aa r X i v : . [ m a t h . C T ] S e p Duality theory for enriched Priestley spaces
Dirk Hofmann and Pedro Nora ∗ Center for Research and Development in Mathematics and Applications, Department of Mathematics,University of Aveiro, Portugal.Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany. [email protected] , [email protected] The term Stone-type duality often refers to a dual equivalence between a categoryof lattices or other partially ordered structures on one side and a category of topo-logical structures on the other. This paper is part of a larger endeavour that aimsto extend a web of Stone-type dualities from ordered to metric structures and, moregenerally, to quantale-enriched categories. In particular, we improve our previouswork and show how certain duality results for categories of [0 , , Naturally, the starting point of our investigation of Stone-type dualities is Stone’s classical 1936duality result(1.i)
BooSp ∼ BA op for Boolean algebras and homomorphisms together with its generalisation Spec ∼ DL op to distributive lattices and homomorphisms obtained shortly afterwards in [Stone, 1938]. Here BooSp denotes the category of Boolean spaces and continuous maps, and Spec the category ofspectral spaces and spectral maps (see also [Hochster, 1969]). In this paper we will often work ∗ This work is supported by the ERDF – European Regional Development Fund through the Operational Pro-gramme for Competitiveness and Internationalisation – COMPETE 2020 Programme, by German ResearchCouncil (DFG) under project GO 2161/1-2, and by The Center for Research and Development in Mathematicsand Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT – Fundaçãopara a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020. Also called Stone spaces in the literature, see [Johnstone, 1986], for instance.
Priestley spaces rather than with spectral spaces , and therefore consider the “equivalentequivalence”(1.ii)
Priest ∼ DL op discovered in [Priestley, 1970, 1972]. There are many ways to deduce the duality result (1.i)from (1.ii), we mention here one possibly lesser known argument: in [Brümmer et al. , 1992] itis observed that BA is the only epi-mono-firm epireflective full subcategory of DL , and, usingthat in both BooSp and
Priest the epimorphisms are precisely the surjective morphisms, an easycalculation shows that
BooSp is the only mono-epi-firm mono-coreflective full subcategory of
Priest .Exactly 20 years later, Halmos gave an extension of (1.i) to categories of continuous relations between Boolean spaces and hemimorphisms between Boolean algebras, and a similar generali-sation of
Priest ∼ DL op is described in [Cignoli et al. , 1991]. Denoting by • PriestDist the category of Priestley spaces and continuous monotone relations, by • FinSup the category of finitely cocomplete partially ordered sets and finite suprema pre-serving maps, and by • FinSup DL the full subcategory of FinSup defined by all distributive lattices;this result can be expressed as(1.iii)
PriestDist ∼ FinSup opDL . We note that
PriestDist is precisely the Kleisli category
Priest H of the Vietoris monad H =( H , w , h ) on Priest , and that the functor
PriestDist −→ FinSup opDL is a lifting of the hom-functor
PriestDist ( − ,
1) into the one-element space. Furthermore, recall that the two structures of aPriestley space – the partial order and the compact Hausdorff topology – can be combinedinto a single topology: the so-called downwards topology (see [Jung, 2004], for instance). Inparticular, the two-element Priestley space = { ≤ } produces the Sierpiński space with { } closed, whereby the dual space op of induces the topology on { , } with { } being theonly non-trivial open subset. With this notation, the elements of the Vietories space H X ofa Priestley space X can be identified with continuous maps ϕ : X −→ , whereby arrows oftype X −◦−→ PriestDist correspond to spectral maps ψ : X −→ H ≃ op . In order todeduce the equivalence (1.iii), it is important to establish that there are “enough” spectral maps ψ : X −→ op ; in fact, by definition, a partially ordered compact Hausdorff space X is Priestleywhenever the cone ( ψ : X −→ op ) ψ is point-separating and initial. Here it does not matter ifwe use or op since ≃ op in Priest ; however, when moving to the quantale-enriched setting,the corresponding property does not necessarily hold and therefore we must identify carefully ifwe refer to or to op .Under the equivalence (1.iii), continuous monotone functions correspond precisely to homo-morphisms of distributive lattices, therefore the equivalence Priest ∼ DL op is a direct conse-quence of (1.iii). Furthermore, other well-known duality results can be obtained from (1.iii) ina categorical way, we mention here the following examples.2 As (1.i) can be deduced from
Priest ∼ DL op , Halmos’s duality BooSpRel ∼ FinSup opBA between the category
BooSpRel of Boolean spaces and Boolean relations and the category
FinSup BA of Boolean algebras and hemimorphisms (that is, the full subcategory of FinSup defined by all Boolean algebras) can be deduced from (1.iii). • Combining
PriestDist ∼ FinSup opDL and
Priest ∼ DL op gives immediately the duality resultfor distributive lattices with an operator (see [Petrovich, 1996; Bonsangue et al. , 2007]). • The equivalence
PriestDist ∼ FinSup opDL has the surprising(?) consequence that
PriestDist isidempotent split complete. Hence, the idempotent split completion of
BooSpRel can be cal-culated as the full subcategory of
PriestDist defined by all split subobjects of Boolean spacesin
PriestDist ; likewise, the idempotent split completion of
FinSup BA can be taken as the fullsubcategory of FinSup DL defined by all split subobjects of Boolean algebras. Now, in theformer case, these split subobjects are precisely the so-called Esakia spaces (see [Esakia,1974]), and in the latter case precisely the co-Heyting algebras (see [McKinsey and Tarski,1946]); and we obtain a “relational version” of Esakia duality. We have described thismore in detail in [Hofmann and Nora, 2014].The situation is depicted in Figure 1. BooSp ∼ BA op Priest ∼ DL op PriestDist ∼ FinSup opDL
CoAlg( H ) ∼ DLO op EsaDist ∼ FinSup opHA
EsaSp ∼ HA op Figure 1: Stone type dualitiesOne might wish to consider all compact Hausdorff spaces in (1.i) instead of only the totallydisconnected ones. Then the two-element space = { , } , respectively the two-element Booleanalgebra, still induces naturally an adjunction CompHaus BA op ; hom( − , )hom( − , ) ⊥ however, its restriction to the fixed subcategories is precisely (1.i) (for the pertinent notionsof duality theory we refer to [Dimov and Tholen, 1989; Porst and Tholen, 1991]). In fact, bydefinition, a compact Hausdorff space X is Boolean whenever the cone ( f : X −→ ) f is point-separating and initial with respect to the forgetful CompHaus −→ Set ; likewise, a partiallyordered compact space X is Priestley whenever the cone ( f : X −→ ) f is point-separatingand initial with respect to the forgetful PosComp −→ Set (equivalently, to the forgetful functor
PosComp −→ CompHaus ). 3n order to obtain a duality result for all compact Hausdorff spaces this way, one needsto substitute the dualising object by a cogenerator in CompHaus , for instance, by the unitinterval [0 ,
1] with the Euclidean topology. Accordingly, one typically considers other types ofalgebras on the dual side; i.e. C ∗ -algebras instead of Boolean algebras. In contrast, our aimis to develop a duality theory where one actually keeps the “type of algebras” in Figure 1 butsubstitutes order by metric everywhere; that is, one considers [0 , ∞ ]-enriched categories insteadof -enriched categories (see [Lawvere, 1973]). Therefore one might attempt to create a networkof dual equivalences CompHaus ∼ (??) op PosComp ∼ (??) op PosCompDist ∼ (??) op CoAlg( H ) ∼ (??) op GEsaDist ∼ (??) op GEsaSp ∼ (??) op Figure 2: Metric Stone type dualitieswhere each “question mark category” should be substituted by its metric counterpart ofFigure 1, or even better, a quantale-enriched counterpart. For instance, for a quantale V ,instead of DL one would expect a category of V -categories with all “finite” weighted limits andcolimits and satisfying some sort of “distributivity” condition. Moreover, these results shouldhave the property that, when choosing the quantale V = , we get the original picture of Figure 1back.Unfortunately, the last requirement above does not make much sense . . . since the pic-ture of Figure 2 is somehow inconsequential: both sides of the equivalences should be gen-eralised to corresponding metric or even quantale-enriched versions. In particular, partially or-dered compact spaces should be substituted by their metric cousins as, for instance, studied in[Hofmann and Reis, 2018]. More specifically, we follow [Tholen, 2009] and consider the category V - Cat U of Eilenberg–Moore algebras and homomorphisms for the ultrafilter monad U on V - Cat .In analogy with the ordered case, we call an U -algebra Priestley whenever the cone of all homo-morphisms X −→ V op in V - Cat U is point-separating and initial. In [Hofmann and Nora, 2018]we made an attempt to create at least parts of this picture, for continuous quantale structureson the quantale V = [0 , BooSp op and Priest op are finitary varieties. It is known since the late 1960’s that also CompHaus op is a variety,not finitary but with rank ℵ (see [Duskin, 1969; Gabriel and Ulmer, 1971]); however, this factmight not be obvious from the classical Gelfand duality result CompHaus op ∼ C ∗ - Alg stating the equivalence between
CompHaus op and the category C ∗ - Alg of commutative C ⋆ -algebras and homomorphisms. Nonetheless, it can be deduced “abstractly” from the following4ell-known results. Theorem 1.1.
A cocomplete category is equivalent to a quasivariety if and only if it has aregular projective regular generator.Proof.
See, for instance, [Adámek, 2004, Theorem 3.6].
Theorem 1.2.
A category is a variety if and only if it is a quasivariety and has effectiveequivalence relations.Proof.
See, for instance, [Borceux, 1994, Theorem 4.4.5]Surprisingly, a similar investigation of
PosComp op was initiated only recently: in [Hofmann et al. ,2018] we show that PosComp op is a ℵ -ary quasivariety, and in [Abbadini, 2019; Abbadini and Reggio,2019] it is shown that PosComp op is indeed a ℵ -ary variety. In Section 4 we investigate the cat-egory V - Priest of V -enriched Priestley spaces and morphisms, with emphasis on those propertieswhich identify V - Priest op as some kind of algebraic category. In this section we recall the notions of quantale-enriched category and its generalisation tocompact Hausdorff spaces, which eventually leads to the notion of quantale-enriched Priestleyspace already studied in [Hofmann and Nora, 2018, 2020]. We recall some of the basic definitionsand properties, for more information we refer to [Kelly, 1982; Lawvere, 1973; Tholen, 2009] and[Hofmann et al. , 2014].
Definition 2.1. A quantale V = ( V , ⊗ , k ) is a complete lattice V equipped with a commutativemonoid structure ⊗ , with identity k , so that, for each u ∈ V , u ⊗ − : V −→ V has a right adjoint hom( u, − ) : V −→ V . Definition 2.2.
Let V = ( V , ⊗ , k ) be a quantale.1. A V -category is a pair ( X, a ) consisting of a set X and a map a : X × X −→ V satisfying k ≤ a ( x, x ) and a ( x, y ) ⊗ a ( y, z ) ≤ a ( x, z ) , for all x, y, z ∈ X . Furthermore, a V -category ( X, a ) is called separated whenever( k ≤ a ( x, y ) and k ≤ a ( y, x )) = ⇒ x = y, for all x, y ∈ X .2. A V -functor f : ( X, a ) −→ ( Y, b ) between V -categories is a map f : X −→ Y such that a ( x, x ′ ) ≤ b ( f ( x ) , f ( x ′ )) , for all x, x ′ ∈ X .3. Finally, V -categories and V -functors define the category V - Cat , and its full subcategorydefined by separated V -categories is denoted by V - Cat sep .5e note that there is a canonical forgetful functor V - Cat −→ Set sending the V -category( X, a ) to the set X . For every V -category X = ( X, a ), the dual V -category X op is defined as X op = ( X, a ◦ ) where a ◦ ( x, y ) = a ( y, x ) , for all x, y ∈ X . In fact, this construction defines a functor ( − ) op : V - Cat −→ V - Cat commutingwith the forgetful functor to
Set . Examples 2.3.
Below we list some of the principal examples, for more details we refer, forinstance, to [Hofmann and Reis, 2018].1. The two element chain = { ≤ } with ⊗ = & and k = 1. Then - Cat ∼ Ord .2. The extended real half line ←−−− [0 , ∞ ] ordered by the “greater or equal” relation > and • the tensor product given by addition +, denoted by ←−−− [0 , ∞ ] + ; • or with ⊗ = max, denoted as ←−−− [0 , ∞ ] ∧ .Then ←−−− [0 , ∞ ] + - Cat ∼ Met is the category of (generalised) metric spaces and non-expansivemaps and ←−−− [0 , ∞ ] ∧ - Cat ∼ UMet is the category of (generalised) ultrametric spaces andnon-expansive maps.3. The unit interval [0 ,
1] with the “greater or equal” relation > and the tensor u ⊕ v =min { , u + v } , denoted as ←−− [0 , ⊕ . Then ←−− [0 , ⊕ - Cat ∼ BMet is the category of (generalised)bounded-by-one metric spaces and non-expansive maps.4. The unit interval [0 ,
1] with the usual order and ⊗ = ∧ the minimum, or ⊗ = ∗ theusual multiplication, or ⊗ = ⊙ the Lukasiewicz sum defined by u ⊙ v = max { , u + v − } .Then [0 , ∧ - Cat ∼ UMet , [0 , ∗ - Cat ∼ Met , and [0 , ⊙ - Cat ∼ BMet . Example 2.4.
The notion of probabilistic metric space goes back to [Menger, 1942]. Here a probabilistic metric on a set X is a map d : X × X × [0 , ∞ ] −→ [0 , d ( x, y, t ) = u means that u is the probability that the distance from x to y is less then t . Similar to a classicmetric, such a map is required to satisfy the following conditions:0. d ( x, y, − ) : [0 , ∞ ] −→ [0 ,
1] is left continuous,1. d ( x, x, t ) = 1 for t > d ( x, y, r ) ∗ d ( y, z, s ) ≤ d ( x, z, r + s ),3. d ( x, y, t ) = 1 = d ( y, x, t ) for all t > x = y ,4. d ( x, y, t ) = d ( y, x, t ) for all t ,5. d ( x, y, ∞ ) = 1.The complete lattice D = { f : [0 , ∞ ] −→ [0 , | f ( t ) = _ s 1] with κ (0) = 0 and κ ( t ) = 1 for t > 0. In theformula above, one may substitute the multiplication ∗ by any other tensor ⊗ : [0 , × [0 , −→ [0 , d : X × X −→ D , and conditions (1) and(2) read as κ ≤ d ( x, x ) and d ( x, y ) ⊗ d ( y, z ) ≤ d ( x, z ) . Hence D - Cat ∼ ProbMet is the category of (generalised) probabilistic metric spaces and non-expansive maps.Before adding a topological component to the theory of V -categories, we collect some well-known properties of V -categories and V -functors. Theorem 2.5. The canonical forgetful functor V - Cat −→ Set is topological. Here a cone ( f i : ( X, a ) −→ ( X i , a i )) i ∈ I in V - Cat is initial respect to V - Cat −→ Set if and only if, for all x, y ∈ X , a ( x, y ) = ^ i ∈ I a i ( f i ( x ) , f i ( y )) . Therefore V - Cat has concrete limits and colimits and a (surjective, initial monocone)-factorisationsystem; moreover, V - Cat −→ Set has a right adjoint Set −→ V - Cat (indiscrete structures) anda left adjoint D : Set −→ V - Cat (discrete structures). Furthermore, a morphism f : X −→ Y in V - Cat is1. a monomorphism if and and only if f is injective,2. a regular monomorphism if and only if f is an embedding with respect to V - Cat −→ Set ,3. an epimorphism if and only if f is surjective. Proposition 2.6. The V -category V = ( V , hom) is injective with respect to embeddings and,for every V -category X , the cone ( f : X −→ V ) f is initial with respect to the forgetful functor V - Cat −→ Set .Remark . Since ( − ) op : V - Cat −→ V - Cat is a concrete isomorphism, Proposition 2.6 appliesalso to the V -category V op in lieu of V .In the remainder of this section we assume that the lattice V is completely distributive , werefer to [Wood, 2004] for the definition and an extensive discussion of properties of this notion.In particular, under this assumption it is useful to consider the totally below relation ≪ onthe lattice V , which is defined by v ≪ u whenever u ≤ _ A = ⇒ v ∈ ↓ A, for every subset A of V . Assumption 2.8. The underlying lattice of the quantale V is completely distributive.7 emark . Regarding the various topologies on V we have the following facts, for more infor-mation see [Gierz et al. , 2003].1. The Lawson topology on the completely distributive lattice V is compact Hausdorff. Withrespect to this topology, as shown in [Gierz et al. , 2003, Proposition VII-3.10], an ultrafilter v in V converges to ξ ( v ) = ^ A ∈ v _ A ∈ V . Moreover, the Scott topology respectively its dual topology have the following conver-gences:Scott topology: v → x ⇐⇒ ξ ( v ) ≥ x ,Dual of Scott topology: v → x ⇐⇒ ξ ( v ) ≤ x .2. By [Gierz et al. , 2003, Lemma VII-2.7] and [Gierz et al. , 2003, Proposition VII-2.10], theLawson topology of V coincides with the Lawson topology of V op , and the set {↑ u | u ∈ V} ∪ {↓ u | u ∈ V} is a subbasis for the closed sets of this topology which is known as the interval topology.3. The sets ↑ v = { u ∈ V | v ≤ u } ( v ∈ V )form a subbase for the closed sets of the dual of the Scott topology of V (see [Gierz et al. ,2003, Proposition VI-6.24]). We denote (the convergence of) this topology by ξ ≤ .4. The convergence ξ : U V −→ V together with the ultrafilter monad U = ( U , m, e ) andthe quantale V defines a topological theory in the sense of [Hofmann, 2007], and thereforeallows for an extension of the ultrafilter monad U to a monad on V - Cat (see [Tholen, 2009]).We denote the corresponding Eilenberg–Moore category V - Cat U by V - CatCH , and refer toits objects as V -categorical compact Hausdorff spaces (see also [Hofmann and Reis,2018]). In more detail, a V -categorical compact Hausdorff space is a triple ( X, a, α ) where • ( X, a ) is a V -category and • α : U X −→ X is the convergence of a compact Hausdorff topology on X such that α : ( U X, U a ) −→ ( X, a ) is a V -functor. Example 2.10. The triple V = ( V , hom , ξ ) is a V -categorical compact Hausdorff space. More-over, for a V -categorical compact Hausdorff space X = ( X, a, α ), also X op = ( X, a ◦ , α ) is a V -categorical compact Hausdorff space. Example 2.11. As it is pointed out in [Tholen, 2009], -categorical compact Hausdorff spacesare precisely Nachbin’s ordered compact Hausdorff spaces. Proposition 2.12. For a quantale V , the sets { u ∈ V | v ≪ u } ( v ∈ V ) form a subbase for its Scott topology. roof. We start by proving that for every v ∈ V the set { u ∈ V | v ≪ u } is open. Let v be anultrafilter in V that converges to u ∈ V such that v ≪ u . The properties of the totally belowrelation guarantee that there exists w ∈ V such that v ≪ w ≪ u . Then, by Remark 2.9 (1),for every A ∈ v , u ≤ W A . Hence, for every A ∈ v there exists a ∈ A such that w ≤ a . Therefore,for every A ∈ v , A ∩ { u ∈ V | v ≪ u } 6 = ∅ . We show now that the sets { u ∈ V | v ≪ u } ( v ∈ V ) induce the convergence of the Scotttopology. Let w be an element of V and v and ultrafilter on V such that, for every v ≪ w in V ,the set { u ∈ V | v ≪ u } belongs to v . Then, since V is completely distributive, we have w = _ v ≪ w v ≤ _ A ∈ v ^ A = ξ ( v ) . Remark . For a point-separating cone ( f i : ( X, a, α ) −→ ( X i , a i , α i )) i ∈ I in V - CatCH , thefollowing assertions are equivalent, for details see [Tholen, 2009].(i) For all x, y ∈ X , a ( x, y ) = ^ i ∈ I a i ( f i ( x ) , f i ( y )).(ii) ( f : ( X, a, α ) −→ ( X i , a i , α i )) i ∈ I is initial with respect to V - CatCH −→ CompHaus .(iii) ( f : ( X, a, α ) −→ ( X i , a i , α i )) i ∈ I is initial with respect to V - CatCH −→ Set .In the sequel we will simply say “initial” when referring to either of these forgetful functors. Wealso note that a cone ( f i : ( X, a, α ) −→ ( X i , a i , α i )) i ∈ I is point-separating if and only if it is amonocone in V - CatCH . Theorem 2.14. The category V - CatCH is monadic over V - Cat and topological over CompHaus ,hence V - CatCH is complete and cocomplete and has a (surjective, initial monocone)-factorisationsystem.Proof. See [Tholen, 2009]. Definition 2.15. A V -categorical compact Hausdorff space X is called Priestley whenever thecone V - CatCH ( X, V op ) is point-separating and initial with respect to V - CatCH −→ CompHaus . Example 2.16. For V = , the notion of Priestley space coincides with the classical one. Remark . By definition, the V -categorical compact Hausdorff space V op is Priestley. More-over, every finite separated V -categorical compact Hausdorff space is Priestley.We denote the full subcategory of V - CatCH defined by all Priestley spaces by V - Priest . Dueto well-known facts about factorisation structures for cones (see [Adámek et al. , 1990]), we havethe following: Proposition 2.18. The full subcategory V - Priest of V - CatCH is reflective. We denote the left adjoint of the inclusion functor V - Priest −→ V - CatCH by π : V - CatCH −→V - Priest . 9 roof. For each X in V - CatCH , its reflection X −→ π ( X ) into V - Priest is given by the (surjec-tive, initial monocone)-factorisation of the cone ( ϕ : X −→ V op ) ϕ of all morphisms from X to V op in V - CatCH . X π ( X ) V op ϕ e ϕ To show that this construction defines indeed a left adjoint to V - Priest −→ V - CatCH , consider f : X −→ Y in V - CatCH where Y is Priestley. Then, for every ϕ : Y −→ V op , there is somearrow e ϕ : π ( X ) −→ V op making the diagram(2.i) X π ( X ) Y V op f e ϕϕ commute. Since the top arrow of (2.i) is surjective and the cone ( ϕ : Y −→ V op ) ϕ is point-separating and initial, there is a diagonal arrow ¯ f : π ( X ) −→ Y in (2.i) making in particularthe diagram X π ( X ) Y f ¯ f commute. Corollary 2.19. The category V - Priest is complete and cocomplete. We already observed in [Hofmann and Nora, 2020, Remark 4.52] that a monocone in V - Priest isinitial with respect to V - Priest −→ Set if and only if it is initial with respect to V - CatCH −→ Set (the same argument as in the proof of [Hofmann and Nora, 2020, Theorem A.6] applies here). Atthis moment we do not know whether, for instance, every separated metric compact Hausdorffspace is Priestley. However, since [0 , op is an initial cogenerator in PosComp (see [Nachbin,1965]), we have the following fact. Proposition 2.20. The inclusion functor PosComp −→ [0 , - CatCH corestricts to PosComp −→ [0 , - Priest . In this section we build on the duality results of [Hofmann and Nora, 2018] for Priestley spacesenriched in the complete lattice [0 , 1] with a continuous quantale structure ⊗ : [0 , × [0 , −→ [0 , 1] and with neutral element 1. We recall some of the principal results, and then, for theŁukasiewicz tensor on [0 , V -relational duality results obtained in[Hofmann and Nora, 2018, Section 9] to categories of functions.In analogy with the classical situation, our starting point is the category [0 , FinSup of finitelycocomplete [0 , , Theorem 3.1. The category [0 , - FinSup is a ℵ -ary quasivariety. roof. See [Hofmann and Nora, 2018, Remark 2.10].In the sequel we consider the Vietoris monad H = ( H , w , h ) on the category PosComp ofpartially ordered compact Hausdorff spaces and monotone continuous maps, more informationon power constructions in topology can be found in [Schalk, 1993a,b]. In our previous work[Hofmann and Nora, 2018; Hofmann et al. , 2019] we used the notation V instead of H ; however,in this paper we think of the classic Vietoris topology [Vietoris, 1922] as an extension of theHausdorff metric and reserve the designation V for the monad based on presheafs X −→ V rather than subsets A ⊆ X . Similarly to the -enriched case mentioned in the Introduction, weobtain the commutative diagram PosComp H [0 , FinSup op PosComp C C =hom( − , [0 , op ) of functors. However, unlike hom( − , 1) : Priest H −→ FinSup op , the functor C : PosComp H −→ [0 , FinSup op is not fully faithful, as the next example shows. Example 3.2. As observed in [Hofmann and Nora, 2018, Example 6.16], for every u ∈ [0 , u ⊗ − : [0 , −→ [0 , 1] is a morphism in [0 , FinSup sending 1 to u . On the other hand,there are only two morphisms of type 1 −◦−→ PosComp H .Therefore we have to consider further structure on the right-hand side. The starting point isthe following observation. Theorem 3.3. The category [0 , - FinSup has a bimorphism representing monoidal structure.Proof. See [Kelly, 1982, Section 6.5].This leads us to the category Mnd([0 , FinSup )of monoids and homomorphisms in [0 , FinSup with respect to the above-mentioned monoidalstructure and with neutral element the top-element , and to the categoryLaxMnd([0 , FinSup )with the same objects as Mnd([0 , FinSup ), but now with morphisms those of [0 , FinSup thatpreserve the monoid structure laxly:Φ( ψ ⊗ ψ ) ≤ Φ( ψ ) ⊗ Φ( ψ ) . We obtain the commutative diagram PosComp H LaxMnd([0 , FinSup ) op PosComp C ⊤ ⊣ C =hom( − , [0 , op ) of functors (represented by solid arrows), and the induced monad morphism j = ( j X ) X is givenby the family of maps 11 X : H X −→ [ C X, [0 , , A Φ A ,with Φ A : C X −→ [0 , , ψ sup x ∈ A ψ ( x ). Proposition 3.4. Let X be in PosComp and A ⊆ X closed and upper. Then A is irreducible ifand only if Φ A satisfies Φ A (1) = 1 and Φ A ( ψ ⊗ ψ ) = Φ A ( ψ ) ⊗ Φ A ( ψ ) .Proof. See [Hofmann and Nora, 2018, Proposition 6.7]. Corollary 3.5. Let ϕ : X −◦−→ Y be a morphism in PosComp H . Then ϕ is a function if andonly if Cϕ is a morphism in Mnd([0 , - FinSup ) . Theorem 3.6. For ⊗ = ∗ or ⊗ = ⊙ , the monad morphism j is an isomorphism. Therefore thefunctors C : PosComp H −→ LaxMnd([0 , - FinSup ) op and C : PosComp −→ Mnd([0 , - FinSup ) op are fully faithful.Proof. See [Hofmann and Nora, 2018, Theorem 6.14 and Corollary 6.15].Theorem 3.6 does not extend to arbitrary continuous quantale structures on [0 , 1] since, byExample 3.2, the functor C : PosComp H −→ LaxMnd([0 , ∧ - FinSup ) op is not full. In fact, thisexample also shows that its restriction C : CompHaus H −→ LaxMnd([0 , ∧ - FinSup ) op to compactHausdorff spaces is not full. However, passing from relations to functions improves the situation:it is shown in [Banaschewski, 1983] that the functor C : CompHaus −→ Mnd([0 , ∧ - FinSup ) op isfully faithful (see also [Hofmann and Nora, 2018, Remark 2.7]). This result generalises to oursetting. Theorem 3.7. The functor C : CompHaus −→ Mnd([0 , - FinSup ) op is fully faithful.Proof. See [Hofmann and Nora, 2018, Theorem 6.23]. Remark . To identify the image of the functor of Theorem 3.7, we can proceed as in Section 7of [Hofmann and Nora, 2018], although with a small adjustment. Since we consider now “initialwith respect to CompHaus −→ Set ” instead of “initial with respect to PosComp −→ Set ” in[Hofmann and Nora, 2018, Lemma 7.3 and 7.4], at the beginning of the proof we do not kownwhether “ ψ ( x ) > ψ ( y ) or ψ ( x ) < ψ ( y )”. We can remedy the situation by requiring that L is alsoclosed in CX under an additional unary operation, which in [0 , 1] is interpreted as u − u .This new operation acts as a “complement”, and introducing it corresponds to the passagefrom distributive lattices to Boolean algebras in the classical case (see also [Hofmann, 2002b,Example 3.5]).However, Banaschewski’s result does not extend to partially ordered compact spaces, as thefollowing example shows. 12 xample 3.9. The functor C : PosComp −→ Mnd([0 , ∧ - FinSup ) op is not full. As pointed out in[Hofmann and Nora, 2018, Example 6.16], for the separated ordered compact space X = { > } , C X = { ( u, v ) ∈ [0 , × [0 , | u ≤ v } . H X contains three elements; however, for every w ∈ [0 , w : C X −→ [0 , , ( u, v ) u ∨ ( w ∧ v )is a morphism in Mnd([0 , ∧ - FinSup ) with Φ w (0 , 1) = w . Theorem 3.10. We consider an additional operation ⊖ in our theory (which is interpreted astruncated minus in [0 , ). Then C : PosComp H −→ LaxMnd ⊖ ([0 , - FinSup ) op is fully faithful.Proof. See [Theorem 6.19 Hofmann and Nora, 2018].As we already observed in [Hofmann and Nora, 2018], the setting above is not really conse-quential since we still consider ordered compact Hausdorff spaces as well as the Vietoris functorbased on subsets , that is, continuous functions into the Sierpiński space . We obtain resultscloser to the classical case by also enriching the topological side. That is, we consider en-riched Priestley spaces and the enriched Vietoris monad V = ( V , w , h ). The latter is intro-duced in [Hofmann, 2014] in the context of U -categories and U -functors, for a topological theory U = ( U , V , ξ ) based on the ultrafilter monad U = ( U , m, e ). For an overview of the backgroundtheory we refer to [Hofmann, 2014, Section 1], and mention here only that • an U -category ( X, a ) is given by a set X and a map a : U X × X −→ V satisfying twoaxioms similar to the ones of a V -category, • the category of U -categories and U -functors is denoted as U - Cat , • by combining the internal hom and the convergence ξ : U V −→ V , the quantale V becomesan U -category where ( v , v ) hom( ξ ( v ) , v ), • the underlying set of V X is the set { all U -functors ϕ : X −→ V} . For V = , U -categories correspond to topological spaces and U -functors to continuous maps(see [Barr, 1970]), and V = is the Sierpiński space where { } is closed. On the other hand,for the multiplication ∗ on [0 , U -category is essentially an approach space (see [Lowen,1997]), thanks to the isomorphism of quantales [0 , ∗ ≃ ←−−− [0 , ∞ ] + . Remark . An interesting connection between topological theories and lax distributive lawsis exposed in [Tholen, 2019]. Theorem 3.12. If ⊗ = ∗ , ⊗ = ∧ or ⊗ = ⊙ , then the monad V = ( V , w , h ) on U - Cat restrictsto [0 , - Priest .Proof. See [Hofmann and Nora, 2018, Corollary 9.7].13e obtain now the commutative diagram[0 , Priest V [0 , FinSup op [0 , Priest C ⊤ ⊣ C =hom( − , [0 , op ) of functors (represented by solid arrows), we stress that here the functor C : [0 , Priest V −→ [0 , FinSup op is a lifting of the hom-functor hom( − , X -component of the inducedmonad morphism j is given by j X : V X −→ [ C X, [0 , , ( ϕ : 1 −◦−→ X ) ψ ψ · ϕ = _ x ∈ X ( ψ ( x ) ⊗ ϕ ( x )) ! . Theorem 3.13. If ⊗ = ∗ or ⊗ = ⊙ , then the monad morphism j is an isomorphism. Conse-quently, the functor (3.i) C : [0 , - Priest V −→ [0 , - FinSup op is fully faithful.Proof. See [Hofmann and Nora, 2018, Theorem 9.10].We recall that, in the classic case PriestDist −→ FinSup op mentioned in the Introduction, wecan first restrict the objects on the right-hand side to (distributive) lattices, and then observethat those continuous distributors coming from continuous monotone maps correspond preciselyto lattice homomorphisms on the right-hand side. We aim now at a similar result for the fullyfaithful functor (3.i). To do so, we wish to identify those [0 , C X −→ [0 , 1] whichcorrespond to “the points of X inside V X ”; that is, to the U -functors of the form a ( (cid:5) x, − ) : X −→ [0 , , 1] employing the factthat the quantale [0 , ⊙ is a Girard quantale : for every u ∈ [0 , u = hom(hom( u, ⊥ ) , ⊥ ).We recall that hom( u, ⊥ ) = 1 − u and put u ⊥ = 1 − u . Also note that ( − ) ⊥ : [0 , −→ [0 , op is an isomorphism in [0 , ⊙ - Priest .In a nutshell, our strategy is the same as in the ordered case: we show that an additionalproperty on Φ translates to “ ϕ : X −→ [0 , 1] is irreducible”, and “soberness” of X guaran-tees ϕ = a ( (cid:5) x, − ), for some x ∈ X . Hence, we need to introduce these notions for U -categories,which fortunately was already done in [Clementino and Hofmann, 2009]. In our context, “sober”means Cauchy-complete (called Lawvere complete in [Clementino and Hofmann, 2009]) and “ir-reducible” means left adjoint U -distributor . We do not introduce these notions here but ratherrefer to the before-mentioned literature; however, we recall the following two results. Theorem 3.14. An U -functor ϕ : X −→ [0 , (viewed as an U -distributor from to X ) is leftadjoint if and only if the representable [0 , -functor [ ϕ, − ] : U - Cat ( X, [0 , −→ [0 , , ϕ ′ ^ x ∈ X hom( ϕ ( x ) , ϕ ′ ( x )) preserves copowers and finite suprema. roof. See [Hofmann and Stubbe, 2011, Proposition 3.5]. Theorem 3.15. Every V -categorical compact Hausdorff space X is Cauchy complete (viewed asan U -category); that is, every left adjoint U -distributor ϕ from to X is of the form ϕ = a ( (cid:5) x, − ) ,for some x ∈ X .Proof. See [Hofmann and Reis, 2018, Corollary 4.18].Let now ϕ : X −→ [0 , 1] be an U ⊙ -functor. To link Theorem 3.14 with our situation, we view ϕ as a [0 , ⊙ -distributor ϕ : 1 −◦−→ X and note that[0 , Dist ( X, 1) [0 , Dist (1 , X ) op [0 , 1] [0 , op( − ) ⊥ ( −· ϕ ) [ ϕ, − ] op ( − ) ⊥ commutes in [0 , ⊙ - Cat (see [Hofmann and Reis, 2018, Proposition 4.35]). Furthermore, we canrestrict the top line of diagram above to the [0 , ⊙ -functor( − ) ⊥ : U ⊙ - Cat ( X, [0 , op ) −→ U ⊙ - Cat ( X, [0 , op , which implies at once: Proposition 3.16. An U ⊙ -functor ϕ : X −→ [0 , is a left adjoint U ⊙ -distributor ϕ : 1 −◦−→ X if and only if the [0 , ⊙ -functor ( − · ϕ ) : U - Cat ( X, [0 , op ) −→ [0 , preserves powers and finiteinfima. Finally, for an object X in [0 , ⊙ - Priest , we will show that the inclusion [0 , ⊙ -functor C X ֒ → U ⊙ - Cat ( X, [0 , op ) is W -dense. This property guarantees that −· ϕ : U ⊙ - Cat ( X, [0 , op ) −→ [0 , − · ϕ : CX −→ [0 , 1] does so.For every U ⊙ -category ( X, a ), the [0 , ⊙ -subcategory(3.ii) { all U ⊙ -functors ϕ : X −→ [0 , } ⊆ [0 , X is closed under weighted limits and finite weighted colimits; we shall show now that this propertycharacterises the collection of all U ⊙ -functors ϕ : X −→ [0 , , R ⊆ [0 , X closed under weighted limits and finite weightedcolimits corresponds to a monad µ : [0 , X −→ [0 , X (1 ≤ µ, µµ ≤ µ )where the [0 , µ preserves finite weighted colimits. Here, given R ⊆ [0 , X , µ ( α ) = ^ { ϕ | ϕ ∈ R , α ≤ ϕ } , and, for a monad µ : [0 , X −→ [0 , X , R = { α ∈ [0 , X | µ ( α ) = α } . For a subset A ⊆ X , we write χ A : X −→ [0 , 1] for the characteristic function of A . The followingkey result is essentially [Lowen, 1997, Proposition 1.6.5].15 roposition 3.17. Let µ, µ ′ : [0 , X −→ [0 , X be monads that preserve finite weighted colimits.Then µ = µ ′ if and only if µ ( χ A ) = µ ′ ( χ A ) , for all A ⊆ X . Note that, for R ⊆ [0 , X closed under weighted limits and finite weighted colimits and withcorresponding monad µ , we have µ ( χ A )( x ) = ^ { ϕ | ϕ ∈ R , χ A ≤ ϕ } = ^ { ϕ | ϕ ∈ R and, for all z ∈ A , ϕ ( z ) = 1 } , for all x ∈ X . For a U -category ( X, a ), the monad µ corresponding to (3.ii) is given by µ ( α )( x ) = _ x ∈ UX a ( x , x ) ⊙ ξU α ( x ) , for all α ∈ [0 , X . In particular, for every A ⊆ X , µ ( χ A )( x ) = _ x ∈ UX a ( x , x ) ⊙ ξU χ A ( x ) , = _ x ∈ UA a ( x , x ) , for all x ∈ X . Lemma 3.18. Let R ⊆ [0 , X be a [0 , ⊙ -subcategory closed under weighted limits and finiteweighted colimits and a : U X × X −→ [0 , be the initial convergence induced by the cone ( ϕ : X −→ [0 , ϕ ∈R in U ⊙ - Cat . Then the following assertions hold.1. a ( x , x ) = V { ϕ ( x ) | ϕ ∈ R , ξU ϕ ( x ) = 1 } , for all x ∈ U X and x ∈ X .2. For all A ⊆ X and x ∈ X , ^ { ϕ ( x ) | ϕ ∈ R and, for all z ∈ A , ϕ ( z ) = 1 } = _ x ∈ UA a ( x , x ) . Proof. To see the first statement, note that a ( x , x ) = ^ { hom( ξU ϕ ( x ) , ϕ ( x )) | ϕ ∈ R} ≤ ^ { ϕ ( x ) | ϕ ∈ R , ξU ϕ ( x ) = 1 } . On the other hand, for every ϕ ∈ R , put u = ξU ϕ ( x ). Then hom( u, ϕ ) ∈ R and ξU (hom( u, ϕ ))( x ) =1, which proves the assertion. Regarding the second statement, the inequality ^ { ϕ ( x ) | ϕ ∈ R and, for all z ∈ A , ϕ ( z ) = 1 } ≥ _ x ∈ UA a ( x , x )is certainly true. To see the opposite inequality, put u = ^ { ϕ ( x ) | ϕ ∈ R and, for all z ∈ A , ϕ ( z ) = 1 } . Let v < u und put ε = u − v , then hom( u, v ) = 1 − ε . For every ϕ ∈ R with ϕ ( x ) < v , there existssome z ∈ A with ϕ ( z ) < − ε . In fact, if ϕ ( z ) ≥ − ε for all z ∈ A , then hom(1 − ε, ϕ ( z )) = 1for all z ∈ A , but hom(1 − ε, ϕ ( x )) = ϕ ( x ) + ε < u . Therefore f = { ϕ − ([0 , − ε ]) | ϕ ∈ R , ϕ ( x ) < v } ∪ { A } is a filter base, let x be an ultrafilter finer than f . Then, for every ϕ ∈ R with ϕ ( x ) < v , ξU ϕ ( x ) ≤ − ε . Therefore a ( x , x ) = ^ { ϕ ( x ) | ϕ ∈ R , ξU ϕ ( x ) = 1 } ≥ v. Corollary 3.19. Let R ⊆ [0 , X be a [0 , ⊙ -subcategory closed under weighted limits and finiteweighted colimits. Then R = { all U ⊙ -functors ϕ : X −→ [0 , } , where we consider on X the initial convergence a : U X × X −→ [0 , induced by R . Corollary 3.20. Let R , R ′ ⊆ [0 , X be [0 , ⊙ -subcategories closed under weighted limits andfinite weighted colimits. If R and R ′ induce the same convergence, then R = R ′ . We return now [0 , ⊙ -enriched Priestley spaces. Corollary 3.21. Let X be in [0 , ⊙ - Priest and R be the closure of [0 , ⊙ - Priest ( X, [0 , in [0 , X under infima. Then the [0 , ⊙ -subcategory R ⊆ [0 , X is closed under weighted limitsand finite weighted colimits.Proof. Since the maps ∨ : [0 , × [0 , −→ [0 , , [0 , −→ [0 , , u , ∧ : [0 , × [0 , −→ [0 , , [0 , −→ [0 , , u u, − ) : [0 , −→ [0 , 1] and u ⊙ − : [0 , −→ [0 , 1] ( u ∈ [0 , , ⊙ - Priest , the [0 , ⊙ -subcategory [0 , ⊙ - Priest ( X, [0 , , X is closedunder finite weighted limits and finite weighted colimits. Clearly, R ⊆ [0 , X is closed under allweighted limits. Since ^ i ∈ I ϕ i ! ∨ ^ i ∈ J ϕ j ! = ^ ( i,j ) ∈ I × J ( ϕ i ∨ ϕ j ) , R is closed in [0 , X under binary suprema, and R is closed in [0 , X under tensors since u ⊙ − preserves non-empty infima. Corollary 3.22. Let X be in [0 , ⊙ - Priest . Then every U ⊙ -functor X −→ [0 , is an infimumof morphisms X −→ [0 , in [0 , ⊙ - Priest .Proof. Since [0 , ≃ [0 , op in [0 , ⊙ - Cat U and X is Priestley, the cone [0 , ⊙ - Priest ( X, [0 , , ⊙ - CatCH −→ CompHaus . Then, since the func-tor K : [0 , ⊙ - CatCH −→ U ⊙ - Cat preserves initial mono-cones, the closure of [0 , ⊙ - Priest ( X, [0 , , X under infima coincides with U ⊙ - Cat ( X, [0 , − ) ⊥ : [0 , −→ [0 , op , we obtain the desired result. Corollary 3.23. For every X in [0 , ⊙ - Priest , the inclusion C X ֒ → U ⊙ - Cat ( X, [0 , op ) is W -dense. Therefore, for every U ⊙ -functor ϕ : X −→ [0 , , the [0 , ⊙ -functor ( − · ϕ ) : U ⊙ - Cat ( X, [0 , op ) −→ [0 , preserves finite weighted limits if and only if the [0 , ⊙ -functor ( − · ϕ ) : C X −→ [0 , does so. 17e let [0 , ⊙ - FinLat denote the category of finitely complete and finitely cocomplete [0 , ⊙ -categories and [0 , ⊙ -functors that preserve finite weighted limits and colimits. We note that[0 , ⊙ - FinLat is a ℵ -ary quasivariety which can be shown as in [Hofmann and Nora, 2018,Remark 2.10] by adding operations and equations for powers and finite infima. From the resultsabove we obtain: Theorem 3.24. The fully faithful functor C : ([0 , ⊙ - Priest V ) op −→ [0 , ⊙ - FinSup restricts to a fully faithful adjoint functor C : ([0 , ⊙ - Priest ) op −→ [0 , ⊙ - FinLat . In Section 3 we presented some duality results for the category [0 , ⊙ - Priest which in particularexpose some algebraic flavour of [0 , ⊙ - Priest op . For a general quantale V , we are still faraway from concrete duality results, and in this section we investigate properties of V -categoricalcompact Hausdorff spaces which help us to recognise (cid:0) V - Priest (cid:1) op as some sort of algebraiccategory.Since we will use it frequently, below we recall an intrinsic characterisation of cofiltered limitsin CompHaus which goes back to [Bourbaki, 1942]. We refer to this result commonly as the Bourbaki-criterion . Theorem 4.1. Let D : I −→ CompHaus be a cofiltered diagram. Then a cone ( p i : L −→ D ( i )) i ∈ I for D is a limit cone if and only if1. ( p i : L −→ D ( i )) i ∈ I is point-separating, and2. for every i ∈ I , \ j → i im D ( j → i ) = im p i . That is, “the image of each p i is as large as possible”.Remark . The second condition above is automatically satisfied if p i : X −→ D ( i ) is surjective. Remark . The Bourbaki-criterion applies also to complete categories A with a limit preservingfaithful functor | − | : A −→ CompHaus . In this case, the first condition above reads as( p i : L −→ D ( i )) i ∈ I is point-separating and initial with respect to the functor | − | : A −→ CompHaus . Example 4.4. From the Bourbaki-criterion it follows at once that, for instance, every Priestleyspace X is a cofiltered limit of finite Priestley spaces. In fact, let ( p i : X −→ X i ) i ∈ I be thecanonical cone for the canonical diagram of X with respect to all finite spaces. Clearly, the cone18 p i : X −→ X i ) i ∈ I is point-separating and initial since is finite. For every index i , consider theimage factorisation of p i . X finite spaces: X i im( p i ) p i Since im( p i ) ֒ → X i belongs to the diagram, the second condition is satisfied.We can deduce in a similar fashion the well-known facts that every Boolean space X is acofiltered limit of finite spaces, every compact Hausdorff space is a cofiltered limit of metrizablecompact Hausdorff spaces, and so on. Remark . The classic Stone/Priestley duality Priest op ∼ DL implies in particular that Priest op is a finitary variety, a fact which can also be seen abstractly using Theorems 1.1 and 1.2. Belowwe explain the argument in some detail as it serves as a motivation for the investigation in theremainder of this section.1. Priest has all limits and colimits. This is well-known, but we stress that it is a special caseof Corollary 2.19.2. Every embedding in Priest is a regular monomorphism; therefore the class of embeddingscoincides with the class of regular monomorphisms. We use the argument of [Hofmann,2002b, Lemma 4.8]: for an embedding m : X −→ Y in Priest , consider a presentation( q i : Y −→ Y i ) i ∈ I as a cofiltered limit of finite Priestley spaces (= finite partially orderedsets). For every i ∈ I , take the (surjetive, embedding)-factorisation X p i −−−−−→ X i m i −−−−−→ Y i of q i · m . Then also ( p i : X −→ X i ) i ∈ I is a limit cone (by the Bourbaki-criterion); moreover, m is the limit of the family ( m i ) i ∈ I .(4.i) X YX i Y imp i q i m i Having finite and hence discrete domain and codomain, each m i : X i −→ Y i is a regularmonomorphism in Pos fin = Priest fin (this is a special case of Theorem 2.5) and thereforealso in Priest . Consequently, also m = lim i m i is a regular monomorphism in Priest .3. By definition and by the above, the two-element space is a regular cogenerator in Priest .4. The two-element space is finitely copresentable in Priest . This is very well-known; for ourpurpose we mention here that it is a consequence of [Hofmann, 2002a, Lemma 2.2]. In thissection we observe that this result generalises beyond the finitary case (see Lemma 4.37).5. The two-element space is regular injective in Priest . This follows immediately from finitecopresentability: Consider a regular monomorphism m : X −→ Y in Priest together with(4.i), and let f : X −→ be a morphism in Priest . Since is finitely copresentable, there19s some i ∈ I and a morphism ¯ f : X i −→ with ¯ f · p i = f . Since is injective in Pos (we stress that this is a special case of Proposition 2.6), there is some ¯ g : X i −→ Y i with¯ g · m i = ¯ f . Hence, ¯ g · q i is an extension of f along m . X YX i Y i mp i f q i m i ¯ f ¯ g Priest has effective equivalence corelations. A direct proof, even for partially orderedcompact Hausdorff spaces in general, can be found in [Abbadini and Reggio, 2019].Note that our treatment of properties of Priest rests on results about Ord and Pos , thereforewe have first a look at V -categories. Theorem 4.6. V - Cat op is a quasivariety.Proof. First recall from Theorem 2.5 that the regular monomorphisms in V - Cat are preciselythe embeddings, and from Proposition 2.6 that V is injective and ( f : X −→ V ) f is initial, forevery V -category X . Moreover, V I (indiscrete structure) is a cogenerator in V - Cat and therefore V × V I is a regular injective regular cogenerator. Since V - Cat is also complete, the assertionfollows. Remark . The observation above should be compared to the fact that “ Top op is a qua-sivariety”, for details see [Barr and Pedicchio, 1995, 1996] and [Adámek and Pedicchio, 1997;Pedicchio and Wood, 1999].On the other hand, the quasivariety V - Cat op does not have any rank. To see this, we recallfirst the following result from [Gabriel and Ulmer, 1971, Page 64] (see also [Ulmer, 1971]). Proposition 4.8. A set is copresentable in Set if and only if it is a singleton. The corresponding result for V - Cat is now an immediate consequence of the following obser-vation. Proposition 4.9. The “discrete” functor D : Set −→ V - Cat preserves non-empty limits, inparticular cofiltered limits. If k = ⊤ is the top-element of V , then D preserves also the terminalobject. Corollary 4.10. If X is copresentable in V - Cat , then | X | = 1 .Proof. By Proposition 4.9, the forgetful functor | − | : V - Cat −→ Set preserves copresentableobjects since, for every V -category X , hom( − , | X | ) ≃ hom( D − , X ).We turn now our attention to separated V -categories (see [Hofmann and Tholen, 2010], forinstance). 20 heorem 4.11. The full subcategory V - Cat sep of V - Cat is closed under initial monocones.Therefore the inclusion functor V - Cat sep −→ V - Cat has a left adjoint; moreover, the canoni-cal forgetful functor V - Cat sep −→ Set is mono-topological with left adjoint D : Set −→ V - Cat sep (discrete structures). Consequently, V - Cat sep is complete and cocomplete, with concrete limits.A morphism f : X −→ Y in V - Cat sep is a monomorphism if and and only if the map f isinjective.Remark . We do not know if Top op0 or V - Cat opsep are quasivarieties. Note that in both casesthe class of regular monomorphisms does not coincide with the class of embeddings, as we alsoexplain below (see also [Baron, 1968]).The description of further classes of morphisms in V - Cat sep is facilitated by the notion of L-closure introduced in [Hofmann and Tholen, 2010]. Lemma 4.13. Let X be a V -category, M ⊆ X and x ∈ X . Then the following assertions areequivalent.(i) x ∈ M .(ii) For all f, g : X −→ Y in V - Cat , if f | M = g | M , then f ( x ) ≃ g ( x ) .(iii) For all f, g : X −→ Y in V - Cat with Y separated, if f | M = g | M , then f ( x ) = g ( x ) .(iv) For all f, g : X −→ V in V - Cat , if f | M = g | M , then f ( x ) = g ( x ) . Corollary 4.14. The epimorphisms in V - Cat sep are precisely the L-dense V -functors, and theregular monomorphisms the closed embeddings.Proof. The assertion regarding epimorphisms is in [Hofmann and Tholen, 2010, Theorem 3.8].However, both claims follow immediately from Lemma 4.13.We denote by V - Cat sep , cc the full subcategory of V - Cat sep formed by all Cauchy completeseparated V -categories. The following two results follow immediately from Corollary 4.14. Corollary 4.15. A separated V -category X is Cauchy-complete if and only if X is a regularsubobject of a power of V in V - Cat sep . Moreover, the regular monomorphisms in V - Cat sep , cc areprecisely the embeddings of V -categories. Corollary 4.16. The V -category V is a regular injective regular cogenerator in V - Cat sep , cc .Hence, (cid:0) V - Cat sep , cc (cid:1) op is a quasivariety.Remark . Clearly, the “discrete” functor D : Set −→ V - Cat sep preserves non-empty limits.Under some conditions (see [Clementino and Hofmann, 2009, Proposition 2.2]), every discrete V -category is Cauchy-complete and the discrete functor D : Set −→ V - Cat sep , cc is left adjoint tothe forgetful functor V - Cat sep , cc −→ Set and preserves codirected limits. Hence, in this case atmost a one-element V -category can be copresentable in V - Cat sep , cc . Remark . In general, the category ( V - Cat sep , cc ) op is not a variety, i.e. does not have effectiveequivalence corelations. A counterexample is already given by the case V = since the dualof Pos ∼ - Cat sep , cc is not a variety. This fact is well-known and follows immediately from thefollowing facts: 21 Pos op is equivalent to the category TAL of totally algebraic lattices and maps preservingall suprema and all infima (see [Rosebrugh and Wood, 1994], for instance), • TAL is a full subcategory of the category CCD of (constructively) completely distributivelattices and maps preserving all suprema and all infima, • the unit interval [0 , 1] is completely distributive but not totally algebraic, • the category CCD is monadic over Set (see [Pedicchio and Wood, 1999], and [Pu and Zhang,2015] for a generalisation to quantaloid-enriched categories). Here the free algebra over aset X is given by the complete lattice of upsets of the powerset of X , and this lattice istotally algebraic and therefore also the free totally algebraic lattice over X .Another important property of V -categories and V -functors is established in [Kelly and Lack,2001]: V - Cat is locally presentable, for every quantale V . Under Assumption 4.19 below, andbased on [Seal, 2005, 2009], we show that V - Cat is locally ℵ -copresentable by describing acorresponding countable limit sketch. This will help us later to identify V - CatCH as the modelcategory of a ℵ -ary limit sketch in CompHaus . To do so, in the remainder of this section weimpose the following conditions on the quantale V . Assumption 4.19. We assume that the underlying lattice of V is completely distributive, andthat there is a countable subset D ⊆ V so that, for all v ∈ V , v = _ { u ∈ D | u ≪ v } . Examples 4.20. The quantales of Examples 2.3 and Example 2.4 satisfy Assumption 4.19. Remark . Under Assumption 4.19, for each v ∈ V , ↑ v = \ {↑ u | u ∈ D, u ≪ v } . Hence, by Remark 2.9 (3), the sets ↑ u ( u ∈ D ) form a subbasis for the closed sets of the dual ofthe Scott topology of V .We start with the following well-known fact. Lemma 4.22. The assignments ( ϕ : X → V ) ( ϕ − ( ↑ u ) u ∈ D ) and ( B u ) u ∈ D ( ϕ : X → V , x _ { u ∈ D | x ∈ B u } ) define a bijection between the sets V X and { ( B u ) u ∈ D | for all u ∈ D , B u ⊆ X & B u = \ v ≪ u B v } . Remark . Under the bijection above, a map a : X × X −→ V corresponds to a family ( R u ) u ∈ D of binary relations R u on X . Proposition 4.24. A V -relation a : X × X −→ V is reflexive if and only if ∆ X ⊆ R k . Moreover, a : X × X −→ V is transitive if and only if, for all u, v ∈ D , R u · R v ⊆ R u ⊗ v . roof. See [Seal, 2009]. Remark . A V -category ( X, a ) is separated if and only if the relation R k on X is anti-symmetric.Therefore the structure of a V -category can be equivalently described by a family of binaryrelations, suitably interconnected. Since a map f : X −→ Y between V -categories is a V -functorif and only if f preserves the corresponding relations, we obtain at once: Corollary 4.26. The categories V - Cat and V - Cat sep are model categories in Set of an ℵ -arycountable limit sketch.Remark . We do not know yet wether V - Cat sep , cc is locally presentable. However, we notethat in [Adámek et al. , 2015] this property is proven for V = [0 , ⊙ , that is, for the case ofbounded metric spaces.We turn now our attention to V -categorical compact Hausdorff spaces. First we observe thatProposition 4.9 as well as some of its consequences generalise directly to the topological case. Proposition 4.28. The “discrete” functors D : CompHaus −→ V - CatCH and D : CompHaus −→V - CatCH sep preserve non-empty limits. If k = ⊤ is the top-element of V , then D preserves alsothe terminal object. Regarding copresentable compact Hausdorff spaces, we recall the following result from [Gabriel and Ulmer,1971, 6.5(c)] (see also [Ulmer, 1971]). Theorem 4.29. 1. The finitely copresentable compact Hausdorff spaces are precisely the fi-nite ones.2. The ℵ -copresentable compact Hausdorff spaces are precisely the metrisable ones. In par-ticular, the unit interval [0 , is ℵ -copresentable in CompHaus . Corollary 4.30. For every regular cardinal λ , the forgetful functors | − | : V - CatCH −→ CompHaus and | − | : V - CatCH sep −→ CompHaus preserve λ -copresentable objects. In particular, everyfinitely copresentable (separated) V -categorical compact Hausdorff space is finite and every ℵ -copresentable (separated) V -categorical compact Hausdorff space has a metrizable topology. We are particularly interested in properties of the space V . We start with the followingobservation. Proposition 4.31. A subbase for the Lawson topology on V is given by the sets { u ∈ V | v ≪ u } and { u ∈ V | v (cid:2) u } ( v ∈ D ) . Proof. By definition, the Lawson topology is the join of the Scott topology and the lower topologyof V (see Remark 2.9); we recall that the latter is generated by the sets ( ↑ v ) ∁ , with v ∈ V . Sincethe lattice V is completely distributive, the Scott topology of V has as subbase the sets (seeProposition 2.12) { u ∈ V | v ≪ u } , v ∈ V . Since “generated topology” defines a left adjoint, the sets { u ∈ V | v ≪ u } and { u ∈ V | v (cid:2) u } ( v ∈ V )form a subbase for the Lawson topology of V . Let now v ∈ V . For each v ≪ u ∈ V , there issome w ∈ D with v ≪ w ≪ u , therefore { u ∈ V | v ≪ u } = [ w ∈ D,v ≪ w { u ∈ V | w ≪ u } . Finally, since v ∈ W { w ∈ D | w ≪ v } , we obtain ↑ v = T {↑ w | w ∈ D, w ≪ v } and therefore( ↑ v ) ∁ = S { ( ↑ w ) ∁ | w ∈ D, w ≪ v } . Corollary 4.32. The Lawson topology makes V a ℵ -copresentable object in CompHaus .Proof. By Proposition 4.31, the Lawson topology on V has a countable subbase and thereforealso a countable base. Hence, V with the Lawson topology is a metrizable compact Hausdorffspace and therefore, by Theorem 4.29, ℵ -copresentable in CompHaus .We shall now extend Corollary 4.26 to the topological context and show that V - CatCH is amodel category of a limit sketch in CompHaus . To prepare this, we recall an alternative wayof expressing the compatibility between topology and V -categories which is closer to Nachbin’soriginal definition. Proposition 4.33. For a V -category ( X, a ) and a U -algebra ( X, α ) with the same underlyingset X , the following assertions are equivalent.(i) α : U ( X, a ) −→ ( X, a ) is a V -functor.(ii) a : ( X, α ) × ( X, α ) −→ ( V , ξ ≤ ) is continuous.Proof. See [Hofmann and Reis, 2018, Proposition 3.22]. Lemma 4.34. Consider V with the dual of the Scott topology. Then, under the correspondenceof Lemma 4.22, ϕ : X −→ V is continuous if and only if, for each u ∈ D , B u is closed in X .Proof. Recall from Remark 4.21 that the sets ↑ u ( u ∈ D ) form a subbase for the closed sets ofthe dual of the Scott topology of V .Applying Lemma 4.34 to the map a : ( X, α ) × ( X, α ) −→ ( V , ξ ≤ ) of Proposition 4.33 givesimmediately: Theorem 4.35. Both V - CatCH and V - CatCH sep are model categories in CompHaus of a count-able ℵ -ary limit sketch. Hence, both categories are locally copresentable.Proof. For the second affirmation, use [Adámek and Rosický, 1994, Remark 2.63]. Remark . At this moment we do not have any information about the rank of the locallypresentable category V - CatCH op ; in particular, we do not know if V - CatCH op is ℵ -ary locallycopresentable. 24n order to obtain more information on copresentable objects in V - CatCH , we adapt now[Hofmann, 2002a, Lemma 2.2] to the case of a general regular cardinal. Here we call a λ -arylimit sketch S = ( C , L , σ ) λ -small whenever there is a set M of morphisms in C of cardinalityless than λ so that every morphism of C is a finite composite of morphisms in M . Hence, for λ > ℵ , we require the category C to be λ -small. Lemma 4.37. Let λ be a regular cardinal and let S = ( C , L , σ ) be a λ -small limit sketch. Thena model of S in a category X is λ -copresentable in Mod( S , X ) provided that each component is λ -copresentable in X .Proof. See [Hofmann, 2002a, Lemma 2.2].By Assumption 4.19, the limit sketch for V - CatCH is countable which allows us to derive thefollowing properties. Corollary 4.38. A V -categorical compact Hausdorff space is ℵ -ary copresentable in V - CatCH (respectively V - CatCH sep ) if and only if its underlying compact Hausdorff space is metrizable. Inparticular, V op is ℵ -ary copresentable in V - CatCH and in V - CatCH sep . Corollary 4.39. If the quantale V is finite, then the finitely copresentable objects of V - CatCH (respectively V - CatCH sep ) are precisely the finite ones.Remark . The conclusion of Lemma 4.37 is not necessarily optimal. For instance, the circleline T = R (cid:14) Z is ℵ -copresentable but not finitely copresentable in CompHaus (see [Gabriel and Ulmer,1971, 6.5]); hence, Lemma 4.37 implies that T is ℵ -copresentable in the category CompHausAb of compact Hausdorff Abelian groups and continuous homomorphisms. However, by the famousPontryagin duality theorem (see [Morris, 1977], for instance), T is even finitely copresentable in CompHausAb which cannot be concluded from Lemma 4.37. Remark . In particular, the finitely copresentable partially ordered compact spaces areprecisely the finite ones. Moreover, a partially ordered compact space is ℵ -copresentable in PosComp if and only if its underlying compact Hausdorff topology is metrisable. This character-isation is slightly different from our result in [Hofmann et al. , 2018] where the ℵ -copresentableobjects in PosComp are characterised as those spaces where both – the order and the topology– are induced by the same (not necessarily symmetric) metric.The results above also imply that the reflector π : V - CatCH −→ V - Priest preserves ℵ -cofiltered limits. In the classical case, the corresponding property is shown in [Gabriel and Ulmer,1971, Page 67] using Stone duality; however, our proof here is based on the Bourbaki-criterion. Proposition 4.42. The reflection functor π : V - CatCH −→ V - Priest preserves ℵ -cofilteredlimits (and even cofiltered limits if V is finite).Proof. Let ( p i : X −→ D ( i )) i ∈ I be a ℵ -cofiltered limit in V - CatCH ( ℵ -cofiltered if V is finite).Since V op is ℵ -ary copresentable ( ℵ -ary copresentable if V is finite) in V - CatCH , the cone ofall morphisms of type X −→ V op is given by the cone of all morphism X p i −−−−−→ D ( i ) ϕ −−−−→ V op i ∈ I and ϕ : D ( i ) −→ V op in [0 , CatCH . Hence, for every i ∈ I and every ϕ : X −→ V op ,we obtain the commutative diagram X π ( X ) D ( i ) π ( D ( i )) V op . p i π ( p i ) e ϕ Therefore the cone ( π ( p i ) : π ( X ) −→ π ( D ( i ))) i ∈ I is initial with respect to the forgetful functor V - CatCH −→ CompHaus .Let now i ∈ I and x ∈ D ( i ) with x ∈ T { im( π ( D ( k ))) | k : j → i in I } . Let A ⊆ X be theinverse image of x under the reflection map D ( i ) −→ π ( D ( i )). Then, for every k : j → i in I , ∅ = A ∩ im( k ). Since the set { im( k ) | k : j → i } is codirected and A is compact, we obtain ∅ = \ k : j → i A ∩ im D ( k ) = A ∩ \ k : j → i im D ( k ) = A ∩ im( p i ) . Therefore x ∈ im( π ( p i )).Combining Corollaries 4.42 and 4.30 we obtain: Corollary 4.43. 1. An object is ℵ -ary copresentable in V - Priest if and only if its underly-ing compact Hausdorff space is metrizable. In particular, V op is ℵ -ary copresentable in V - Priest .2. Assume that V is finite. Then an object is finitely copresentable in V - Priest if and only ifit is finite. In particular, V op is finitely copresentable in V - Priest .Proof. Since the left adjoint π : V - CatCH −→ V - Priest of V - Priest −→ V - CatCH preserves ℵ -codirected limits, the inclusion functor V - Priest −→ V - CatCH preserves ℵ -copresentable objects.Furthermore, since V - Priest is closed in V - CatCH under limits, V - Priest −→ V - CatCH reflects ℵ -copresentable objects. The second affirmation follows similarly. Theorem 4.44. The category V - Priest is locally ℵ -ary copresentable. If V is finite, then V - Priest is even locally ℵ -ary copresentable.Proof. By the Bourbaki-criterion, every X in V - Priest is a limit of the canonical diagram of X with respect to the full subcategory of V - Priest defined by all ℵ -copresentable objects. Since V - Priest is complete, we conclude that V - Priest is locally ℵ -ary copresentable. If V is finite, thesame argument works with ℵ in lieu of ℵ . Remark . By Corollary 4.43, the fully faithful functor C : [0 , ⊙ - Priest −→ [0 , ⊙ - FinLat op of Theorem 3.24 preserves ℵ -filtered limits which allows for an alternative proof of Corol-lary 4.42 for ⊗ = ⊙ : Firstly, the dualising object [0 , 1] induces a natural dual adjunction (see[Porst and Tholen, 1991]) [0 , ⊙ - CatCH [0 , ⊙ - FinLat op C =hom( − , [0 , − , [0 , ⊥ , ⊙ - Priest . Then the func-tor π : [0 , ⊙ - CatCH −→ [0 , ⊙ - Priest is the composite of the functor C : [0 , ⊙ - CatCH −→ [0 , ⊙ - FinLat op and the right adjoint functor [0 , ⊙ - FinLat op −→ [0 , ⊙ - Priest above (see [Lambek and Rattray,1979, Theorem 2.0], and note that, for every L in [0 , ⊙ - FinLat , the space hom( L, [0 , V -categorical compact Hausdorff spaces with compact V -categories. To do so,we also impose now the following condition . Assumption 4.46. For the neutral element k of V , the set { u ∈ V | u ≪ k } is directed.Then ⊥ < k and, for all u, v ∈ V , k ≤ u ∨ v = ⇒ ( k ≤ u ou k ≤ v );which guarantees that the L-closure is topological (see [Hofmann and Tholen, 2010, Propo-sition 3.3]). Moreover, under this condition, a separated V -category X induces a Hausdorfftopology; if this topology is compact, X becomes a V -categorical compact Hausdorff space (see[Hofmann and Reis, 2018, Theorem 3.28 and Propositions 3.26 and 3.29]). We let V - Cat sep , comp denote the full subcategory of V - Cat sep defined by those V -categories with compact topology,then this construction defines a fully faithful functor V - Cat sep , comp −→ V - CatCH sep . From Lemma 4.13 and Corollary 4.14 we obtain immediately: Corollary 4.47. Let f : X −→ Y be in V - Cat sep , comp . Then1. f is a regular monomorphism in V - CatCH sep if and only if f is an embedding, and2. f is an epimorphism in V - CatCH sep if and only if f is surjective. Lemma 4.48. If the V -category V is compact, then the L-topology on V coincides with theLawson topology.Proof. By [Hofmann and Nora, 2020, Remark 4.27], for every u ∈ V , the sets ↑ u and ↓ u areclosed in V with respect to the L-closure. Example 4.49. In particular, the L-closure on the [0 , ⊙ -category [0 , 1] induces the Euclideantopology with convergence ξ . Corollary 4.50. Assume that the V -category V is compact. Then we have a fully faithful functor V - Cat sep , comp −→ V - Priest , and every V -enriched Priestley space is a cofiltered limit of compact separated V -categories.Moreover: every embedding f : X −→ Y in V - Priest is a regular monomorphism, and • therefore the epimorphisms in V - Priest are precisely the surjective morphisms.Consequently, V op is a regular cogenerator in V - Priest .Proof. Regarding embeddings, we use the same argument as in Remark 4.5 (2). Every epimor-phism e in V - Priest factorises as e = m · g where g is surjective and m is a regular monomorphism,hence m is an isomorphism and therefore e is surjective. Remark . If V is finite, then V op is finitely copresentable in V - Priest and, with the sameargument as in Remark 4.5 (5), we deduce that V op is regular injective in V - Priest . Unfortunately,the same argument does not seem to work if V is infinite since in this case • V op is countably but in general not finitely copresentable in V - Priest , but • we are not able to prove that every V -enriched Priestley space is a ℵ -cofiltered limit ofcompact separated V -categories.We finish this paper by bringing another well-known result from order theory into the enrichedrealm: every V -categorical compact Hausdorff space is a quotient of a Priestley space. We shallmake use of the free V -categorical compact Hausdorff space, for U -category ( X, a ), and thereforeassume that our topological theory U = ( U , V , ξ ) is strict in the sense of [Hofmann, 2007]: Assumption 4.52. The complete lattice V is completely distributive, and we consider theLawson topology ξ : U V −→ V (see Remark 2.9). Furthermore, the tensor ⊗ : V × V → V iscontinuous with respect to the Lawson topology.We consider the free V -categorical compact Hausdorff space( U X, b a, m X )of a U -category ( X, a ) where b a = U a · m ◦ X (see [Hofmann et al. , 2014, Theorem III.5.3.5]).Moreover, by [Hofmann, 2007, Lemma 6.7 and Proposition 6.11], the map δ A : X −→ V , x _ { a ( x , x ) | x ∈ U A } is an U -functor, for every A ⊆ X , since it can be written as the composite X V U A V . p a q δ A W For our next result we need to consider a stronger version of Assumption 4.46 which we assumefrom now on : Assumption 4.53. The set { u ∈ V | u ≪ v } is directed, for every v ∈ V . Lemma 4.54. For every U -category ( X, a ) and all x , y ∈ U X , b a ( x , y ) = _ { u ∈ V | ∀ A ∈ x . δ − A ( ↑ u ) ∈ y } . roof. Same as in [Hofmann, 2013, page 83], which in turn relies on [Hofmann, 2006, Corol-lary 1.5]. Lemma 4.55. For every U -category ( X, a ) , the cone ( U X ξ · U δ A −−−−−−−→ V ) A ⊆ X is initial in V - CatCH .Proof. For all x , y ∈ U X , we show that b a ( x , y ) ≥ ^ { ξ · U δ A ( y ) | A ∈ x } , and observe that δ A ( x ) ≥ k , for every A ∈ x . Let u ≪ ^ { ξ · U δ A ( y ) | A ∈ x } . Then, for every A ∈ x , u ≪ ξ · δ A ( y ), and therefore ↑ u ∈ U δ A ( y ), which is equivalent to δ − A ( ↑ u ) ∈ y . Therefore u ≤ b a ( x , y ), by Lemma 4.54. Corollary 4.56. For every U -category ( X, a ) , the V -categorical compact Hausdorff space ( U X ) op is Priestley. Corollary 4.57. Every V -categorical compact Hausdorff space is a regular quotient of a Priestleyspace.Proof. With α : U X −→ X denoting the convergence of X (and X op ), α : U ( X op ) −→ X op is a regular quotient in V - CatCH , and hence also α : U ( X op ) op −→ X . References Abbadini, M. 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