Kantorovich Functors and Characteristic Logics for Behavioural Distances

Behavioural distances measure the deviation between states in quantitative systems, such as probabilistic or weighted systems. There is growing interest in generic approaches to behavioural distances. In particular, coalgebraic methods capture variations in the system type (nondeterministic, probabilistic, game-based etc.), and the notion of quantale abstracts over the actual values distances take, thus covering, e.g., two-valued equivalences, (pseudo-)metrics, and probabilistic (pseudo-)metrics. Coalgebraic behavioural distances have been based either on liftings of SET-functors to categories of metric spaces, or on lax extensions of SET-functors to categories of quantitative relations. Every lax extension induces a functor lifting but not every lifting comes from a lax extension. It was shown recently that every lax extension is Kantorovich, i.e. induced by a suitable choice of monotone predicate liftings, implying via a quantitative coalgebraic Hennessy-Milner theorem that behavioural distances induced by lax extensions can be characterized by quantitative modal logics. Here, we essentially show the same in the more general setting of behavioural distances induced by functor liftings. In particular, we show that every functor lifting, and indeed every functor on (quantale-valued) metric spaces, that preserves isometries is Kantorovich, so that the induced behavioural distance (on systems of suitably restricted branching degree) can be characterized by a quantitative modal logic.


Introduction
Qualitative transition systems, such as standard labelled transition systems, are typically compared under two-valued notions of behavioural equivalence, such as Park-Milner bisimilarity. For quantitative systems, such as probabilistic, weighted, or metric transition systems, notions of behavioural distance allow for a more fine-grained comparison, in particular give a numerical measure of the deviation between inequivalent states, instead of just flagging them as inequivalent [14,6,2,26]. The variation found in the mentioned system types calls for unifying methods, and correspondingly has given rise to generic notions of behavioural distance based on universal coalgebra [35], a framework for state-based systems in which the transition type of systems is encapsulated as an (endo-)functor on a suitable base category. Coalgebraic behavioural distances have been defined on the one hand using liftings of given set functors to the category of metric spaces [5], and on the other hand using lax extensions, i.e. extensions of set functors to categories of quantitative relations [13,40]. Since every lax extension induces a functor lifting in a straightforward way [40] but on the other hand not every functor lifting is induced by a lax extension, the approach via liftings is more widely applicable. On the other hand, it has been shown that every lax extension is Kantorovich, i.e. induced by a suitable choice of modalities, modelled as predicate liftings in the spirit of coalgebraic logic [30,36]. Using quantitative coalgebraic Hennessy-Milner theorems, it follows that under expected conditions on the functor and the lax extension, behavioural distance coincides with logical distance.
Roughly speaking, our main contribution in the present paper is to show that the same holds for functor liftings and their induced behavioural distances. In more detail, we have the following (cf. Figure 1 for a graphical summary): -Every lifting of a set functor is topological, i.e. induced by a generalized form of predicate liftings in which one may need to switch to non-standard spaces of truth values for the predicates involved (Theorem 3.1). -Functor liftings that preserve isometries are Kantorovich, i.e. induced by (possibly polyadic) predicate liftings. (Here, we understand predicate liftings as involving only the standard space of truth values -that is, the unit interval, in the case of 1-bounded metric spaces). In fact, preservation of isometries is also necessary (Theorem 3.9). -Lastly, we detach the technical development from set functors, and show that a functor on (pseudo)metric spaces is Kantorovich, in the sense that the distance of its elements can be characterized by predicate liftings, iff it preserves isometries (Theorem 5.3).
By a recent coalgebraic quantitative Hennessy-Milner theorem that fits this level of generality [12], it follows that given a functor F on (pseudo¡)metric spaces that preserves isometries, acts non-expansively on morphisms, and admits a dense finitary subfunctor, behavioural distance can be characterized by quantitative modal logic (Corollary 5.10). In additional results, we further clarify the relationship between functor liftings and lax extensions, and in particular characterize the functor liftings that are induced by lax extensions (Theorem 3.18). Indeed, we conduct the main technical development not only in coalgebraic generality, but also parametric in a quantale, hence abstracting both over distances and over truth values. One benefit of this generality is that our results cover the two-valued case, captured by the two-element quantale. In particular, one instance of our results is the fact that every finitary set functor has a separating set of finitary predicate liftings, and hence admits a modal logic having the Hennessy-Milner property [36]. Moreover, we do not restrict to symmetric distances, and hence cover also simulation preorders and simulation distances [26]. monotone predicate liftings predicate liftings lax extensions functor liftings [40], Corollary 3.17 [5], Theorem 3.8 Theorem 3.16 [40] Theorem 3.18 Related Work Quantale-valued quantitative notions of bisimulation for functors that already live on generalized metric spaces (rather than being lifted from functors on sets) have been considered early on [42]. We have already mentioned previous work on coalgebraic behavioural metrics, for functors originally living on sets, via metric liftings [5] and via lax extensions [13,40]. Existing work that combines coalgebraic and quantalic generality and accommodates asymmetric distances, like the present work, has so far concentrated on establishing so-called van Benthem theorems, concerned with characterizing (coalgebraic) quantitative modal logics by bisimulation invariance [41]. There is a line of work on Kantorovich-type coinductive predicates at the level of generality of topological categories [22,23] (phrased in fibrational terminology), with results including a game characterization and expressive logics for coinductive predicates already assumed to be Kantorovich in a general sense, i.e. induced by variants of predicate liftings. In this work, the condition of preserving isometries already shows up as fiberedness, and indeed the condition already appears in work on metric liftings [5]. As mentioned in the above discussion, we complement existing work on quantitative coalgebraic Hennessy-Milner theorems [24,40,12] by establishing the Kantorovich property they assume. [28,21,38]. Recall that a coalgebra for a functor F : C → C consists of an object X of C, thought of as an object of states, and a morphism α : X → FX, thought of as assigning structured collections (sets, distributions, etc.) of successors to states.
We will focus on concrete categories over Set, that is categories that come equipped with a faithful functor |−| : C → Set, which allows speaking about individual states as elements of |X|. A lifting of an endofunctor F : Set → Set to C is an endofunctor F : Example 2.1. Some functors of interest for coalgebraic modelling are as follows.
1. The finite powerset functor P ω : Set → Set maps each set to its finite powerset, and for a map g, P ω (g) takes direct images under g. Given a set A (of labels), coalgebras for the the functor P ω (A × −) are finitely branching A-labelled transition systems.
2. The finite distribution functor D ω : Set → Set maps a set X to the set D ω X of finitely supported probability distributions on X. Given a finite set A, coalgebras for the functor (1 + D ω ) A , are probabilistic transition systems [27,10].
Finitary functors are those which are determined by their action on finite sets. More precisely, a functor is finitary if for every set X and every x ∈ FX, there is a finite subset inclusion m : Standard examples of non-finitary functors are as follows.
3. The (unbounded) powerset functor P : Set → Set. 4. The neighbourhood functor N : Set → Set sends a set X to the set PPX, and a function f : X → Y to the function Nf : NX → NY that assigns to every element x ∈ NX the set {B ⊆ Y | f −1 B ∈ x}.

Quantales and Quantale-Enriched Categories
A central notion of our development is that of a quantale, which will serve as a parameter determining the range of truth values and distances. A quantale (V, ⊗, k), more precisely a commutative and unital quantale, is a complete lattice V -with joins and meets denoted by and , respectively -that carries the structure of a commutative monoid with tensor ⊗ and unit k, such that for every u ∈ V, the map u ⊗ − : V → V preserves suprema. This entails that every u ⊗ − has a right adjoint hom(u, −) : V → V, characterized by the property u ⊗ v ≤ w ⇐⇒ v ≤ hom(u, w). We denote by and ⊥ the greatest and the least element of a quantale, respectively. A quantale is non-trivial if ⊥ = , and integral if = k.
1. Every frame (i.e. a complete lattice in which binary meets distribute over infinite joins) is a quantale with ⊗ = ∧ and k = . In particular, every finite distributive lattice is a quantale, prominently 2, the two-element lattice {⊥, } and 1, the trivial quantale.
2. Every left continuous t-norm [3] defines a quantale on the unit interval equipped with its natural order.
, inf, max, 0) of non-negative real numbers with infinity, ordered by the greater or equal relation, and with tensor given by maximum. (c) The quantale [0, 1] ⊕ = ([0, 1], inf, ⊕, 0) of the unit interval, ordered by the greater or equal order, and with tensor given by truncated addition. (Note that the quantalic order here is dual to the standard numeric order).
4. Every commutative monoid (M, ·, e) generates a quantale on PM (the free quantale over M ) w.r.t. set inclusion and with the tensor A ⊗ B = {a · b | a ∈ A and b ∈ B}, for all A, B ⊆ M . The unit of this multiplication is the set {e}.
A V-category is pair (X, a) consisting of a set X and a map a : X × X → V such that k ≤ a(x, x) and a(x, y) ⊗ a(y, z) ≤ a(x, z) for all x, y, z ∈ X. We view a as a (not necessarily symmetric) distance function, noting however that objects with 'greater' distance should be seen as being closer together. A V-category (X, a) is symmetric if a(x, y) = a(y, x) for all x, y ∈ X. Every V-category (X, a) carries a natural order defined by x ≤ y whenever k ≤ a(x, y), which induces a faithful functor ). V-categories and V-functors form the category V-Cat, and we denote by V-Cat sym the full subcategory of V-Cat determined by the symmetric V-categories and by V-Cat sym,sep the full subcategory of V-Cat sym determined by the separated symmetric V-categories. 1. The Category 1-Cat is equivalent to the category Set of sets and functions.
2. The category 2-Cat is equivalent to the category Ord of preordered sets and monotone maps.
3. Metric, ultrametric and bounded metric spacesà la Lawvere [28] can be seen as quantale-enriched categories: (a) The category [0, ∞] + -Cat is equivalent to the category GMet of generalized metric spaces and non-expansive maps. We focus on V = 2 and V = [0, 1] ⊕ , which we will use to capture classical (qualitative) and metric (quantitative) aspects of system behaviour, respectively.. Table 1 provides some instances of generic quantale-based concepts (either introduced above or to be introduced presently) in these two cases, for further reference.
preorder bounded-by-1 hemimetric space symmetric V-category equivalence bounded-by-1 pseudometric space V-functor monotone map non-expansive map initial V-functor order-reflecting monotone map isometry Table 1. V-categorical notions in the qualitative and the quantitative setting. The prefix 'pseudo' refers to absence of separatedness, and the prefix 'hemi' additionally indicates absence of symmetry.
A V-category (X, a) is discrete if a = 1 X , and indiscrete if a(x, y) = for all x, y ∈ X. The dual of (X, a) is the V-category (X, a) op = (X, a • ) given by a • (x, y) = a(y, x). Given a set X and a structured cone, i.e. a family (f i : X → |(X i , a i )|) i∈I of maps into V-categories (X i , a i ), the initial structure a : X × X → V on X is defined by a(x, y) = i∈I a i (f i (x), f i (y)), for all x, y ∈ X. A cone ((X, a) → (X i , a i )) i∈I is said to be initial (w.r.t. the forgetful functor |−| : V-Cat → Set) if a is the initial structure w.r.t. the structured cone (X → |(X i , a i )|) i∈I ; a V-functor is initial if it forms a singleton initial cone. For every Vcategory (X, a) and every set S, the S-power (X, a) S is the V-category consisting of the set of all functions from S to X, equipped with the V-category structure [−, −] given by [f, g] = x∈X a(f (x), g(x)), for all f, g : S → X. By equipping its hom-sets with the substructure of the appropriate power, the category V-Cat becames V-Cat-enriched and, hence, also Ord-enriched w.r.t to the corresponding natural order of V-categories. We say that an endofunctor on V-Cat is locally monotone if it preserves this preorder.
Remark 2.4. Let us briefly outline the connections between V-Cat and V-Cat sym , which for real-valued V correspond to hemimetric and pseudometric spaces, respectively. By virtue of the above construction of initial structures, the categories V-Cat and V-Cat sym are topological over Set [1]; in particular, both categories are complete and cocomplete. Moreover, V-Cat sym is a (reflective and) coreflective full subcategory of V-Cat. The coreflector (−) s : V-Cat → V-Cat sym is identity on morphisms and sends every (X, a) to its symmetrization, the V-category (X, a s ) where a s (x, y) = a(x, y) ∧ a(y, x) (keep in mind that in Example 2.2.3, the order is the dual of the numeric order).
Finally, we note that for every quantale V, (V, hom) is a V-category, which for simplicity we also denote by V. The following result records two fundamental properties of the V-category V.
Proposition 2.5. The V-category V = (V, hom) is injective w.r.t. initial morphisms, and for every V-category X, the cone (f : X → V) f is initial.

Predicate Liftings
Given a cardinal κ and a V-category X, a κ-ary X-valued predicate lifting for a functor F : V-Cat → V-Cat is a natural transformation λ : V-Cat(−, X κ ) → V-Cat(F−, X). When V is the trivial quantale, we identify an X-valued predicate lifting with a natural transformation λ : Set(−, X κ ) → Set(F−, X) via the isomorphism Set ∼ = 1-Cat. In this case, we are primarily interested in predicate liftings valued in the underlying set of another quantale, and we say that such predicate liftings are monotone if each of its components is a monotone map w.r.t. the pointwise order induced by that quantale.
Remark 2.6. By the Yoneda lemma, every κ-ary X-valued predicate lifting for a functor F : V-Cat → V-Cat is determined by a V-functor FX κ → X. In particular, the collection of all X-valued κ-ary predicate liftings for a functor is a set.
1. The Kripke semantics of the standard diamond modality ♦ of the modal logic K is induced (in a way recalled in Section 5) by the unary predicate lifting ♦ X (A) = {B ⊆ X | A ∩ B = ∅} for the (finite) powerset functor (modulo the isomorphism PX ∼ = Set(X, 2)).
2. Computing the expected value for a given [0, 1]-valued function with respect to each probability distribution defines a unary [0, 1]-valued predicate lifting for the functor D ω : Set → Set, which we denote by E.

Quantale-Enriched Relations and Lax Extensions
The structure of a quantale-enriched category is a particular kind of "enriched relation". For a quantale V and sets X and With this composition, the collection of all sets and V-relations between them form a category, denoted V-Rel. The identity morphism on a set X is the V-relation 1 X : X − − → X that sends every diagonal element to k and all the others to ⊥.
The category V-Rel comes with an involution (−) • : V-Rel op → V-Rel that maps objects identically and sends a V-relation r : , the converse of r. Moreover, by equipping its hom-sets with the pointwise order induced by V, V-Rel is made into a quantaloid (e.g. [33]), i.e. enriched over complete join semilattices. This entails that there is an optimal way of extending a V-relation r : A lax extension 3 of a functor F : Set → Set to V-Rel is a lax functor F : V-Rel → V-Rel that agrees with F on sets and whose action on functions is compatible with F. To make the latter requirement precise, we note that a function is interpreted as the V-relation that sends every element of its graph to k and all the others to ⊥; then, a lax extension of F to V-Rel, or simply a lax extension, is a map (r : Example 2.9. The generalized "lower-half" Egli-Milner order between powersets, which for a relation r : defines a lax extension of the powerset functor P : Set → Set to Rel. Similarly, the generalized "upper-half" and the generalized Egli-Milner order define lax extensions of the powerset functor to Rel.
Lax extensions are deeply connected with monotone predicate liftings. To realize this, it is convenient to think of the X-component of a κ-ary predicate lifting as Definition 2.10. A κ-ary predicate lifting λ for a functor F : Set → Set is induced by a lax extension F : Example 2.11. By interpreting a subset of a set X as a relation from 1 to X, the unary predicate lifting ♦ (see Example 2.7) for the powerset functor P : Set → Set is induced by the lax extension of Example 2.9; indeed, it is determined by the map 1 → P1 that selects the set 1.
Remark 2.12. Every predicate lifting induced by a lax extension is monotone.
Lax extensions have been instrumental in coalgebraic notions of behavioural distance (e.g. [13,40,41]), and the notion of Kantorovich extension has been crucial to connect such notions with coalgebraic modal logic [7].
Definition 2.13. Let F : Set → Set be a functor, and Λ a class of monotone predicate liftings for F.
Example 2.14. The Kantorovich extension of the powerset functor P : Set → Set to Rel w.r.t the ♦ predicate lifting coincides with the extension given by the "lower-half" of the Egli-Milner order (Example 2.9).
As suggested by the previous example, the Kantorovich extension leads to a representation theorem that plays an important role in Section 3.2.
Theorem 2.15 ([16]). Let F : V-Rel → V-Rel be a lax extension, and let Λ be the class of all predicate liftings induced by F. Then, F = F Λ .

Topological Liftings
It is well-known that every lax extension F : V-Rel → V-Rel of a functor F : Set → Set gives rise to a lifting (which we denote by the same symbol) of F to V-Cat (for instance, see [39]). By definition, liftings are completely determined by their action on objects. In particular, the lifting induced by a lax extension F : V-Cat → V-Cat sends a V-category (X, a) to the V-category (FX, Fa). Of course, it does not make sense to talk about functor liftings to the category V-Cat when V is trivial, hence we assume from now on that V is non-trivial. Predicate liftings also induce functor liftings, via a simple construction available on all topological categories that goes back, at least, to work in categorical duality theory [11,31]: To lift a functor G : A → Y along a topological functor |−| : B → Y, it is enough to give, for every object A in A, a structured cone so that, for every h in C(A) and every f : A → A, the composite h · Gf belongs to the cone C(A ). Then, for an object A in A, one defines G I A by equipping GA with the initial structure w.r.t. the structured cone (1). It is easy to see that the assignment X → G I X indeed defines a functor G I : A → B such that |−| · G I = G. This technique has been previously applied in the context of codensity liftings [22,23,37,20] and Kantorovich liftings [5]. We apply this to our situation as follows. Given a functor F : Set → Set, take G = F · |−|; then a lifting of F to V-Cat can be specified by a class of natural transformations (which may be thought of as generalized predicate liftings, in that they lift A λvalued predicates to B λ -valued ones). Namely, given a V-category X, we consider the structured cone consisting of all maps where λ ranges over the given natural transformations and f over all V-functors X → A λ . As described above, we obtain a V-category F I X by equipping F|X| with the initial structure w.r.t. this cone. We call functor liftings constructed in this way topological. Indeed, it turns out that every functor lifting is topological, even when one restricts B λ in (2) to be the V-category (V, hom): In examples, we usually construct a generalized predicate lifting (2)   1. The discrete lifting of the identity functor Id : Set → Set, which sends every V-category to the discrete V-category with the same underlying set, can be obtained as a topological lifting constructed from the identity V-valued predicate lifting for Id by choosing A to be the V-category consisting of the set V equipped with the indiscrete structure.
2. The lifting of the identity functor Id : Set → Set to Ord that computes the smallest equivalence relation that contains a given preorder can be obtained as a topological lifting constructed from the 2-valued identity predicate lifting for Id by choosing A to be the discrete preordered set with two elements.
3. It is well-known that the total variation distance between finite distributions µ, υ on a set X coincides with the Kantorovich distance on the discrete boundedby-1 metric space X (e.g. [15] (2)). Therefore, the total variation distance defines a lifting of the finite distribution functor to BHMet that can be obtained as the topological lifting constructed from the predicate lifting E by choosing A to be the indiscrete space [0, 1]. This example is closely related to the first one. Indeed, this lifting is the composite of the Kantorovich lifting of the finite distribution functor to BHMet (see Example 3.5) and the discrete lifting of the identity functor to BHMet. By Theorem 3.9 below, precomposing functor liftings with the discrete lifting of the identity functor can be used to derive non-Kantorovich s. Remark 3.3. Theorem 3.1 can be fine-tuned to show that the discrete lifting F d : Ord → Ord of a finitary functor F : Set → Set is a topological lifting constructed from a set Λ of finitary 2-valued predicate liftings for F. Hence, for every set X, considered as a discrete preordered set, we have that the cone of all maps λ(f ) : F d (X, 1 X ) → 2, for κ-ary predicate liftings λ ∈ Λ and maps X → 2 κ , is initial. Thus, as F d (X, 1 X ) is antisymmetric, this cone is mono. In this sense, our results subsume the result that every finitary Set-functor admits a separating set of finitary predicate liftings [36].

Kantorovich Liftings
For our present purposes, we are primarily interested in topological liftings induced by predicate liftings in the standard sense, i.e. the natural transformations (2) are of the shape λ : V-Cat(−, V κ ) −→ Set(F|−|, |V|), and thus employ V, equipped with its standard V-category structure, as the object of truth values throughout.
In particular, this format is needed to use predicate liftings as modalities in existing frameworks for quantitative coalgebraic logic (Section 5). Many functor liftings considered in work on coalgebraic behavioural distance can be understood as topological liftings constructed in this way (e.g. [5,23,40,41,12]). To simplify notation, in the sequel we often omit the forgetful functor to Set. where λ ∈ Λ is κ-ary and f : (X, a) → V κ is a V-functor. Generally, a lifting We go on to exploit the universal property of initial lifts of cones to characterize the liftings that are Kantorovich. In the following, fix a functor F : Set → Set and a quantale V. Consider the partially ordered conglomerate Pred(F) of classes of Vvalued predicate liftings for F ordered by containment, i.e. Λ ≤ Λ ⇐⇒ Λ ⊇ Λ ; and the partially ordered class Lift(F) of liftings of F to V-Cat ordered pointwise, i.e. F ≤ F ⇐⇒ Fa ≤ F a, for every V-category (X, a). Definition 3.6. Let F : V-Cat → V-Cat be a lifting of F. A κ-ary V-valued predicate lifting λ for F is compatible with F if it restricts to a predicate lifting for F: where the vertical arrows denote set inclusions -that is, if λ lifts V-functorial predicates on X to V-functorial predicates on FX. The class of all predicate liftings compatible with F is denoted by P(F).
The Kantorovich lifting defines a universal construction: The following result shows that Kantorovich liftings are characterized by a pleasant property that is required in multiple results in the context of coalgebraic approaches to behavioural distance (e.g. [5,23,12,42]).
Theorem 3.9. A lifting of a Set-functor to V-Cat is Kantorovich iff it preserves initial morphisms.
Corollary 3.11. The composite of Kantorovich liftings is Kantorovich.
Example 3.12. The characterization of Theorem 3.9 makes it easy to distinguish Kantorovich liftings.
1. It is an elementary fact that every lifting induced by a lax extension preserves initial morphisms (e.g. [18,Proposition 2.16]). In particular, the Wasserstein lifting [5] is Kantorovich.
2. The identity functor on Set has a lifting (−) • : V-Cat → V-Cat that sends every V-category to its dual. Clearly, this lifting preserves initial morphisms, and hence it is Kantorovich. Indeed, one can show that it is the Kantorovich lifting of the identity functor w.r.t. the set of V-valued predicate liftings determined by the representable V-functors V op → V.
3. The functor (−) s : V-Cat → V-Cat sym that symmetrizes V-categories gives rise to a lifting (−) s : V-Cat → V-Cat of the identity functor on Set. Clearly, this functor preserves initial morphisms, and hence it is Kantorovich. Indeed, one can show that it is the Kantorovich lifting of the identity functor w.r.t. the set of all V-valued predicate liftings determined by the representable V-functors V s → V.
4. The discrete lifting of the identity functor on Set to V-Cat is not Kantorovich, as it fails to preserve initial morphisms.
5. The lifting of the identity functor on Set to V-Cat that sends a V-category (X, a) to the V-category given by the final structure w.r.t. the structured cospan of identity maps |(X, a)| → X ← |(X, a • )| is not Kantorovich. This lifting generalizes Example 3.2(2).
6. The lifting of the finite distribution functor on Set to BHMet given by the Kantorovich distance is Kantorovich, while the lifting given by the total variation distance is not Kantorovich.

Liftings Induced by Lax Extensions
We show next that lax extensions, functor liftings, and predicate liftings are linked by adjunctions, and characterize the liftings induced by lax extensions. We begin by showing that the Kantorovich extension and the Kantorovich lifting are compatible. Example 3.14. The identity functor on Ord is the lifting induced by the identity functor on Rel as a lax extension of the identity functor on Set. The constant map into is a monotone map 2 → 2 and, hence, determines a predicate lifting that is compatible with the identity functor on Ord. It is easy to see that this predicate lifting is induced by the largest extension of the identity functor, however, it is not induced by the identity functor on Rel [16, Example 3.12].
It should also be noted that the predicate liftings compatible with a functor lifting that preserves initial morphisms are not necessarily monotone. That is, the map P : Lift(F) I → Pred(F) does not necessarily corestrict to Pred(F) M .   Therefore, the interplay between lax extensions, liftings and predicate liftings is captured by the diagram which commutes when only the right adjoints or only the left adjoints are considered. Finally, we characterize the liftings induced by lax extensions.
Theorem 3.18. A lifting F of a Set-functor F to V-Cat is induced by a lax extension of F to V-Rel iff F preserves initial morphisms and is locally monotone.
V-enriched lax extensions have proved to be crucial to deduce quantitative van Benthem and Hennessy-Milner theorems [40,41]. We recall that a lax extension of a functor F : Set → Set to V-Rel is V-enriched [41,16] if, for all u ∈ V, u ⊗ 1 FX ≤ F(u ⊗ 1 X ); where u ⊗ r denotes the V-relation "r scaled by u", that is, (u ⊗ r)(x, y) = u ⊗ r(x, y).

Behavioural Distance
One main motivation for lifting functors to metric spaces was to obtain coalgebraic notions of behavioural distance [5,40]. Indeed, every functor F : V-Cat → V-Cat gives rise to a notion of distance on a F-coalgebras: Definition 4.1. [12] Let (X, a, α) be a coalgebra for a functor F : Notice the analogy with the standard notion of behavioural equivalence: Two states are behaviourally equivalent if they can be made equal under some coalgebra morphism; and according to the above definition, two states in a metric coalgebra have low behavioural distance if they can be made to have low distance under some coalgebra morphism. Kantorovich liftings and lax extensions are key ingredients in mentioned alternative coalgebraic approaches to behavioural distance on Set-based coalgebras. Let F : Set → Set be a functor. A Kantorovich lifting F Λ : V-Cat → V-Cat induces a notion of behavioural distance on an F-coalgebra α : X → FX as the greatest V-categorical structure (X, a) that makes α a V-functor of type (X, a) → F Λ (X, a) [5,23]. From Theorem 3.9 and [12, Proposition 12] (generalized to V-Cat, with the same proof), we obtain that this distance coincides with behavioural distance as defined above. On the other hand, every lax extension F : V-Rel → V-Rel of F also induces a behavioural distance on an F-coalgebra α : X → FX as the greatest simulation on α [34,42,13,40], i.e. the greatest V-relation s : X − − → X such that α · s ≤ Fs · α. It follows by routine calculation that this distance coincides with the distance defined via the lifting induced by the lax extension and, hence, Theorem 3.13 ensures that, if we start with a collection of monotone predicate liftings, then the corresponding Kantorovich extension and Kantorovich lifting yield the same notion of behavioural distance. This allows including the approach to behavioural distance via lax extensions in the categorical framework for indistinguishability introduced recently by Komorida et al. [23]. On the other hand, there are notions of behavioural distance defined via Kantorovich liftings that do not arise via lax extensions. Indeed, it has been shown that the neighbourhood functor N : Set → Set does not admit a lax extension to Rel that preserves converses ( F(r • ) = ( Fr) • ) whose (2-valued) notion of behavioural distance coincides with behavioural equivalence [29,Theorem 12]. However, from [12, Theorem 34, Proposition A.6] (see also [17]), we can conclude that the (2-valued) notion of behavioural distance defined by the canonical Kantorovich lifting of N to Equ w.r.t. to the predicate lifting induced by the identity natural transformation N → N coincides with behavioural equivalence. (It is easy to see that Marti and Venema's result holds even if one allows lax extensions of N that do not preserve converses, and that the situation remains the same in the asymmetric case.)

Expressivity of Quantitative Coalgebraic Logics
We proceed to connect the characterization of Kantorovich functors with existing expressivity results for quantitative coalgebraic logic, focusing from now on on symmetric V-categories. Therefore, we interpret the V-categorical notions and results also with V-Cat sym instead of V-Cat and V s instead of V.
We recall a variant [12] of (quantitative) coalgebraic logic [30,36,7,25,40] that follows the paradigm of interpreting modalities via predicate liftings, in this case of V-valued predicates for a V-Cat-functor (Section 2.3). Let Λ be a set of finitary predicate liftings for a functor F : V-Cat sym → V-Cat sym . The syntax of quantitative coalgebraic modal logic is then defined by the grammar where Λ is a set of modalities of finite arity, which we identify, by abuse of notation, with the given set Λ of predicate liftings. We view all other connectives as propositional operators. Let L(Λ) be the set of modal formulas thus defined.
We then obtain a notion of logical distance: Definition 5.1. Let Λ be a set of predicate liftings for a functor F : V-Cat → V-Cat. The logical distance ld Λ α on an F-coalgebra (X, a, α) is the initial structure on X w.r.t. the structured cone of all maps φ α : X → |(V, hom s )| with φ ∈ L(Λ). More explicitly, for all x, y ∈ X, In the remainder of the paper, we establish criteria under which a V-Cat symfunctor admits a set of predicate liftings for which logical and behavioural distances coincide. Recall that a (quantitative) coalgebraic logic is expressive if ld Λ α ≤ bd F α , for every F-coalgebra (X, α). (It is easy to show that the reverse inequality holds universally [12,Theorem 16]).
Existing expressivity results for quantitative coalgebraic logics for Set-functors depend crucially on Kantorovich liftings (e.g. [40,41,23,12]). However, it has been shown [12] that the Kantorovich property can be usefully detached from the notion of functor lifting.
Definition 5.2. Let Λ be a class of predicate liftings for a functor F : V-Cat → V-Cat. The functor F is Λ-Kantorovich if for every V-category X, the cone of all V-functors λ(f ) : FX → V, with λ ∈ Λ κ-ary and f ∈ V-Cat(X, V κ ), is initial. A functor F : V-Cat → V-Cat is said to be Kantorovich if it is Λ-Kantorovich for some class Λ of predicate liftings for F.
Clearly, every Kantorovich lifting of a Set-functor to V-Cat w.r.t. a class Λ of predicate liftings is Λ-Kantorovich. Moreover, Theorem 3.9 is easily generalized to Kantorovich functors. 1. The inclusion functor V-Cat sym,sep → V-Cat sym has a left adjoint (−) q : V-Cat sym → V-Cat sym,sep that quotients every X by its natural preorder, which for symmetric X is an equivalence, and gives rise to a Kantorovich functor on V-Cat sym .
2. Given a bounded-by-1 pseudometric space (X, d), i.e. an object of [0, 1] ⊕ -Cat sym BPMet, the Prokhorov distance [32] for probability measures on the measurable space of Borel sets of (X, d) is defined by It is straightforward to verify that this distance defines a BPMet-functor (which acts on morphisms by measuring preimages) that preserves isometries and, therefore, it is Kantorovich.
3. For every V-category (X, a), the functor (X, a) × − : V-Cat → V-Cat is Kantorovich. If the underlying lattice of V is Heyting, then under certain conditions this functor has a right adjoint [8,9] which is Kantorovich as well.
Here, for X = (X, a) exponentiable, the right adjoint (−) X of X × − sends a V-category Y = (Y, b) to the V-category Y X = (Y X , c) with underlying set {all V-functors (1, k) × (X, a) → (Y, b)} and, for h, k ∈ Y X , To ensure that a Kantorovich functor is represented by finitary predicate liftings, we need to impose a size constraint: Definition 5.6. A functor F : V-Cat sym → V-Cat sym is ω-bounded if for every symmetric V-category X and every t ∈ FX, there exists a finite subcategory X 0 ⊆ X and t ∈ FX 0 such that t = Fi(t ) where i is the inclusion X 0 → X. Finally, from [12, Theorem 31] we obtain: Corollary 5.9. Let V be a finite quantale, and let F : V-Cat sym → V-Cat sym be a lifting of a finitary functor that preserves initial morphisms. Then there is a set Λ of predicate liftings for F of finite arity such that the coalgebraic logic L(Λ) is expressive.
Corollary 5.10. Let F : BPMet → BPMet be a functor that preserves isometries, is locally non-expansive, and admits a dense ω-bounded subfunctor. Then there is a set Λ of predicate liftings for F of finite arity such that the coalgebraic logic L(Λ) is expressive.
These instantiate to results on concrete system types, e.g. ones induced by (sub)functors listed in Example 5.5, such as probabilistic transition systems equipped with a behavioural distance induced by the functor that sends a bounded metric space X to the subspace of the space of all probability measures on X equipped with the Prokhorov distance (see Example 5.5(2)) determined by the closure of the set of finitely supported probability measures.

Conclusions and Future Work
Quantitative coalgebraic Hennessy-Milner theorems [24,40,12] assume that the functor (on metric spaces) describing the system type is Kantorovich, i.e. canonically induced by a suitable choice of -not necessarily monotone -predicate liftings, which then serve as the modalities of a logic that characterizes behavioural distance. We have shown as one of our main results that a functor on (quantale-valued) metric spaces is Kantorovich iff it preserves initial morphisms (i.e. isometries). As soon as such a functor additionally adheres to the expected size and continuity constraints (which replace the condition of finite branching found in the classical Hennessy-Milner theorem for labelled transition systems), one thus has a logical characterization of behavioural distance in coalgebras for the functor, in the sense that behavioural distance equals logical distance.
In fact we have shown that every functor on metric spaces can be captured by a generalized form of predicate liftings where the object of truth values may change along the lifting. A simple example is the discretization functor, which is characterized by a predicate lifting in which the truth value object for the input predicates is equipped with the indiscrete pseudometric, so that the lifting accepts all predicates instead of only non-expansive ones. This hints at a perspective to design heterogeneous modal logics that characterize behavioural distance for such functors, with modalities connecting different types of formulas (e.g. non-expansive vs. unrestricted), which we will pursue in future work. One application scenario for such a logic are behavioural distances on probabilistic systems involving total variation distance, which may be seen as a composite of the usual probabilistic Kantorovich functor and the discretization functor. that factor as for some λ ∈ Λ coincides with F on X. Therefore, the claim is consequence of the fact that the cone V-Cat(FX, V) is initial (see Remark 2.5).
The Yoneda lemma guarantees that there is an epi-cocone of natural transformations determined by every V-category A and every V-functor λ : FA → V.

A.2 Details of Example Example 3.2(2)
Consider the discrete preordered set with two elements (2 = {0, 1}, 1 2 ) and the preordered set (3 = {0, 1, 2}, ∨) with three elements generated by 2 ≤ 1 and 2 ≤ 0. Then, the inclusion (2, 1 2 ) → (3, ∨) is initial, however its image under the lifting it is not, since the lifting acts as identity on (2, 1 2 ) but sends (3, ∨) to the indiscrete preordered set with three elements. To see that this is a topological lifting of identity functor on Set constructed from the 2-valued identity predicate lifting for Id : Set → Set by choosing A = (2, 1 2 ), note that for every preordered set (X, r), x is equivalent to y w.r.t to the smallest equivalence relation that contains r if and only if there is a zigzag between x and y in (X, r) if and only if for every monotone map f : (X, r) → (2, 1 2 ), f (x) = f (y).

A.3 Proof of Theorem 3.8
Let F : V-Cat → V-Cat be a lifting of F and S be a set of predicate liftings for F. We claim that P(F) ⊇ S iff F ≤ F S . Suppose that P(F) ⊇ S. Then, since every κ-ary predicate lifting in S is compatible with F, for every V-functor f : (X, a) → V κ , the map On the other hand, suppose that F ≤ F S . By definition of Kantorovich lifting, every predicate lifting in S is compatible with F S . Hence, for every λ : F S (V κ ) → V determined by a κ-ary predicate lifting λ in S, A.4 Proof of Theorem 3.9 Firstly, we show that every Kantorovich lifting of a functor F : Set → Set preserves initial morphisms. Let i : (X, a) → (Y, b) be an initial morphism, Λ a class of predicate liftings for F, j : (Z, c) → F Λ (Y, b) a V-functor, and h : Z → FX a map such that j = Fi · h.
By definition of F Λ it is sufficient to show that for every κ-ary predicate lifting for F in Λ and every V-functor f : (X, a) → V κ , λ(f ) · h is a V-functor. Since V is injective in V-Cat w.r.t. initial morphisms, V κ is also injective in V-Cat with respect to initial morphisms. Hence, for every V-functor f : (X, a) → V κ there is a V-functor g : (Y, b) → V κ such that f = g · i. Consequently, Secondly, we show that the converse statement holds. Suppose that F is a lifting that preserves initial morphisms. We already know from Theorem 3.8 that F ≤ F P(F) . To prove that under our assumption the reverse inequality also holds, let (X, a) be a V-category and κ = |X|. Then, the Yoneda embedding (X, a) → [(X, a) op , V] gives us an initial morphism y ∼ Hence, since F preserves initial morphisms, F y : F(X, a) → F(V κ ) is initial. Now, let P κ (F) denote the set of all κ-ary predicate liftings in P(F). Given that the cone of all V-functors is initial and the composition of initial cones is initial, the cone is initial. Therefore, F P(F) (X, a) ≤ F(X, a).

A.5 Proof of Theorem 3.13
To obtain this result it is useful to express the Kantorovich lifting in the language of V-relations [16]. In the sequel, given a V-functor f : X → V κ , we denote by , for all k ∈ κ and x ∈ X.
Also, we recall that a V-relation r : Proposition A.2. Let (X, a) be a V-category, κ a cardinal and f : X → V κ a function. The following propositions are equivalent: Corollary A.3. Let C = (f i : X → |V κ |) i∈I be a structured cone. The initial V-category on X w.r.t. C is given by Proposition A.4. Let λ be a κ-ary V-valued predicate lifting for a functor F : Set → Set. The Kantorovich lifting F λ : V-Cat → V-Cat of F w.r.t. λ sends a V-category (X, a) to the V-category (FX, F λ a), where Therefore, the Kantorovich lifting of a class of predicate liftings Λ sends a V-category (X, a) to the V-category (FX, F Λ a), where F Λ a = λ∈Λ F λ a. Now, Theorem 3.13 is an immediate consequence of the following lemma.
Lemma A.5. Let λ be a κ-ary V-valued monotone predicate lifting for a functor F : Set → Set, and let (X, a) be a V-category. Then, Proof. Let g : κ − − → X be a V-relation. Since 1 X ≤ a, we have g ≤ a · g. Also, given that a · a ≤ a, we obtain that a · g : (κ, 1 κ ) − • − → (X, a) is a V-distributor. Therefore, because λ is monotone, The other inequality is an immediate consequence of Proposition A.2(3).
A.6 Proof of Theorem 3.16 In the sequel, given a V-relation r : X − − → Y , we denote by r : Y → V X the function that sends each y ∈ Y to the function r(−, y) : X → V. Moreover, for every set S, we denote the structure of the V-category V S by h S .
Proposition A.6. Every predicate lifting induced by a lax extension of a Setfunctor to V-Rel is compatible with the lifting to V-Cat induced by the lax extension.
Proof. Let λ be a κ-ary V-valued predicate lifting for a functor F : Set → Set that is induced by a lax extension F : V-Rel → V-Rel of F. Then, by [16,Theorem 3.11], Therefore, the claim follows from Propositions A.2 and 3.7.
Proposition A.7. Let F : Set → Set be a functor. The monotone map I : Lax(F) → Lift(F) I is order-reflecting.
Proof. Note that every V-relation r : X − − → Y can be factorized as r = (r ) • · h X · 1 X .
Let F : V-Rel → V-Rel be a lax extension of F. Then, by Propositions A.6 and Remark 2.12, MPI( F) contains the all predicate liftings induced F. Hence, by Theorem 2.15, On the other hand, let F : V-Cat → V-Cat be a lifting of F that preserves initial morphisms. Then, by Theorem 3.13 and the fact that MP :

A.7 Proof of Theorem 3.18 and Theorem 3.19
We will show Theorem 3.18 as Theorem 3.19 follows identically by taking into account [16,Theorem 2.16] and [16,Theorem 4.1]. We already observed that every lifting induced by a lax extension preserves initial morphisms, and the fact that it is also locally monotone follows immediately from L1 and L3 since f ≤ g in V-Cat((X, a), On the other hand, suppose that F : V-Cat → V-Cat is a lifting of F that preserves initial morphisms and is locally monotone. Then, since every V-functor is monotone, we have that every κ-ary V-valued predicate lifting λ for F compatible with F is monotone, as each X-component is given by the composite of monotone maps V-Cat((X, 1 X ), V κ ) V-Cat(F(X, 1 X ), F(V κ )) V-Cat(F((X, 1 X ), V).

A.8 Details of section 4
Proposition A.8. Let F : V-Cat → V-Cat be a lifting of a functor F : Set → Set that preserves initial morphisms and corestricts to V-Cat sym . Then, for every F-coalgebra α : X → FX, bd F α is symmetric.
Proof. Let α : X → FX be an F-coalgebra. Since the lifting preserves initial morphisms, then by [12,Proposition 12] (the same proof holds for the non-symmetric case), bd F α is the greatest V-categorical structure that makes α : (X, bd F α ) → F(X, bd F α ) a V-functor. In particular, this means that α : (X, bd F α ) → F(X, bd F α ) is initial. Therefore, as F corestricts to V-Cat sym and this category is closed under initial cones in V-Cat, we conclude that bd F α is symmetric.
Proposition A.9. Let F : V-Rel → V-Rel be a lax extension of a functor F : Set → Set and let α : X → FX be an F-coalgebra. Every symmetric Fsimulation on α is also a F s -(bi)simulation.

A.10 Details of Example 5.5(3)
For every V-category (X, a), the functor (X, a) × − : V-Cat → V-Cat is Kantorovich. Under certain conditions, this functor has a right adjoint which is Kantorovich as well. To see that, assume now that the underlying lattice of V is Heyting, and we denote the right adjoint of u ∧ − : V → V by (−) u : V → V. It has been shown [8,9] that (X, a) × − is left adjoint if and only if, for all x, z ∈ X and u, v ∈ V, y∈Y (a(x, y) ∧ u) ⊗ (a(y, z) ∧ v) ≥ a(x, z) ∧ (u ⊗ v).
For such X = (X, a), the right adjoint (−) X of X × − sends a V-category Y = (Y, b) to the V-category Y X = (Y X , c) with underlying set b(h(x 1 ), k(x 2 )) a(x1,x2) .
Let F : V-Cat sym → V-Cat sym be an ω-bounded functor that preserves initial morphisms, and let Λ be the set of all finitary predicate liftings for F. We claim that F is Λ-Kantorovich. Suppose that X is a symmetric V-category and x, y are elements of FX. Then, since F is ω-bounded, there is a finite subcategory i : X 0 → X and x , y ∈ FX 0 such that x = Fi(x ) and y = Fi(y ). Moreover, given that i is initial and for every cardinal κ the V-category V κ s is injective in V-Cat sym w.r.t. to initial morphisms, every V-functor f : X 0 → V κ can be factorized as f · i, for some V-functor f : X → V κ s . Hence, for every κ-ary λ in Λ, and every V-functor f : X 0 → V κ s , λ(f ) = λ(f ) · Fi. Therefore, as the cone of all V-functors λ(f ) : FX 0 → V s determined by each κ-ary λ in Λ and every V-functor f : X 0 → V κ is initial (see the proof of Theorem 3.9), by Lemma 5. where ar(λ) denotes the arity of the predicate lifting λ.