Convergence and quantale-enriched categories

Generalising Nachbin's theory of"topology and order", in this paper we continue the study of quantale-enriched categories equipped with a compact Hausdorff topology. We compare these $\mathcal{V}$-categorical compact Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that the presence of a compact Hausdorff topology guarantees Cauchy completeness and (suitably defined) codirected completeness of the underlying quantale enriched category.

1. Introduction 1.1. Motivation. This paper continuous a line of research initiated in [Tho09] which combines Nachbin's theory of "topology and order" [Nac50] with the setting of monad-quantale enriched categories [HST14]. Over the past century, the combination of order structures with compact Hausdorff topologies has proven to be very fruitful in various parts of mathematics: in the form of spectral spaces they appear in Stone duality for distributive lattices [Sto38] and Hochster's characterisation of prime spectra of commutative rings [Hoc69], the connection between spectral spaces and certain partially ordered compact spaces was made explicit in [Pri70,Pri72] (see also [Cor75,Fle00]), and was further extended to an equivalence between all partially ordered compact spaces and stably compact topological spaces in the 1970's (see [GHK + 80]). Subsequently, stably compact spaces have also played a central role in the development of domain theory, see [Law11] for details. In a more general context, compact Hausdorff spaces combined with the structure of a quantale-enriched category have been essential in study of topological structures as categories: they appear in the definition of "dual space", still implicitly in [CH09] and more explicitly in [Hof13,GH13,Hof14]. This notion turned out to be an essential ingredient in the study of (co)completeness properties of monad-quantale enriched categories. In [CCH15] we also explain the connection of Nachbin's work with the theory of multicategories [Her00,Her01].
Motivated by this development, we focus here on the ultrafilter monad and study quantale-enriched categories equipped with a compact Hausdorff topology; our examples include ordered, metric, and probabilistic metric compact Hausdorff spaces. We show that the presence of a compact Hausdorff topology guarantees Cauchy completeness and (suitably defined) codirected completeness of the underlying quantale enriched category. Our investigation relies on a connection between these V-categorical compact Hausdorff spaces and monad-quantale enriched categories which generalises the equivalence between partially ordered compact spaces and stably compact topological spaces (see [HST14,Section III.5]). Another important ingredient is the concept of Cauchy completeness à la Lawvere for monad-quantale enriched categories as introduced in [CH09]. In order to include probabilistic metric spaces in our study, our setting is slightly weaker than the one considered in [CH09]. Due to this weaker assumptions, we have to overcome some technical difficulties which force us to revise and extend some notions and results of [CH09].
In order to explain our motivation more in detail, we find it useful to place it in a historical context.

Historical background.
Right from its origins at the beginning of the 20 th century, one major concern of set-theoretic topology was the development of a satisfactory notion of convergence. This in turn was motivated by the increasing use of abstract objects in mathematics: besides numbers, mathematical theories deal with sequences of functions, curves, surfaces,. . . . To the best of our knowledge, a first attempt to treat convergence abstractly is presented in [Fré06]. Whereby the main contribution of [Fré06] is the concept of a (nowadays called) metric space, the starting point of [Fré06] is actually an abstract theory of sequential convergence. Fréchet considers a function associating to every sequence of a set X a point of X, its convergence point, subject to the following axioms: (A): Every constant sequence (x, x, . . . ) converges to x; (B): If a sequence (x n ) n∈N converges to x, then also every subsequence of (x n ) n∈N converges to x.
Under these conditions, Fréchet gave indeed a generalisation of Weierstraß's theorem [Fré04]; however, these constrains seem to be to weak in general since the limes axioms In particular, Hausdorff already presents metric spaces as a direct generalisation of ordered sets where now f associates to each pair (a, b) the distance between a and b. This point of view was taken much further in [Law73]: not only the structure but also the axioms of an ordered set and of a metric space are very similar and, moreover, can be seen as instances of the definition of a category. Furthermore, Hausdorff sees also the definition of a topological space as a generalisation of the concept of a partially ordered set: instead of a relation between points, sequential convergence relates sequences with their convergence points, and a neighbourhood system relies on a relation between points and subsets. Surprisingly, also here the relevant axioms on such relations can be formulated so that they resemble the ones of a partially ordered set. We refer the reader to the monograph [HST14] for an extensive presentation of this theory and for further pointers to the literature. Clearly, Hausdorff considers topologies as generalised partial orders; however, a more direct relation between the two concepts was only given more than twenty years later. In [Ale37], Alexandroff observes that every partial order on a set X defines a topology, and from this topology one can reconstruct the given order relation via (1.i) x ≤ y ⇐⇒ y belongs to every neighbourhood of x.
Furthermore, Alexandroff characterises the topological spaces obtained this way as the so called "diskrete Räume", namely as those T0 spaces where the intersection of open subsets is open. These spaces, without assuming the T0 separation axiom, are nowadays called Alexandroff spaces. In this paper we depart from Hausdorff's nomenclature since partial orders seem to be more frequent than total ones. Therefore we call a binary relation ≤ on a set X an order relation whenever ≤ is reflexive and transitive, and speak of a total order whenever all elements are comparable. Furthermore, we think of the anti-symmetry condition as a(n often unnecessary) separation axiom. We write Ord for the category of ordered sets and monotone maps and, with Top denoting the category of topological spaces and continuous maps, Alexandroff's construction extends to a functor Top −→ Ord which commutes with the underlying forgetful functors to the category Set of sets and functions. The order relation defined by (1.i) is now known as the specialisation order of the space X. This order looses most of the topological information of a space X and does not seem to be very useful for the study of topological properties. Nevertheless, there are some properties of a space X which are reflected in the specialisation order, in particular the lower separation axioms: • X is T0 if and only if the specialisation order of X is separated (=anti-symmetric); and • X is T1 if and only if the specialisation order of X is discrete.
The latter equivalence might be the reason why this order relation does not play a dominant role in general topology. More interesting seems to be the reverse question: which order properties are guaranteed by certain topological properties? For instance, the following observation is very relevant for our paper: • if X is sober, then the specialisation order of X is directed complete (see [Joh86, Lemma II.1.9]).
The specialisation order plays also a role in Hochster's study of ring spectra: [Hoc69] characterises the prime spectra of commutative rings as precisely Stone's spectral spaces [Sto38]. Here, for a commutative ring R, the order of the topology on spec(R) should match the inclusion order of prime ideals; by that reason Hochster considers the dual of the specialisation order. Motivated by the convergence theoretic approach described below, in this paper we will also consider this underlying order of a topological space instead of the specialisation order. A deep connection between topological properties and order properties is made in [Sco72] where injective topological T0 spaces are characterised in terms of their underlying partial order.
Whereby in the considerations above the order relation is the one induced by a given topology, a different road was taken in [Nac50] in his study of ordered topological spaces where topology and order are two independent structures, subject to a mild compatibility condition. This combination allows for an substantial extension of the scope of various important notions and results in topology, we mention here the concept of order-normality and the Urysohn Lemma. Of special interest to us is a particular class of separated ordered topological spaces, namely the compact ones, which are described in [Jun04] as "precisely the T0 analogues of compact Hausdorff spaces". These spaces can be equivalently described in purely topological terms: firstly, there is a comparison functor between the category PosComp of separated ordered compact spaces and monotone continuous maps and Top; secondly, this functor restricts to an equivalence PosComp ≃ StablyComp where StablyComp denotes the category of stably compact spaces and spectral maps. These facts are known since the beginning of the 1970's and were first published in [GHK + 80]. To explain this connection better, we find it useful to return to the story of convergence.
After Hausdorff's fundamental book [Hau65], the notion of convergence does not seem to have played a prominent role in the development of topology. The notion of sequence proved to be insufficient, and only in the 1930's [Bir37] appeared a characterisation of topological T1 spaces in terms of an abstract concept of convergence based on the notion of Moore-Smith sequence [MS22,Moo15]. At the same time, Cartan introduced the concept of filter convergence [Car37a,Car37b], and this idea was met with enthusiasm within the Bourbaki group [Bou42]. However, it seems to us that this enthusiasm was not shared by most treatments of topology as convergence plays often only a secondary role. We refer to [Cla13] for more information on convergence and its history.
Using either filters or nets (as Moore-Smith sequence are typically called nowadays), convergence finally conquered its appropriate place in topology. This also led to the consideration of abstract (ultra)filter convergence structures, we mention here the papers [Gri60,Gri61,CF67] where topological convergence structures are characterised among more general ones. In our opinion, the most useful descriptions were obtained around 1970: firstly, Manes characterises compact Hausdorff spaces as precisely the Eilenberg-G G X hold in the ordered category Rel of sets and relations. Elementwise, the latter axioms read as for all x ∈ X, x ∈ U X and X ∈ U U X. Note that the second condition talks about the convergence of an ultrafilter of ultrafilters X to an ultrafilter x, which comes from applying the ultrafilter functor U to the relation a : U X − → X. Hence, these description involves the additional difficulty of extending the functor U : Set → Set in a suitable way to a locally monotone endofunctor on Rel; but it is extremely useful since it does not only provide axioms but also a calculus to deal with these axioms since they are formulated within the structure of the ordered category Rel. Barr's characterisation gives also new evidence to Hausdorff's intuition that topological spaces are generalised orders, as the two axioms are clearly reminiscent to the reflexivity and the transitivity condition defining an order relation. We also note that the underlying order of a topology a : U X − → X is simply the composite a · e X : X − → X. Using this language, Tholen [Tho09] shows that an ordered compact Hausdorff space can be equivalently described as a set X equipped with an order relation ≤: X − → X and a compact Hausdorff topology α : U X → X which must be compatible in the sense that is monotone. Moreover, the object part of the functor K : PosComp → Top mentioned above can now be simply described by relational composition a simple calculation shows that ≤ ·α : U X − → X satisfies indeed the two axioms of a topology. More importantly, as already initiated in [Tho09], this approach paves the way to mix topology with metric structures or other "generalised orders" in the spirit of Hausdorff; or better: enriched categories in the spirit of Lawvere [Law73]. Undoubtedly, topology is already omnipresent in the study of metric spaces; however, there does not seem to exist a systematic account in the literature thinking of metric and topology as a generalisation of Nachbin's ordered topological spaces. This motivation brings us to the following considerations.
• Instead of analysing a metric space (X, d) using the topology induced by d, we ask what properties of d are ensured by a compact Hausdorff topology compatible with d.
• To answer this question, we look back and ask the same question for the ordered case. Surprisingly, there is a quick answer: since every separated ordered compact space corresponds to a stably compact space which is in particular sober, every separated ordered compact space has codirected infima and, by duality, also directed suprema. • To transport this argumentation back to the metric case, we need a metric variant of sober topological spaces, which is provided by the notion of sober approach space [Low97,BLVO06,VO05]. • we also consider the notion of codirected completeness for metric spaces which implies Cauchy completeness. We compare this notion to other concepts of (co)directedness in the literature.
The principal aim of this paper is to present a theory which encompasses the steps above, for quantaleenriched categories equipped with a compact Hausdorff topology; our examples include ordered, metric, and probabilistic metric compact Hausdorff spaces. We place this study in the general framework of topological theories [Hof07] and monad-quantale-enriched categories (see [HST14]), for the ultrafilter monad Í on Set.

Basic notions
In this section we recall various aspects of the theory of quantale-enriched categories. All results presented here are well-known, for more information about enriched category theory we refer to the classic [Kel82] and to [Stu14]. 2.1. Quantale. A quantale V = (V, ⊗, k) is a complete lattice V, with the order relation denoted by ≤, equipped with a monoidal structure given by a commutative binary operation ⊗, with identity k, which distributes over joins: Thus, by Freyd's Adjoint Functor Theorem, for each u ∈ V, the monotone map u ⊗ − : V → V has a right adjoint hom(u, −) characterised by Our principal examples include the following.
(1) The two element chain 2 = {0, 1} of truth values with 0 ≤ 1 is a quantale for ⊗ = & being the logical operation "and"; in this case hom(u, v) is just the implication u =⇒ v. More general, every Heyting algebra with ⊗ = ∧ being infimum and the identity given by the top element ⊤ is a quantale.
(2) The extended real half line ← −− − [0, ∞] ordered by the "greater or equal" relation becomes a quantale with the tensor product given by the usual addition +, denoted by The corresponding operation hom is given, respectively, by for u, v ∈ [0, 1]. We will denote these quantales by [0, 1] ∧ , [0, 1] * , and [0, 1] ⊙ , respectively. Then, through the map (5) Another way to equip the unit interval [0, 1] with a quantale structure is considering the usual order and ⊗ is given by the nilpotent minimum v}. This is a classical example of a tensor in [0, 1] that is left continuous but not continuous. (6) The set For each of the tensor products ⊗ on [0, 1] defined in (4), D becomes a quantale with for all γ ∈ [0, ∞]; the identity is given by For more information about this quantale we refer to [Fla97,HR13,CH16].
2.2. Completely distributive lattices. In this subsection we recall some properties of complete lattices and quantales which will be useful in the sequel. First of all, we call a quantale V = (V, ⊗, k) non-trivial whenever k > ⊥. More generally: Definition 2.2. The neutral element k of a quantale V = (V, ⊗, k) is called ∨-irreducible whenever k > ⊥ and, for all For an ordered set X, we denote by P ↓ X the complete lattice of down sets of X ordered by inclusion. The ordered set X can be embedded into P ↓ X by and X is complete if and only if ↓ X : X → P ↓ X has a left adjoint X : P ↓ X → X. In this paper we will often require that the complete lattice V is completely distributive (see [Ran52,Woo04]), therefore we recall now: Definition 2.3. A complete ordered set X is called completely distributive whenever the map X : P ↓ X → X preserves all infima.
Hence, since P ↓ X is complete, the lattice X is completely distributive if and only if X has a left adjoint ⇓ X : X → P ↓ X. We recall that Definition 2.4. Let X be a complete ordered set X. For all x, y ∈ X, x is totally below y (x ≪ y) whenever, for all A ∈ P ↓ X, Proposition 2.5. Let ≪ be the totally below relation in a complete ordered set X with order relation ≤.
Then, for all x, y, z ∈ X: If X is a completely distributive lattice, then, for every y ∈ X, therefore x ∈⇓ y if and only if x ≪ y.
Theorem 2.6. A complete lattice X is completely distributive if and only if every y ∈ X can be expressed as y = {x ∈ X | x ≪ y}.
Remark 2.7. A complete ordered set X is completely distributive if and only if X op is so (see [Woo04]).

Examples 2.8.
(1) The complete lattice 2 is completely distributive where x ≪ y if and only if y = 1. (3) In order to show that the complete lattice D is complete distributive, it is useful to introduce some special elements that will allow a more simplified description of D and of its properties.
The step functions f n,ε , with n ∈ [0, ∞] and ε ∈ [0, 1], are elements of D, defined by for all α ∈ [0, ∞]. It is shown in [Fla97] that, for all f, f n,ε ∈ D, f n,ε ≪ f if and only if ε < f (n). This observation allows to write every element f ∈ D as the supremum of those step functions totally bellow f : f = {f n,ε ∈ D | f n,ε ≪ f }. A complete description of the totally below relation on D can be found in [CH16].
Definition 2.9. For a quantale V = (V, ⊗, k), we say that k is approximated whenever the set We note that in each of the quantales of Examples 2.8 the neutral element is approximated.
Proof. Assume that k is approximated. First note that k > ⊥ since, being directed, ⇓k is non-empty. Furthermore, k is ∨-irreducible by [HR13,Remark 4.21], and the second assertion follows from [Fla92, Theorem 1.12].
and the identity on X is the V-relation 1 X : X − → X given by The resulting category of sets and V-relations is denoted by V-Rel. Similarly to the case of the identity relation, every map f : X → Y can be seen as a V-relation f : this construction defines a functor Set → V-Rel. We note that this functor is faithful if and only if V is non-trivial. The set V-Rel(X, Y ) of V-relations from X to Y is actually a complete ordered set where the supremum of a family (ϕ i : X − → Y ) i∈I is calculated pointwise. Since the tensor product of V preserves suprema, for every V-relation r : X − → Y , the maps (−) · r : V-Rel(Y, Z) → V-Rel(X, Z) and r · (−) : V-Rel(Z, X) → V-Rel(Z, Y ) preserve suprema as well; which tells as that V-Rel is actually a quantaloid (see [Ros96]). In particular, both maps have right adjoints in Ord.
Here a right adjoint − r of − · r must give, for each t : X − → Z, the largest V-relation of type Y − → Z whose composite with r is less or equal t, and we call t r the extension of t along r. Explicitly, Similarly, a right adjoint r − of r · − must give, for each t : Z − → Y , the largest V-relation of type Z − → X whose composite with r is less or equal t.
The V-relation r t is called the lifting of t along r, and can be calculated as Another important feature which comes from the fact that V-Rel is locally ordered, is the possibility to define adjoint V-relations: r : X − → Y is left adjoint to s : Y − → X if and only if 1 X ≤ s · r and r · s ≤ 1 Y , which in pointwise notation, gives, for all x ∈ X, and For each V-relation r : X − → Y one can consider its opposite r • : Y − → X given by r • (x, y) = r(y, x), for all x ∈ X and all y ∈ Y . This operation satisfies and 2.4. V-categories. We introduce now categories enriched in a quantale V.
Definition 2.11. Let V = (V, ⊗, k) be a quantale. A V-category is a pair (X, a) consisting of a set X and a V-relation a : X − → X satisfying 1 X ≤ a and a · a ≤ a; in pointwise notation: and With the usual composition of maps and the identity maps, V-categories and V-functors provide the category V-Cat. Note that 1 X ≤ a implies a ≤ a · a, which implies a · a = a, for every V-category (X, a). The quantale V is itself a V-category with structure given by hom : V × V → V. To every V-category (X, a) one can associate its dual V-category X op = (X, a • ), and this construction defines a functor Due to the fact that the forgetful functor O V : V-Cat → Set is topological (see [AHS90,CH03]), the category V-Cat admits all limits and colimits. Moreover, O V : V-Cat → Set has a left adjoint and the free V-category over the one-element set 1 = {⋆} is given by G = (1, k), where k(⋆, ⋆) = k. For every set X, we have the X-fold power V X of the V-category V whose elements are maps ϕ : X → V and, for maps . However one can also consider the structure a ⊗ b on X × Y : a ⊗ b((x, y), (x ′ , y ′ )) = a(x, x ′ )⊗b(y, y ′ ). Both products are commutative and associative but the neutral objects differ in general: (1, ⊤) is the neutral object for the first product while G = (1, k) is the neutral object for the second.
We consider now the quantales of Examples 2.1.

Examples 2.13.
(1) The objects of 2-Cat are ordered sets (that is, sets equipped with a reflexive and transitive binary relation) and the morphisms are monotone maps; thus, 2-Cat ≃ Ord.
(2) A ← −− − [0, ∞] + -category is a generalised metric space in the sense of [Law73] and a ← −−− [0, ∞] + -functor is a non-expansive map. We write Met for the resulting category, that is, 1] ⊕ -categories and [0, 1] ⊙ -categories as boundedby-1 metric spaces and [0, 1] ⊙ -functors as non-expansive maps, so that (3) A D-category consists of a set equipped with a structure a : X × X → D such that, for all x, y, z ∈ X and t ∈ [0, ∞]: and a D- , for x, y ∈ X and t ∈ [0, ∞]. Therefore the category D-Cat is isomorphic to the category of (generalised) probabilistic metric spaces ProbMet. The classical definition of probabilistic metric space (see [Men42,SS83]) demands that (X, a) is separated (see Definition 2.16), symmetric and finitary (a(x, y) ∈ D should be finite for all x, y ∈ X). A detailed study of probabilistic metric spaces as enriched categories can be found in [HR13].
Definition 2.14. Let V 1 and V 2 be quantales, we write ⊗ for the multiplication in both V 1 and V 2 , and k 1 denotes the neutral element of V 1 and k 2 the neutral element of V 2 . A lax quantale morphism ϕ : V 1 → V 2 is a monotone map between the underlying ordered sets satisfying These properties ensure that the mapping (X, a) −→ (X, ϕa) sends V 1 -categories to V 2 -categories; hence, this construction defines a functor which commutes with the forgetful functors to Set.

Examples 2.15.
(1) The identity map on [0, ∞] defines a lax quantale morphism The corresponding functors produce the canonical chain of functors (2) The quantale This map is a lax quantale morphism and it admits a right and a left adjoint , for all f ∈ D with P ∞ and O ∞ being lax quantale morphisms. These lax morphisms induce adjoint functors between the categories Met and ProbMet.
For every quantale V, the canonical map is a lax quantale morphism, which induces the functor The monotone map i : 2 → V has a right adjoint which is a lax morphism of quantales too and induces the functor B p : V-Cat → Ord; explicitly, for all elements x, y of a V-category X.
Besides V-functors, there is another important type of morphisms between categories, called V-distributors. The notion of distributor was introduced by Bénabou in the 1960's and provides "a generalisation of relations between sets to 'relations between (small) categories' " (see [Bén00]).
In fact, these inequalities are equalities due to the reflexivity of a and b. Thus the identity distributor on (X, a) is actually a and, considering the composition of V-relations, we obtain the category V-Dist. We also note that a V-relation ϕ : For V-categories X and Y , the subset is closed under suprema; hence, the supremum of a family (ϕ i : X − → • Y ) i∈I can be calculated pointwise. As in Subsection 2.3, for a V-distributor ϕ : X − → • Y , both maps (−) · ϕ and ϕ · (−) have right adjoint given, respectively, by the extension and lifting along ϕ.
For the particular case of a V-functor of the form x : G → X we obtain x * = a(x, −) and x * = (−, x).
The underlying order of V-categories extends point-wise to an order relation between V-functors. This order relation can be equivalently described using V-distributors: Furthermore, the composition from either sides preserves this order, and therefore V-Cat is actually an ordered category. An important consequence is the possibility to define adjoint V-functors: a pair of 2.6. Cauchy complete V-categories. In 1973, Lawvere [Law73] proved that a metric space X is Cauchy complete if and only if every adjunction ϕ ⊣ ψ : This observation motivates the following nomenclature.
Although the definition requires the representability of every adjunction, it is enough to consider the case Y = G. Thus, a V-category X is Cauchy complete if and only if every adjunction (ϕ : Subsequent developments established conditions under which results relating Cauchy sequences, convergence of sequences, adjunctions of distributors and representability can be generalised to V-Cat (see [Fla92,HT10,HR13,CH09,Cha09]). In this subsection we will present some of these notions and results in order to recall that, under some light conditions on the quantale V, Lawvere's notion of complete V-categories can be equivalently expressed with Cauchy sequences.
In order to talk about convergence, we need to introduce first some topological notions. Here, for a quantale V = (V, ⊗, k), a V-category (X, a) and a subset M ⊆ X, the L-closure M of M is given by the collection of all x ∈ X which represent adjoint distributors on M . More precisely, x ∈ M whenever i * · x * ⊣ x * · i * , where i : M ֒→ X is the inclusion U-functor. In more elementary terms, we have: Proposition 2.20. Let (X, a) be a V-category, M ⊆ X and x ∈ X. Then the following assertions are equivalent.
The proposition above also shows that (X, a) and (X, a) op induce the same closure operator on the set X.
Corollary 2.22. If k is ∨-irreducible in V, then the L-closure operator defines a topology on X such that every V-functor becomes continuous. Hence, in this case the L-closure defines a functor V-Cat → Top.
Equipped with a closure operator, there is a notion of convergence in a V-category. In particular, a sequence s = ( We recall the following definition from [Wag94]. and for all x ∈ X. The relation between these concepts is stated in the following result (see [HR13]).
Theorem 2.25. Let V be a quantale where k is terminal and assume that there is a sequence (u n ) n∈N in V satisfying:

Combining convergence and V-categories
In this section we study V-categories equipped with a compatible convergence structure. As we explained in Section 1, this study has its roots in Nachbin's "Topology and Order" [Nac50] as presented in [Tho09]. We recall the notion of topological theory U [Hof07], which provides enough structure to extent the ultrafilter monad Í to a monad on V-Cat; the algebras for this monad we designate as V-categorical compact Hausdorff spaces. We also recall the notions of U-category and U-functors and the comparison between V-categorical compact Hausdorff spaces and U-categories, which can be already found in [Tho09].
commute. For a monad Ì = (T, m, e), a Ì-algebra (X, α) is an object X of A together with a map α : T X → X satisfying commutes. Ì-algebras and their morphisms compose the Eilenberg-Moore category A Ì of Ì. The forgetful functor G Ì : A Ì → A has a right adjoint F Ì : A → A Ì that takes an object X of A to the Ì-algebra (T X, m X ). For more information on monads we refer to [MS04]. Here the functor U : Set → Set takes a set X to the set U X of ultrafilters on X and, for a map f : The unit e X : X → U X on X sends x ∈ X to the principal ultrafilter x on X, and the multiplication m X : U 2 X → U X is characterised, for every X ∈ U 2 X, by The Eilenberg-Moore category of Í, Set Í , is equivalent to the category of compact Hausdorff topological spaces and continuous maps (see [Man69]).
The following result (see [Sto38]) ensures the existence of certain ultrafilters and will be very important for later usage.
Lemma 3.1. Let f be a filter and j be an ideal on a set X such that f ∩ j = ∅. Then there is an ultrafilter r that extends f and excludes j; that is, f ⊆ r and j ∩ r = ∅.
3.2. Ultrafilter theories. In this paper we consider a particular case U = (Í, V, ξ) of a topological theory (in the sense of [Hof07]) based on the ultrafilter monad Í = (U, m, e), a quantale V = (V, ⊗, k) and a map ξ : U V → V. Here we require (Í, V, ξ) to satisfy all the axioms of the definition of a strict topological theory with the exception of the axiom regarding the tensor product ⊗ of V, for which it is enough to have lax continuity. We call such a theory an ultrafilter theory. More in detail: • the map ξ : U V → V is the structure of an Eilenberg-Moore algebra on V, that is, ξ : U V → V is the convergence of a compact Hausdorff topology on V; • The tensor product is "laxly continuous": • ξ is "compatible with suprema in V" as specified in condition (Q ) in [Hof07].
We call a theory U satisfying even equality in the axiom involving the tensor product a strict ultrafilter theory. We note that every ultrafilter theory where ⊗ = ∧ is strict. Furthermore, U is called compatible with finite suprema whenever the diagram commutes. Note that we do not need to impose a condition on the empty supremum since, for every ultrafilter theory, the diagram An ultrafilter theory U = (Í, V, ξ) is called pointwise strict whenever, for all u ∈ V, the diagram Somehow surprisingly, the formula above depends only on the lattice structure of V; moreover, it is self-dual in the sense that Proof. Just apply Theorem 3.2 to the quantale V op with tensor product given by binary suprema ∨ in V.
Here we use that also the lattice V op is completely distributive and therefore in particular a frame. (1) For V = 2, the convergence of Theorem 3.2 corresponds to the discrete topology on 2. We denote this theory as U 2 . Definition 3.6. Let U 1 = (Í, V 1 , ξ 1 ) and U 2 = (Í, V 2 , ξ 2 ) be ultrafilter theories and ϕ : V 1 → V 2 be a lax quantale morphism. Then ϕ is compatible with U 1 and U 2 whenever, for all For instance, for every ultrafilter theory U = (Í, V, ξ), the canonical map i : 2 → V (see Subsection 2.4) is a lax quantale morphism making the diagram commutative; hence i is compatible with U 2 and U. Proof. Let v ∈ U V and assume that ξ 2 (U p(v)) = 1. Then U p(v) = 1 and therefore ↑k ∈ v. If k is the top-element of V, then {k} ∈ v and consequently ξ(v) = k.
The result above applies in particular when ϕ : V 1 → V 2 is right adjoint. For instance, if U = (Í, V, ξ) is an ultrafilter theory where V is completely distributive and ξ : U V → V is as in Theorem 3.2, then p : V → 2 is compatible with U and U 2 since i ⊣ p. 3.3. Extending the monad. Given an ultrafilter theory U = (Í, V, ξ), we extend the functor U : Set → Set to a lax functor U ξ on V-Rel by putting U ξ X = U X for each set X and (1) For each V-relation r : (3) For each V-relation r : X − → Y and functions f : A → X and g : Y → Z, . We have even equality if U is a strict theory. (5) Then e becomes an op-lax natural transformation e : 1 → U ξ and m a natural transformation m :

V-categorical compact Hausdorff spaces.
Based on the lax extension of the Set-monad Í = (U, m, e) to V-Rel described in Subsection 3.3, the Set-monad Í admits a natural extension to a monad on V-Cat, in the sequel also denoted as Í = (U, m, e) (see [Tho09]). Here the functor U : V-Cat → V-Cat sends a V-category (X, a 0 ) to (U X, U ξ a 0 ), and with this definition e X : X → U X and m X : U U X → U X become V-functors for each V-category X.
Definition 3.11. Let U = (Í, V, ξ) be an ultrafilter theory. An Eilenberg-Moore algebras for the monad Í on V-Cat is called V-categorical compact Hausdorff space.
Hence, a V-categorical compact Hausdorff space can be described as a triple (X, a 0 , α) where (X, a 0 ) is a V-category and α : U X → X is the convergence of a compact Hausdorff topology on X such that β) is continuous. Since the extension U ξ of U commutes with the involution (−) • , with X = (X, a 0 , α) also (X, a • 0 , α) is a V-categorical compact Hausdorff space. It follows from [Hof07, Lemma 3.2] that the V-category (V, hom) combined with the Í-algebra structure ξ induces the V-categorical compact Hausdorff space V = (V, hom, ξ).

Examples 3.12.
(1) Our motivating example is produced by U = U 2 . In this case, the objects of the Eilenberg-Moore category for the monad Í on Ord are precisely the ordered compact Hausdorff spaces introduced in [Nac50], and the homomorphisms are the monotone continuous map. We denote this category by OrdCH. We recall that an ordered compact Hausdorff space X is a set equipped with an order relation ≤ and a compact Hausdorff topology so that is closed with respect to the product topology. It is shown in [Tho09] that this condition is equivalent to being an Eilenberg-Moore algebra for the ultrafilter monad on Ord. (3) Similarly, for U = U D , the objects of ProbMet Í are called probabilistic metric compact Hausdorff spaces. The category ProbMet Í will be represented by ProbMetCH.
Proposition 3.13. Let U 1 = (Í, V 1 , ξ 1 ) and U 2 = (Í, V 2 , ξ 2 ) be ultrafilter theories and ϕ : V 1 → V 2 be a lax quantale morphism compatible with U 1 and U 2 . Then, for every V 1 -category X, the identity map on the set U X is a V 2 functor of type Proof. Let (X, a 0 ) be a V 1 -category. Then, since ϕ is compatible with the ultrafilter theories U 1 and U 2 , for all x, y ∈ U X we have which proves the claim.
Hence, the family of these maps defines a natural transformation and, together with B ϕ : V 1 -Cat → V 2 -Cat, a monad morphism (see [Pum70]) from the ultrafilter monad Í on V 1 -Cat to the ultrafilter monad Í on V 2 -Cat. As a result, we obtain the functor sending (X, a 0 , α) to (X, ϕa 0 , α) and making the diagram commutative. In particular, for every completely distributive quantale V and ξ given by the formula in Theorem 3.2, the lax quantale morphism p : V → 2 induces the functor Examples 3.14. We have seen in Example 3.9 that the lax quantale morphisms introduced in Example 2.15 (2) are compatible with the ultrafilter theories U← −− − [0,∞]+ and U D . As a consequence one has the adjoint functors 3.5. U-categories and U-functors. We have already mentioned in Section 1 that there is a close connection between ordered compact Hausdorff spaces and certain topological spaces. In this subsection we recall the definition of U-categories as enriched substitutes of topological spaces. This notion has its roots in Barr's "relational algebras" [Bar70], an extensive presentation of the theory of (Ì, V)-categories (also called (Ì, V)-algebras), for a monad Ì and a quantale V, can be found in [HST14].
Definition 3.15. A U-category is a pair (X, a) consisting of a set X and a V-relation a : T X − → X satisfying the lax Eilenberg-Moore axioms 1 X ≤ a · e X and a · U ξ a ≤ a · m X .
Expressed elementwise, these two conditions read as for all X ∈ U U X, x ∈ U X and x ∈ X.
Since f ⊣ f • in V-Rel, this condition is equivalent to a ≤ f • · b · U f , and in pointwise notation the latter inequality becomes for all x ∈ U X, x ∈ X. The category of U-categories and U-functors is denoted by U-Cat.

Examples 3.17.
(1) For V = 2, a U 2 -category is a set X equipped with a relation →: U X − → X such that e x (x) → x and, if X → r and r → x, then m X (X) → x. It is shown in [Bar70] that these are precisely the convergence relations induced by topologies; in fact, the main result of [Bar70] states that U 2 -Cat is isomorphic to the category Top of topological spaces and continuous maps.
(2) The concept of approach space was introduced by Lowen in 1989 (see [Low89,Low97]). It involves a set X and a map δ : P X × X → [0, ∞], called approach distance or distance map, satisfying:  a(X, r), a(r, x)) a(m X (X), x), for all X ∈ U U X, r ∈ U X and x ∈ X.
(5) For V = D, U-categories can be identified with probabilistic approach spaces. This is an example of a quantale-valued approach space studied in [LT16]. For the sake of simplicity we will represent U D -Cat by ProbApp.
Similarly to the situation for V-categories, the canonical forgetful functor O U : U-Cat → Set is topological (see [CH03]); which implies that the category U-Cat is complete and cocomplete and O U preserves limits and colimits. We denote the free U-category over the one-element set 1 by G = (1, k); here k : U 1 − → 1 is the V-relation which sends the unique element of U 1 × 1 to k.
There are several other functors connecting U-categories with V-categories and topological spaces.
Firstly, we have a functor Set Í ֒→ U-Cat interpreting an Eilenberg-Moore algebra as a lax one. Furthermore, there is a forgetful functor (−) 0 : U-Cat → V-Cat sending (X, a) to (X, a 0 = a · e X ) and leaving maps unchanged. We notice that the diagram and By Proposition 3.18, for every ultrafilter theory U = (Í, V, ξ), the canonical map i : 2 → V (see Subsection 2.4) induces the functor interpreting a topological space X as the U-category with structure for x ∈ U X and x ∈ X. If, moreover, the right adjoint p : V → 2 of i is compatible with U and U 2 (see Lemmas 3.7 and 3.8), then p defines a functor and only if k ≤ a(x, x).
3.6. Comparison with U-categories. It is shown in [Tho09] that there is a canonical functor which associates to each X = (X, a 0 , α) in (V-Cat) Í the U-category KX = (X, a) where a = a 0 · α. Note that (a 0 · α) 0 = a 0 , that is, the diagram Example 3.20.
(1) For U = U 2 , one obtains the commutative diagram Here every ordered compact Hausdorff space maps to a weakly sober, locally compact and stable topological space; assuming also the T0-axiom, these spaces are called stably compact (see [GHK + 03]). It is also shown in [GHK + 03] that the full subcategory of OrdCH defined by the separated orders is isomorphic to the category StablyComp of stably compact topological spaces and spectral maps. We also note that the space K2 is the Sierpiński space 2 = {0, 1} with {1} closed.
(3) For U = U D , we obtain the diagram The functor K : (V-Cat) Í → U-Cat is right adjoint, its left adjoint assigns to every U-category the V-categorical compact Hausdorff space (U X,â, m X ). Regarding this construction, we recall here from [CH09]: Lemma 3.21. For every U-category (X, a),â := U ξ a · m • X is a V-category structure on U X.
We give now an alternative characterisation of the compatibility between the convergence and the V-categorical structure of an Eilenberg-Moore algebra (X, a 0 , α) in V-Cat Í , which resembles the classical condition stating that "the order relation is closed in the product space".
Proposition 3.22. For a V-category (X, a 0 ) and a Í-algebra (X, α) with the same underlying set X, the following assertions are equivalent.
On the other hand, the second statement translates to α(y))))), which proves the equivalence.
From Proposition 3.22 we conclude immediately: Lemma 3.23. Let U be an ultrafilter theory so that p : V → 2 is compatible with U and U 2 , and (X, a 0 , α) be a V-categorical compact Hausdorff space. Then, for all x ∈ X and u ∈ V, the closed balls with center x and radius u are closed with respect to the compact Hausdorff topology.
Proof. Applying the forgetful functor B p : U-Cat → Top, we obtain that a 0 : (X, α) × (X, α) → B p (V, hom ξ ) is continuous. For every x ∈ X, consider the continuous maps Then, for every u ∈ V, the closed balls with center x and radius u are the preimages of the closed set ↑u.
3.7. Convergences from V-categories. We recall from Corollary 2.22 that, for every quantale V = (V, ⊗, k) where k is ∨-irreducible, we have the functor sending a V-category (X, a 0 ) to X equipped with the L-closure of (X, a 0 ). We investigate now connections between V-categorical and topological properties. Note that, if k is terminal in the quantale V, then the projection maps π 1 : X × Y → X and π 2 : X × Y → Y are V-functors for all V-categories (X, a 0 ) and (Y, b 0 ). Therefore, with the same proof as for [HT10, Corollary 5.8], we obtain: Lemma 3.24. Let V be a quantale where k is terminal and (X, a 0 ) be V-category. For all x, y ∈ X, Hence, (X, a 0 ) is separated if and only if ∆ is closed in (X, a 0 ) ⊗ (X, a 0 ).
In the sequel we will often require that the functor T V is monoidal.
Definition 3.25. The functor T V : V-Cat → Top is monoidal if, for all V-categories (X, a 0 ) and (Y, b 0 ), the identity map on X × Y is continuous of type (1) The identity map on V is continuous of type T V (V, hom) → B p (V, hom ξ ).
(2) Assume that T V is monoidal. Then, for every V-category (X, a 0 ), the topological space T V (X, a 0 ) is Hausdorff if and only if (X, a 0 ) is separated.
Proof. To see (1), we note that an ultrafilter x converges to x in T V (V, hom) if and only if, for all A ∈ x, x ∈ A; that is, On the other hand, .
Regarding (2), by Lemma 3.24, a V-category (X, a 0 ) is separated if and only a 0 )). Hence, since T V is monoidal, the assertion follows.
y y CompHaus to V-Cat and CompHaus.
Proof. Since the composite is continuous, every separated compact V-category becomes a V-categorical compact Hausdorff space when equipped with the topology of T V (X).
Proposition 3.29. Let V = (V, ⊗, k) be a quantale where k is approximated. Then T V is monoidal.
Proof. Under the condition that k is approximated, the topology of a V-category (X, a 0 ) is generated by the "symmetric open balls" B S (x, u) = {y ∈ X | u ≪ a 0 (x, y) and u ≪ a 0 (y, x)}, for x ∈ X and u ≪ k (see Proposition 2.10 and [HR13, Remark 4.21]). Let now (X, a 0 ) and (Y, b 0 ) be V-categories, (x, y) ∈ X × Y and u ≪ k. By Proposition 2.10, there is some ((x, y), u); which proves the claim.
Examples 3.30. By the results of this subsection, every compact separated (probabilistic) metric space is a (probabilistic) metric compact Hausdorff space.

Completeness from compactness
The central topic of this section is Cauchy completeness for U-categories, a notion defined in terms of adjoint U-distributors (see [CH09]). As much as possible we avoid assuming that the ultrafilter theory U is strict; as a consequence, we cannot assume associativity of the composition of U-distributors. The lack of associativity forces us to be more careful in our treatment of adjunctions; in particular, adjoints need not be unique. We show, under some conditions on the quantale V, that the corresponding U-category of a V-categorical compact Hausdorff space is Cauchy complete. Moreover, for strict theories U, we prove that the forgetful functor U-Cat → V-Cat preserves Cauchy-completeness. Combining both results shows that the underlying V-category of a V-categorical compact Hausdorff space is Cauchy complete. In the last subsection we go a step further and study codirected completeness for V-categories.

U-distributors.
A V-relation of the form ϕ : U X − → Y is called a U-relation and think of ϕ as an arrow from X to Y , and write ϕ : X − ⇀ Y . Composition is given by Kleisli composition: for all U-relations ϕ : X − ⇀ Y ; that is, e • X is a lax identity for the Kleisli composition. Moreover: with equality if U is a strict theory.
Remark 4.2. In the language of U-relations, an U-category (X, a) consists of a set X and an U-relation a : X − ⇀ X satisfying e • X ≤ a and a • a ≤ a.
In other words, an U-distributor ϕ : X − ⇀ • Y comes with a right action of the U-relation a and a left action of b. This perspective motivates the designations bimodule or module used by some authors. Note that we always have ϕ • a ≥ ϕ and b • ϕ ≥ ϕ, so that the U-distributor conditions above are in fact equalities which make the U-structures identities for the composition of U-distributors.
Remark 4.4. In general, U-distributors do not compose. However, this property is guaranteed by assuming that the ultrafilter theory is strict.
The following result establishes a connection between U-distributors and U-functors and generalises slightly [CH09, Theorem 4.3].
Theorem 4.5. Let (X, a) and (Y, b) be U-categories, and ϕ : U X − → Y be a V-relation. Then the following assertions are equivalent.
Proof. First note that ϕ is an U-distributor if and only if The first inequality above means precisely that, for all y ∈ Y and all x, y ∈ U X, ϕ(x, y) ⊗â(y, x) ≤ ϕ(y, y), which in turn is equivalent toâ (y, x) ≤ hom(ϕ(x, y), ϕ(y, y)).
Consequently, ϕ ·â ≤ ϕ if and only if, for all y ∈ Y , is a V-functor; which is the case if and only if ϕ : and this inequality is equivalent to ϕ(x, y)).

The latter holds if and only if, for all
In the sequel we will consider in particular U-distributors with domain or codomain G. For an Ucategory X = (X, a), an U-relation ϕ : 1 − ⇀ X is an U-distributor ϕ : G − ⇀ • X if and only if, for all x ∈ X and all r ∈ U X, Similarly, an U-relation ψ : X − ⇀ 1 is an U-distributor ψ : X − ⇀ • G if and only if, for all r ∈ U X and all X ∈ U U X, U ξ a(X, r) ⊗ ψ(r) ≤ ψ(m X (X)) and U ξ ψ(X) ≤ ψ(m X (X)).
Let (X, a) and (Y, b) be U-categories. Each map f : X → Y induces U-relations These U-relations are actually U-distributors precisely when f is an U-functor.
Lemma 4.6. The following are equivalent, for U-categories (X, a) and (Y, b) and a map f : X → Y .
Similarly to the case of V-categories, the local order of V-Rel allows us to consider U-Cat as an ordered category: for U-functors f, g : In particular, every U-category X has an underlying order where x ≤ y whenever x ⊛ ≤ y ⊛ , for all x, y ∈ X; which in turn is equivalent to k ≤ a( x, y). This construction defines a functor O U : U-Cat −→ Ord, and the diagrams a) is separated (see [HT10]) whenever the underlying ordered set O U (X, a) is separated. We note that (−) 0 : U-Cat → V-Cat sends separated U-categories to separated V-categories.

Adjoint U-distributors.
In this subsection we study the important notion of adjoint U-distributor. We employ here the usual definition of adjunction in an ordered category; however, some extra caution is needed since U-distributors in general do not compose. We hasten to remark that f ⊛ ⊣ f ⊛ , for every U-functor f : (X, a) → (Y, b). In fact, by [CH09, Proposition 3.6 (2), p. 188], f ⊛ • f ⊛ and f ⊛ • f ⊛ are U-distributors and Similarly to the nomenclature for V-categories, we call an U-functor f : (X, a) → (Y, b) fully faithful whenever f ⊛ • f ⊛ = a, and fully dense whenever In general, we are not able to prove unicity of left adjoints since composition of U-distributors does not need to be associative. However, we can still prove that right adjoints are unique: Similarly, ψ ≤ ψ ′ , and we conclude that ψ = ψ ′ .
We turn now our attention to U-distributors with domain or codomain G.
Therefore, when studying adjunctions of the form we do not need to worry about the composites ϕ • ψ and ψ • ϕ. Elementwise, ϕ ⊣ ψ translates to for all x ∈ U X and x ∈ X. We also point out that Proposition 4.11. Let ψ : Proof. We calculate

Cauchy complete U-categories.
With the notion of adjunction of U-distributors at our disposal, we come now to the concept of Cauchy completeness (called Lawvere completeness in [CH09]).

Definition 4.12.
A U-category X = (X, a) is called Cauchy complete whenever every adjunction Note that x ⊛ = a( x, −) and that x ⊛ = a(−, x), so that x ⊛ ⊣ x ⊛ means, for all x ∈ U X and x ′ ∈ X, Examples 4.13. Various examples of Cauchy complete U-categories are described in [CH09], we sketch here the principal facts.
(1) We have already seen that Top ≃ U 2 -Cat. In this context, a U 2 -distributor ϕ : In particular, U 2 -distributors of the form ϕ : G − ⇀ • X can be identified with relations ϕ : 1 − → X satisfying and a relation ψ : for all X ∈ U U X and r ∈ U X. Using Theorem 4.5, a U 2 -distributor ϕ : G − ⇀ • X can be also seen as a continuous map X → 2 into the Sierpiński space, which in turn can be interpreted as a closed subset A ⊆ X. A U-distributor ψ : X − ⇀ • G is a map U X → 2 which is continuous with respect to the Zariski closure on U X (an ultrafilter x ∈ U X belongs to the closure of B ⊆ U X whenever B ⊆ r) and anti-monotone with respect to the order relation where for all x, y ∈ U X. Such maps correspond precisely to subsets A ⊆ U X which are Zariski closed and down-closed with respect to the order relation defined above.
A pair of U -distributors forms an adjunction X In terms of the corresponding subsets A ⊆ X and A ⊆ U X, these conditions read as for x ∈ X and r ∈ U X, and a U← −− − [0,∞]+ -distributor of the type ψ : for all r ∈ U X and X ∈ U U X.
if, for all x ∈ X and X ∈ U U X,

Such a variable set A corresponds to a right adjoint of ϕ if and only if
. Therefore an approach space X is Cauchy complete if and only if each irreducible variable set is representable. Finally, this condition is equivalent to X being weakly sober in the sense of [BLVO06]. for all x ∈ X. We can view ϕ M as an U-relation ϕ M : 1 − ⇀ X given by ϕ M = a · U i · U ! • (here i : M ֒→ X and ! : M → 1). It is easy to see that ϕ M is actually an U-distributor ϕ M : G − ⇀ • X, hence, ϕ M : X → V is an U-functor. We also not that ϕ ∅ = ⊥ and ϕ A∪B = ϕ A ∨ ϕ B .
We import now from [HS11, Lemma 3.2 and Corollary 3.3]: Proposition 4.14. Let U = (Í, V, ξ) be an ultrafilter theory where V is completely distributive and ξ : U V → V is as in Theorem 3.2. For every U-category X = (X, a), x ∈ U X and x ∈ X, a(x, x) = A∈x ϕ A (x).
Proof. First note that from {u ∈ V | u ≪ k} is directed it follows that k is ∨-irreducible (see [HR13,Remark 4.21]). For every u ≪ k, put A u = {x ∈ X | u ≤ ϕ(x)}. By hypothesis, A u = ∅. We claim that ϕ ≤ ϕ Au . To see this, put The for all x ∈ X. For our next result, we recall that the forgetful functor (−) 0 : U-Cat → V-Cat has a left adjoint F : V-Cat → U-Cat sending a V-category (X, a 0 ) to the U-category (X, e • X · U ξ a 0 ), and leaving maps unchanged.
Proposition 4.19. Let U be an ultrafilter theory. Then the following assertions hold.
(1) F sends fully faithful V-functors to fully faithful U-functors.
(2) If U is strict, then F sends fully dense V-functors to fully dense U-functors.
for the corresponding U-structures. Assume first that f : (X, a 0 ) → (Y, b 0 ) is fully faithful. Then Assume now that U is strict and f is fully dense. Now we calculate: Theorem 4.20. Let U be a strict ultrafilter theory. Then (−) 0 : U-Cat → V-Cat sends Cauchy complete U-categories to Cauchy complete V-categories.
Proof. Just note that a V-category (resp. U-category) is Cauchy complete if and only if it is injective with respect to fully faithful and fully dense V-functors (resp. U-functors) as it was proven in [HT10, Theorems 3.10 and 5.11]. For U = U 2 , the result above is vacuous since every ordered set is Cauchy complete. As we already pointed out in Section 1, a stronger result holds in this case: the underlying order of a sober space is codirected complete. In the next subsection we proof a similar result for U-categories, under additional conditions on the quantale V.
Remark 4.22. A related study of properties of metric spaces via approach spaces can be found in [LZ16]. Among other results, it is shown there that in the underlying metric of an approach space every forward Cauchy sequence converges (see [BvBR98,Wag97]). We will come back to this notion in the next subsection.
4.5. Codirected complete V-categories. In this subsection we look at Cauchy completeness of Vcategories from a different perspective, namely as (co)completeness with respect to some choice of (co)limit weights. In this paper we need only very particular limits and colimits, therefore we refer for more information to [KS05,Stu14] and recall here only what we believe is essential for our paper.
As the starting point, we assume that a saturated class Φ of limit weights ϕ : G − → • X is given; examples of such choice are given below. For each V-category X, we write Φ(X) to denote the weights with codomain X. Moreover, we consider V-Dist(G, X) as a V-subcategory of V-Rel(1, X) ≃ V X and Φ(X) as a V-subcategory of V-Dist(G, X) op , this way the mapping h Φ X : X −→ Φ(X), x −→ x * is a V-functor. A V-category X is called Φ-complete whenever h Φ X has a right adjoint inf Φ X : Φ(X) −→ X.
Intuitively, inf Φ X calculates the infimum of a limit weight ϕ : G − → • X. The assumption that Φ is saturated guarantees that each Φ(X) is Φ-complete; in fact, it is the free Φ-completion of X. Dually, notions of cocompleteness depend on a choice of a saturated class Ψ of colimit weights ψ : X − → • G. Then a V-category X is Ψ-cocomplete if and only if the V-functor X −→ Ψ(X), x −→ x * has a left adjoint. Here we consider Ψ(X) as a V-subcategory of V-Dist(X, G).
Remark 4.23. For a saturated class Φ of limit weights, a V-category X is Φ-complete if and only if there exists a V-functor I : Φ(X) → X with Ih Φ X ≃ 1 X ; such a V-functor is necessarily right adjoint to h Φ X .
For instance, is a saturated class of limit weights, and a V-category X is Φ-complete if and only if X is Cauchy complete.
The following definition provides another important example of a saturated class of limit weights. preserves finite suprema and tensors; that is, for all ϕ, ϕ ′ : G − → • X and all u ∈ V, We note that the class Φ ∆ of all codirected V-distributors ϕ : G − → • X is saturated (see [KS05]).

Definition 4.25.
A V-category X is called codirected complete whenever X is Φ ∆ -complete.
Example 4.26. For V = 2, we can interpret every 2-distributor ϕ : G − → • X as an upclosed subset A ⊆ X of X. Then A is codirected in the sense of Definition 4.24 if and only if A is codirected in the usual sense; that is, A = ∅ and, for all x, y ∈ A, there is some z ∈ A with z ≤ x and z ≤ y.
We recall now that, by Theorem 4.5, U-distributors of type G − ⇀ • X correspond to U-functors X → V; and with this perspective we can consider U-Dist(G, X) as a V-subcategory of V-Dist(G, X 0 ).  Proof. We recall first from Proposition 2.10 that, under these assumptions, k is ∨-irreducible. Using the adjunction of Proposition 4.27, the V-functor  This proves that y X : X → Φ ∆ (X) has a right adjoint in V-Cat.
We finish this subsection by exhibiting a connection with other accounts of "codirected complete metric spaces" which appear in the literature. Firstly, non-symmetric versions of Cauchy sequences and their limits are introduced in [Smy88] and further studied in [Rut96,BvBR98]: a sequence (x n ) n∈N in a metric space (X, d) is called forward-Cauchy whenever ∀ε > 0 ∃N ∈ N ∀n ≥ m ≥ N . d(x m , x n ) < ε, and (x n ) n∈N is called backward-Cauchy whenever ∀ε > 0 ∃N ∈ N ∀n ≥ m ≥ N . d(x n , x m ) < ε.
The definitions above extend naturally to nets (see [FSW96]), and in [Vic05] it is shown that that forward-Cauchy nets in metric spaces correspond precisely to those ← −− −  [Vic05]; however, in this paper we deviate slightly from the notation of [Vic05]. In order to compare these two notions of "directedness", we restrict our study to a certain type of quantales.
Definition 4.31. We call a quantale V = (V, ⊗, k) a Girard quantale whenever V has a dualising element D ∈ V; that is, for every u ∈ V, u = hom(hom(u, D), D).
This type of quantales is introduced in [Yet90], we also refer to [Was09] for a study of categories enriched in a Girard quantale. For V = (V, ⊗, k) being a Girard quantale with dualising element D, we write u ⊥ = hom(u, D). As shown in [Yet90], the operations (−) ⊥ and ⊗ allow us to determine the internal hom of V: for all u, v ∈ V, hom(u, v) = (u ⊗ v ⊥ ) ⊥ .