ) defined by U ⊢ UL V def ⇐⇒ ∃ B ∈ V∀ A ∈ U ∗ ∃ a ∈ A ( a ≤ W B ) ⇐⇒ ∃ B ∈ V∃ A ∈ U ( W A ≤ W B ) ,A ≺ UL B def ⇐⇒ A ≺ L B. Thus
U ≪ ⊢ UL V ⇐⇒ ∃ A ∈ U∃ B ∃ C ∈ V ( A ≺ L B & W B ≤ W C ) ⇐⇒ ∃ A ∈ U∃ C ∈ V ( W A ≺ W C ) . As for the strong proximity lattice ( S ∨ , ≺ ∨ ) defined above, its double power-locale P D ( S ∨ ) = ( S ∨ , ⊢ ∨ , ≺ ∨ ) characterised in Lemma 7.8 satisfies U ≪ ⊢ ∨ V ⇐⇒ ∃ A ∈ U∃ B ∈ V ( W A ≺ W B ) . Clearly, the entailment systems ( S ∨ , ≪ ⊢ ∨ ) and ( Fin ( S ) , ≪ ⊢ UL ) are isomorphic.Since P D is functorial and the embedding SContEnt
Prox ֒ → Entsys is faithful, wehave P D ( S ) ∼ = P D ( S ∨ ) ∼ = P U (P L ( S )). Corollary 7.14.
For any strong proximity lattice S , we have Σ(Σ( S )) d ∼ = Σ(Σ( S d )) . Proof.
By Proposition 7.9 and Proposition 7.13. In [23], Σ( S ) is expressed as the exponential over the Sierpinski locale. roposition 7.15. For any strong proximity lattice S , we have1. P L (Σ( S )) ∼ = Σ(P U ( S )) ,2. P L (P U ( S )) ∼ = P D ( S ) .Proof.
1. By Corollary 7.14, and item 2 of Theorem 7.5.2. Apply item 1 to S d and use Theorem 7.5 and Proposition 7.13.Item 2 of Proposition 7.6 and Proposition 7.15 are known for locally compactlocales; see Vickers [23]. Item 1 of Proposition 7.6 and Proposition 7.15 say thatthe locales of the form Σ( S ) for a stably compact S is closed under the lowerand upper powerlocales. Moreover, the lower and upper powerlocales of Σ( S )are obtained by the upper and the lower powerlocales of S , respectively. Let ( S, ≺ ) be a strong proximity lattice. The Vietoris pow-erlocale of Spec( S ) is presented by a strong continuous entailment relationP V ( S ) = ( S V , ⊢ V , ≺ V ) on the set S V def = { ✸ a | a ∈ S } ∪ { ✷ a | a ∈ S } , where ⊢ V is generated by the following axioms: ✸ ⊢ V ✸ ( a ∨ b ) ⊢ V ✸ a, ✸ b ✸ a ⊢ V ✸ b (if a ≤ b ) ⊢ V ✷ ✷ a, ✷ b ⊢ V ✷ ( a ∧ b ) ✷ a ⊢ V ✷ b (if a ≤ b ) ✷ a, ✸ b ⊢ V ✸ ( a ∧ b ) ✷ ( a ∨ b ) ⊢ V ✷ a, ✸ b The idempotent relation ≺ V is defined by ✸ a ≺ V ✸ b def ⇐⇒ a ≺ b, ✷ a ≺ V ✷ b def ⇐⇒ a ≺ b. One can easily verify that P V ( S ) satisfies the condition in Lemma 5.27.Moreover, it is straightforward to show that the locale presented by P V ( S ) isisomorphic to the Vietoris powerlocale of Spec( S ); see Johnstone [11] for theconstruction of Vietoris powerlocales. Thus, the Vietoris powerlocale of a stablycompact locale is stably compact.The construction P V ( S ) extends to a functor P V : SProxLat → SContEnt ,which sends each join-preserving proximity relation r : ( S, ≺ ) → ( S ′ , ≺ ′ ) to ajoin-preserving proximity map P V ( r ) : P V ( S ) → P V ( S ′ ) defined by ✸ A ✷ B P V ( r ) ✸ C ✷ D def ⇐⇒ ∃ a ∈ A ( a ∧ V B r W C ) or ∃ d ∈ D ( V B r d ∨ W C ) , where ✸ A ✷ B def = { ✸ a | a ∈ A } ∪ { ✷ b | b ∈ B } for each A, B ∈ Fin ( S ). Theorem 7.17.
For any strong proximity lattice S , we have P V ( S ) d ∼ = P V ( S d ) . Proof. P V ( S ) d and P V ( S d ) are identical except that ✸ and ✷ are swapped.Goubault-Larrecq proved the result corresponding to Theorem 7.17 for sta-bly compact spaces using A -valuations [8, Corollary 5.24].26 .2 Patch topologies Coquand and Zhang [4] gave a construction of patch topologies for entailmentrelations with the interpolation property. The same construction carries over tothe setting of strong continuous entailment relation.
Definition 7.18 (Coquand and Zhang [4, Section 4]) . Given a strong continu-ous entailment relation ( S, ⊢ , ≺ ), the patch topology of S is a strong continuousentailment relation Patch( S ) = ( S P , ⊢ P , ≺ P ) on the set S P def = S ∪{ a ◦ | a ∈ S } , where ⊢ P is generated by the following axioms: A ⊢ P B (if A ⊢ B ) (7.2) a, b ◦ ⊢ P (if a ≺ b ) (7.3) ⊢ P a ◦ , b (if a ≺ b ) (7.4)The idempotent relation ≺ P is defined by a ≺ P b def ⇐⇒ a ≺ b, a ◦ ≺ P b ◦ def ⇐⇒ b ≺ a. Let Patch ′ ( S ) = ( S P , ⊢ ′ P , ≺ P ) be the strong continuous entailment relationwhich is obtained from Patch( S ) by adjoining the following axioms: B ◦ ⊢ ′ P A ◦ (if A ⊢ B ) (7.5)where A ◦ def = { a ◦ | a ∈ A } for each A ∈ Fin ( S ). Proposition 7.19.
For any strong continuous entailment relation ( S, ⊢ , ≺ ) , wehave Patch ′ ( S ) ∼ = Patch( S ) . Proof.
We show that the entailment systems associated with Patch ′ ( S ) andPatch( S ) coincide, i.e., ≪ ⊢ P = ≪ ⊢ ′ P . Since ⊢ ′ P is generated by the extra axioms,we have ≪ ⊢ P ⊆ ≪ ⊢ ′ P . To prove the converse inclusion, it suffices to show that X ⊢ ′ P Y = ⇒ ∀ Z ( ≺ P ) U X ( Z ≪ ⊢ P Y ) . This is proved by induction on the derivation of X ⊢ ′ P Y (see Lemma 5.25).The case (R ′ ) is obvious, so it suffices to check the case (AxL) for each axiom ofPatch( S ′ ). We only deal with (7.5). Suppose that B ◦ , X ⊢ ′ P Y is derived from B ◦ ⊢ ′ P A ◦ and ∀ a ∈ A (cid:0) X, a ◦ ⊢ ′ P Y (cid:1) where A ⊢ B . Let Z ( ≺ P ) U B ◦ , X . Then,there exists C ∈ Fin ( S ) such that C ◦ ⊆ Z and B ≺ L C , and so there exists C ′ such that B ≺ L C ′ ≺ L C . Thus, there exists D such that A ≺ U D ⊢ C ′ by(5.5). For each d ∈ D , there exist a ∈ A and d ′ such that a ≺ d ′ ≺ d . Then, Z, d ′◦ ( ≺ P ) U X, a ◦ so Z, d ′◦ ≪ ⊢ P Y by induction hypothesis. Thus, for each d ∈ D , there exist d ′ ≺ d and W d ∈ Fin ( S ) such that Z, d ′◦ ⊢ P W d ( ≺ P ) L Y so that Z ⊢ P W d , d by (7.4). Since D ⊢ C ′ ≺ L C , we get Z, C ◦ ⊢ P S d ∈ D W d by (7.2)and successive applications of (T) and (7.3). Hence, Z = Z ∪ C ◦ ≪ ⊢ P Y .In terms of SContEnt , we have proximity maps r : Patch( S ) → Patch ′ ( S )and s : Patch ′ ( S ) → Patch( S ) defined by A r B def ⇐⇒ A ≪ ⊢ P B, A s B def ⇐⇒ A ≪ ⊢ ′ P B, which are inverse to each other. 27 heorem 7.20. For any strong continuous entailment relation S , we have Patch( S ) ∼ = Patch( S d ) . Proof.
By Proposition 7.19, we may identify Patch( S ) with Patch ′ ( S ). Then,we have Patch ′ ( S ) ∼ = Patch ′ ( S d ) by exchanging the roles of a and a ◦ . Remark . Combining Patch and the equivalence between
SProxLat
Perf and
SContEnt
Perf , we get a functor Patch:
SProxLat
Perf → SContEnt
Perf , which sendsan adjoint pair ( s, r ) of proximity relations r : ( S, ≺ ) → ( S ′ , ≺ ′ ) and s : ( S ′ , ≺ ′ ) → ( S, ≺ ) to an adjoint pair ( P ( s ) , P ( r )) of proximity maps P ( r ) : Patch( G ( S )) → Patch( G ( S ′ )) and P ( s ) : Patch( G ( S ′ )) → Patch( G ( S )) defined by A, B ◦ P ( r ) C, D ◦ def ⇐⇒ ∃ a, b ∈ S ( V A ∧ b ≺ W B ∨ a & a r W C & V D s b ) ,C, D ◦ P ( s ) A, B ◦ def ⇐⇒ ∃ a, b ∈ S ( V B ∧ a ≺ W A ∨ b & b r W D & V C s a ) . The space of valuations is a localic analogue of the probabilistic power domain by Jones and Plotkin [13, 14]. We first recall several notions of real numberswhich are needed for its definition.1. A lower real is a rounded downward closed subset of rationals Q .2. An upper real is a rounded upward closed subset of Q .3. A Dedekind real is a disjoint pair (
L, U ) of an inhabited lower real L andan inhabited upper real U which is located: p < q implies p ∈ L or q ∈ U .Let −−−→ [0 , ∞ ] and ←−−− [0 , ∞ ] denote the lower and the upper reals greater than 0 re-spectively (including infinity). We follow Vickers [25, Section 4 and Section 6]for the definition of spaces of valuations and covaluations. Definition 7.22. A valuation on a locale X is a Scott continuous function µ : Ω( X ) → −−−→ [0 , ∞ ] satisfying µ (0) = 0 , µ ( x ) + µ ( y ) = µ ( x ∧ y ) + µ ( x ∨ y ) , where the second condition is called the modular law. A covaluation is a Scottcontinuous function ν : Ω( X ) → ←−−− [0 , ∞ ] satisfying ν (1) = 0 and the modular law.The space of valuations V ( X ) on a locale X is the locale whose models arevaluations on X . The space of covaluations C ( X ) is defined similarly.For a strong proximity lattice ( S, ≺ ), the locale V (Spec( S )) is presented bya geometric theory T V over the set S V def = {h p, a i | p ∈ Q & a ∈ S } with the following axioms: ⊤ ⊢ h p, a i (if p < h p, i ⊢ ⊥ (if 0 < p )28 p, a i ⊢ h q, b i (if q ≤ p and a ≤ b ) h p, a i ∧ h q, b i ⊢ _ p ′ + q ′ = p + q h p ′ , a ∧ b i ∧ h q ′ , a ∨ b ih p, a ∧ b i ∧ h q, a ∨ b i ⊢ _ p ′ + q ′ = p + q h p ′ , a i ∧ h q ′ , b ih p, a i ⊢ h q, b i (if q < p and a ≺ b ) h p, a i ⊢ _ p Under the other axioms of T V (or T C ), the axioms h p, a i ∧ h q, b i ⊢ _ p ′ + q ′ = p + q h p ′ , a ∧ b i ∧ h q ′ , a ∨ b i (7.6) h p, a ∧ b i ∧ h q, a ∨ b i ⊢ _ p ′ + q ′ = p + q h p ′ , a i ∧ h q ′ , b i (7.7) are equivalent to the following axioms: h p, a i ∧ h q, b i ⊢ h r, a ∧ b i ∨ h s, a ∨ b i ( if p + q = r + s ) (7.8) h r, a ∧ b i ∧ h s, a ∨ b i ⊢ h p, a i ∨ h q, b i ( if p + q = r + s ) (7.9)Here, the equivalence of two axioms means that one axiom holds in the localepresented by the other axiom and the rest of the axioms of T V (or T C ). Proof. The proof is inspired by Coquand and Spitters [3, Lemma 2], which weelaborate below. We identify generators S V with the corresponding elements of29p( T V ) (or the locale presented by (7.8) and (7.9) in place of (7.6) and (7.7)).We write ≤ V for the orders in these locales.First, assume (7.6). Let p, q, r, s ∈ Q such that p + q = r + s . Take any p ′ , q ′ ∈ Q such that p ′ + q ′ = p + q . If p ′ ≥ r , then h p ′ , a ∧ b i ≤ V h r, a ∧ b i . If p ′ < r , then q ′ = s + ( r − p ′ ), and thus h q ′ , a ∨ b i ≤ V h s, a ∨ b i . Hence h p ′ , a ∧ b i ∧ h q ′ , a ∨ b i ≤ V h r, a ∧ b i ∨ h s, a ∨ b i for all p ′ , q ′ ∈ Q such that p ′ + q ′ = p + q . Applying (7.6), we obtain (7.8).Similarly, we obtain (7.9) from (7.7).Conversely, assume (7.8). By the last two axioms of T V , we have h q, a i ∧ h r, b i ≤ V _ q ′ + r ′ >q + r h q ′ , a i ∧ h r ′ , b i . (7.10)Let q ′ , r ′ ∈ Q such that q ′ + r ′ > q + r . Let θ ∈ Q such that q ′ + r ′ = q + r + θ ,and choose N ∈ N so large that q + r + θ − N θ < 0. By (7.8), we have h q ′ , a i ∧ h r ′ , b i ≤ V h q + r + θ − ( − θ + nθ ) , a ∧ b i ∨ h− θ + nθ, a ∨ b i for all n ∈ N . For each n ∈ N , define ϕ n = h q + r + 2 θ − nθ, a ∧ b i , ϕ n = h− θ + nθ, a ∨ b i . Then, we have h q ′ , a i ∧ h r ′ , b i ≤ V ^ n ≤ N +1 ϕ n ∨ ϕ n ≤ V _ f ∈ Ch ( N +1) ^ n ≤ N +1 ϕ nf n , (7.11)where Ch ( N + 1) is the set of choice functions f : { , . . . , N + 1 } → { , } . Foreach f ∈ Ch ( N + 1), one of the following cases occurs: Case 1 : ∀ n ≤ N + 1 f n = 0. Since − θ < 0, we have ϕ f ≤ V h q + r + θ, a ∧ b i ≤ V h q + r + θ, a ∧ b i ∧ h− θ, a ∨ b i . Case 2 : ∀ n ≤ N + 1 f n = 1. Since q + r + θ − N θ < 0, we have ϕ N +1 f N +1 ≤ V h q + r − N θ, a ∧ b i ∧ h N θ, a ∨ b i . Case 3 : ∃ n ≤ N f n = 0 & f n +1 = 1. ϕ nf n ∧ ϕ n +1 f n +1 ≤ V h q + r − nθ, a ∧ b i ∧ h nθ, a ∨ b i . Case 4 : ∃ n ≤ N f n = 1 & f n +1 = 0. ϕ nf n ∧ ϕ n +1 f n +1 ≤ V h q + r + θ − nθ, a ∧ b i ∧ h− θ + nθ, a ∨ b i . Thus, in any case ^ n ≤ N +1 ϕ nf n ≤ V _ q ′ + r ′ = q + r h q ′ , a ∧ b i ∧ h r ′ , a ∨ b i . Hence, by (7.11) and (7.10), we have (7.6). Similarly, (7.9) implies (7.7).30 roposition 7.24. For any strong proximity lattice ( S, ≺ ) , the locale V (Spec( S )) can be presented by a strong continuous entailment relation V ( S ) = ( S V , ⊢ V , ≺ V ) where ⊢ V is generated by the following axioms: ⊢ V h p, a i ( if p < h p, i ⊢ V ( if < p ) h p, a i ⊢ V h q, b i ( if q ≤ p and a ≤ b ) h p, a i , h q, b i ⊢ V h r, a ∧ b i , h s, a ∨ b i ( if p + q = r + s ) h r, a ∧ b i , h s, a ∨ b i ⊢ V h p, a i , h q, b i ( if p + q = r + s ) The idempotent relation ≺ V is defined by h p, a i ≺ V h q, b i def ⇐⇒ q < p & a ≺ b. The locale C (Spec( S )) can be presented by a strong continuous entailment rela-tion C ( S ) = ( S V , ⊢ C , ≺ C ) where ⊢ C is generated by the following axioms: h p, a i ⊢ C ( if p < ⊢ C h p, i ( if < p ) h p, a i ⊢ C h q, b i ( if p ≤ q and a ≤ b ) h p, a i , h q, b i ⊢ C h r, a ∧ b i , h s, a ∨ b i ( if p + q = r + s ) h r, a ∧ b i , h s, a ∨ b i ⊢ C h p, a i , h q, b i ( if p + q = r + s ) The idempotent relation ≺ C is defined by h p, a i ≺ C h q, b i def ⇐⇒ p < q & a ≺ b. In particular, the spaces of valuations and covaluations on a stably compactlocale are stably compact.Proof. One can check that V ( S ) and C ( S ) satisfy the condition in Lemma 5.27.Then, the claim of the proposition follows from Lemma 7.23.The constructions V ( S ) and C ( S ) extend to functors V : SProxLat → SContEnt and C : SProxLat → SContEnt , which send each join-preserving proximity re-lation r : ( S, ≺ ) → ( S ′ , ≺ ′ ) to join-preserving proximity maps V ( r ) : V ( S ) → V ( S ′ ) and C ( r ) : C ( S ) → C ( S ′ ) defined by A V ( r ) B def ⇐⇒ ∃ C ∈ Fin ( S V ) ( A ⊢ V C & ∀h p, c i ∈ C ∃h q, b i ∈ B ( p > q & c r b )) ,A C ( r ) B def ⇐⇒ ∃ C ∈ Fin ( S V ) ( A ⊢ C C & ∀h p, c i ∈ C ∃h q, b i ∈ B ( p < q & c r b )) . Theorem 7.25. For any strong proximity lattice S , we have V ( S ) d ∼ = C ( S d ) and C ( S ) d ∼ = V ( S d ) . Proof. Immediate from Proposition 7.24 and Lemma 7.1.We now focus on probabilistic valuations and covaluations, i.e., those valua-tions µ and covaluations ν satisfying µ (1) = 1 and ν (0) = 1.31or a strong proximity lattice ( S, ≺ ), the space V P (Spec( S )) of probabilisticvaluations is presented by a geometric theory T V P which extends the theory T V with the following extra axioms: h p, a i ⊢ ⊥ (if 1 < p ) , ⊤ ⊢ h p, i (if p < . The space C P (Spec( S )) of probabilistic covaluations is presented by a geometrictheory T C P which extends the theory T C with the following extra axioms: ⊤ ⊢ h p, a i (if 1 < p ) , h p, i ⊢ ⊥ (if p < . Proposition 7.24 restricts to probabilistic valuations and covaluations. Proposition 7.26. For any strong proximity lattice ( S, ≺ ) , the locale V P (Spec( S )) can be presented by a strong continuous entailment relation V P ( S ) = ( S V , ⊢ V P , ≺ V ) where ⊢ V P is generated by the axioms of ⊢ V and the following extra axioms: h p, a i ⊢ V P ( if < p ) , ⊢ V P h p, i ( if p < . The locale C P (Spec( S )) can be presented by a strong continuous entailment re-lation C P ( S ) = ( S V , ⊢ C P , ≺ C ) where ⊢ C P is generated by the axioms of ⊢ C andthe following extra axioms: ⊢ C P h p, a i ( if < p ) , h p, i ⊢ C P ( if p < . In particular, the spaces of probabilistic valuations and probabilistic covalu-ations on a stably compact locale are stably compact. As a corollary we obtain the probabilistic version of Theorem 7.25. Theorem 7.27. For any strong proximity lattice S , we have V P ( S ) d ∼ = C P ( S d ) and C P ( S ) d ∼ = V P ( S d ) . For probabilistic valuations and covaluations, we have the following duality. Lemma 7.28. For any strong proximity lattice S , we have A ⊢ V P B ⇐⇒ A • ⊢ C P B • for all A, B ∈ Fin ( S V ) , where A • def = {h − p, a i | h p, a i ∈ A } . Proof. The direction ⇒ is proved by induction on the derivation of A ⊢ V P B .Note that each axiom A ⊢ V P B of V P ( S ) corresponds to an axiom A • ⊢ C P B • of C P ( S ). The direction ⇐ is similarly proved by induction on A • ⊢ C P B • .Since “1” in the lower and upper reals form a Dedekind real, the followingproposition is analogous to Vickers [25, Proposition 6.3], which holds for anarbitrary locale. We give a proof for the special case of stably compact locales. Proposition 7.29. For any strong proximity lattice S , we have V P ( S ) ∼ = C P ( S ) . roof. Define proximity maps r : V P ( S ) → C P ( S ) and s : C P ( S ) → V P ( S ) by A r B def ⇐⇒ A ≪ ⊢ VP B • , B s A def ⇐⇒ B ≪ ⊢ CP A • . Using Lemma 7.28, it is straightforward to show that r and s are indeed prox-imity maps which are inverse to each other. Theorem 7.30. For any strong proximity lattice S , we have V P ( S ) d ∼ = V P ( S d ) and C P ( S ) d ∼ = C P ( S d ) . Proof. By Theorem 7.27 and Proposition 7.29.Goubault-Larrecq [8, Theorem 6.11] proved the corresponding result for sta-bly compact spaces. Although his proof is classical and the space of covaluationsis implicit in his proof, the essential idea seems to be similar. Acknowledgements I thank the referees for numerous suggestions which help me improve the paperin an essential way. In particular, their suggestion to use entailment systemsallows me to simplify and constructivise many parts of the paper. I also thankSteve Vickers and Daniel Wessel for helpful discussions. This work was carriedout while I was in the Hausdorff Research Institute for Mathematics (HIM),University of Bonn, for their trimester program “Types, Sets and Constructions”(May–August 2018). I thank the institute for their support and the organisersof the program for creating a stimulating environment for research. 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