2. Preliminaries

We briefly review the theory of quantale-enriched categories from the point of view of quantale-enriched relations, as well as lax extensions and predicate liftings, and establish some notation and basic results. We warn the reader that there are several approaches to these topics in the literature, often with particular notations and nomenclature; we follow the conventions of Hofmann et al. (Reference Hofmann, Seal, Tholen, Clementino, Colebunders, Hofmann, Lowen, Lucyshyn-Wright, Seal and Tholen2014).

2.1 Quantale-enriched relations and categories

A quantale, more precisely a commutative unital quantale, is a complete lattice $\mathcal{V}$ that carries the structure of a commutative monoid $(\mathcal{V},\otimes,k)$ such that for every $u \in \mathcal{V}$ the map $u \otimes - \colon \mathcal{V}\to \mathcal{V}$ preserves suprema. Therefore, in a quantale, every map $u \otimes -\colon \mathcal{V} \to \mathcal{V}$ has a right adjoint $\hom(u,-) \colon \mathcal{V} \to \mathcal{V}$ , which is characterized by:

\[ u \otimes v \le w \iff v \le \hom(u,w),\]

for all $v, w \in \mathcal{V}$ . A quantale is nontrivial if the least element $\bot$ of $\mathcal{V}$ does not coincide with the greatest element $\top$ . Moreover, a quantale is integral if $\top$ is the unit of the monoid operation $\otimes$ of $\mathcal{V}$ , which we refer to as tensor or multiplication.

Remark 3. In categorical parlance, a quantale is a commutative monoid in the monoidal category of complete lattices and suprema-preserving maps.

Examples: 4. Quantales are common in mathematics and computer science.

1. (1) Every frame becomes a quantale with $\otimes = \wedge$ and $k = \top$ .

2. (2) Every left continuous t-norm (Alsina et al. Reference Alsina, Frank and Schweizer2006) defines a quantale on the unit interval equipped with its natural order. The following are some typical examples.

   1. a. The product t-norm has tensor given by multiplication and

      \[ \hom(u,v) = \begin{cases} \min(\frac{v}{u},1) & \text{if $u \neq 0$,} \\ 1 & \text{otherwise.} \end{cases} \]
      Via the map $[0,\infty]\to[0,1],u \mapsto e^{-u}$ , this quantale is isomorphic to the quantale $[0,\infty]$ of extended nonnegative real numbers used by Lawvere (Reference Lawvere1973) to define (generalized) metric spaces.

   2. b. The infimum t-norm has tensor given by infimum and

      \[ \hom(u,v) = \begin{cases} 1 & \text{if $u \le v$,} \\ v & \text{otherwise.} \end{cases} \]

   3. c. The Łukasiewicz t-norm has tensor given by $u \otimes v = \max(0,u+v-1)$ and $\hom(u,v) = \min(1,1-u+v)$ . This quantale is isomorphic to the quantale that is used implicitly in the usual treatment of 1-bounded metric spaces. More concretly, this quantale is isomorphic to the quantale based on the unit interval equipped with the dual of the natural order and with tensor given by truncated addition, $u\oplus v = \min(u+v,1)$ ; the map $[0,1] \to [0,1] \colon u \mapsto 1-u$ provides an isomorphism.

3. (3) Every commutative monoid $(M, \cdot, e)$ generates a quantale structure on $(\mathsf{P} M, \bigcup)$ , the free quantale on M. The tensor $\otimes$ on $\mathsf{P} M$ is defined by:

   \[ A \otimes B = \{ a \cdot b \mid a \in A \text{ and } b \in B\}, \]
   for all $A,B \subseteq M$ . The unit of this multiplication is the set $\{e\}$ .

For a quantale $\mathcal{V}$ and sets X, Y, a $\mathcal{V}$ -relation – an enriched relation – from X to Y is a map $X \times Y \to \mathcal{V}$ ; we let denote the space of such maps, and in particular write to indicate that r is a $\mathcal{V}$ -relation from X to Y. As for ordinary relations, $\mathcal{V}$ -relations can be composed via “matrix multiplication.” That is, for and , the composite is calculated pointwise by:

\[ (s \cdot r)(x,z)=\bigvee_{y \in Y} r(x,y) \otimes s(y,z),\]

for $x \in X$ and $z \in Z$ . The collection of all sets and $\mathcal{V}$ -relations between them forms the category $\mathcal{V}\text{-}\mathsf{Rel}$ . For each set X, the identity morphism on X is the $\mathcal{V}$ -relation that sends every diagonal element to k and all the others to $\bot$ .

Examples: 5. The category of relations enriched in the two-element frame is the usual category $\mathsf{Rel}$ of sets and relations. Relations enriched in left continuous t-norms are often called fuzzy or quantitative relations.

We can compare $\mathcal{V}$ -relations of type using the pointwise order:

\[ r \leq s \iff \forall (x,y) \in X \times Y,\, r(x,y) \leq s(x,y).\]

Every hom-set of $\mathcal{V}\text{-}\mathsf{Rel}$ becomes a complete lattice when equipped with this order, and an easy calculation shows that $\mathcal{V}$ -relational composition preserves suprema in each variable. Therefore, $\mathcal{V}\text{-}\mathsf{Rel}$ is a quantaloid and enjoys pleasant properties inherited from $\mathcal{V}$ . In particular, precomposition and poscomposition with a $\mathcal{V}$ -relation define maps with right adjoints that compute Kan lifts and extensions, respectively. The lift of a $\mathcal{V}$ -relation along -relation defined by the property:

for every

We can compute lifts explicitly as:

Dually, the extension of a $\mathcal{V}$ -relation along is the $\mathcal{V}$ -relation defined by the property:

for every

Elementwise, we obtain

Being a quantaloid, $\mathcal{V}\text{-}\mathsf{Rel}$ is a 2-category and, therefore, it makes sense to talk about adjoint $\mathcal{V}$ -relations; as usual, is left adjoint to , written $l \dashv r$ , if $l \cdot r \leq 1_Y$ and $1_X \leq r \cdot l $ .

Proposition 6. Consider $\mathcal{V}$ -relations , , and .

1. (1)

2. (2)

3. (3)

4. (4)

Proof. All claims follow straightforwardly from the universal properties of lifts and extensions.

If $\mathcal{V}$ is nontrivial, we can see $\mathcal{V}\text{-}\mathsf{Rel}$ as an extension of $\mathsf{Set}$ through the faithful functor ${(-)}_\circ \colon \mathsf{Set} \to \mathcal{V}\text{-}\mathsf{Rel}$ that acts as identity on objects and interprets a function $f \colon X \to Y$ as the $\mathcal{V}$ -relation that sends every element of the graph of f to k and all the others to $\bot$ . To avoid unnecessary use of subscripts usually, we write f instead of $f_\circ$ .

The canonical isomorphism $X \times Y \simeq Y \times X$ in $\mathsf{Set}$ induces a contravariant involution in $\mathcal{V}\text{-}\mathsf{Rel}$ :

\[ {(-)}^\circ \colon {\mathcal{V}\text{-}\mathsf{Rel}}^\mathrm{op} \longrightarrow \mathcal{V}\text{-}\mathsf{Rel}\]

that maps objects identically and sends a $\mathcal{V}$ -relation to the $\mathcal{V}$ -relation defined by $r^\circ(y,x) = r(x,y)$ , the converse of r.

Remark 7. The converse of a function $f \colon X \to Y$ yields an adjunction $f \dashv f^\circ$ in $\mathcal{V}\text{-}\mathsf{Rel}$ (and this property of functions distinguishes them among $\mathcal{V}$ -relations precisely when the quantale $\mathcal{V}$ is integral and lean Hofmann et al. Reference Hofmann, Seal, Tholen, Clementino, Colebunders, Hofmann, Lowen, Lucyshyn-Wright, Seal and Tholen2014, Proposition III.1.2.1). For every set Z, this adjunction extends to the following ones:

1. (1) $ f \cdot (-) \dashv f^\circ \cdot (-) \colon \mathcal{V}\text{-}\mathsf{Rel}(Z,X) \to \mathcal{V}\text{-}\mathsf{Rel}(Z,Y)$ ;

2. (2) $(-) \cdot f^\circ \dashv (-) \cdot f \colon \mathcal{V}\text{-}\mathsf{Rel}(X,Z) \to \mathcal{V}\text{-}\mathsf{Rel}(Y,Z)$ .

The next results collect some useful facts about the interplay between extensions, functions, relations, and involution.

Proposition 8. Let and be $\mathcal{V}$ -relations, and let $f \colon A \to X$ and $g \colon B \to Y$ be functions. Then the following holds.

1. (1)

2. (2) ;

3. (3) .

Proof. All claims are consequences of the uniqueness of adjoints. For example, to show 2 it is sufficient to observe that from Remark 7, it follows that both and are right adjoint to $(s\cdot g)\cdot-$ .

Proposition 9. Let and be $\mathcal{V}$ -relations. Then,

Where $a^\circ$ denotes the converse of the map $a \colon 1 \to A$ that selects the element a.

Corollary 10. Let , be $\mathcal{V}$ -relations, and $y \colon 1 \to Y$ a function. Then,

Proposition 11. Let and be $\mathcal{V}$ -relations, and let $f \colon X \to A$ and $g \colon Y \to B$ be functions such that $g \cdot r \leq s \cdot f $ . Then, $r \cdot f^\circ \leq g^\circ \cdot s$ .

Proof. $r \cdot f^\circ \leq g^\circ \cdot g \cdot r \cdot f^\circ \leq g^\circ \cdot s \cdot f \cdot f^\circ \leq g^\circ \cdot s$ .

Category theory underlines preordered sets as the fundamental ordered structures. For an arbitrary quantale $\mathcal{V}$ , the same role is taken by $\mathcal{V}$ -categories. Analogously to the classical case, we say that a $\mathcal{V}$ -relation is reflexive if $1_X \leq r$ , and transitive if $ r \cdot r \leq r$ . A $\mathcal{V}$ -category is a pair (X,a) consisting of a set X of objects and a reflexive and transitive $\mathcal{V}$ -relation ; a $\mathcal{V}$ -functor $(X,a) \to (Y,b)$ is map $f \colon X \to Y $ such that $f \cdot a \leq b \cdot f$ . Clearly, $\mathcal{V}$ -categories and $\mathcal{V}$ -functors define a category, denoted as $\mathcal{V}\text{-}\mathsf{Cat}$ .

Remark 12. As the nomenclature suggests, the notions of $\mathcal{V}$ -category and $\mathcal{V}$ -functor come from enriched category theory (Kelly Reference Kelly1982; Lawvere Reference Lawvere1973; Stubbe Reference Stubbe2014). In fact, unravelling the definition of reflexive and transitive $\mathcal{V}$ -relation yields the typical definition of quantale-enriched category: a pair (X,a) consisting of a set X and a map $a \colon X \times X \to \mathcal{V}$ that satisfies the inequalities:

\begin{align*} k \le a(x,x) && \text{and} && a(x,y) \otimes a(y,z) \leq a(x,z) \end{align*}

for all $x,y,z \in X$ . Similarly, a $\mathcal{V}$ -functor $f \colon (X,a) \to (Y,b)$ is a map $f \colon X \to Y$ such that, for all $x,y \in X$ ,

\[ a(x,y) \le b(f(x),f(y)). \]

Examples: 13. The following are some familiar examples of quantale-enriched categories.

1. (1) The category $\mathsf{2}\text{-}\mathsf{Cat}$ is equivalent to the category $\mathsf{Ord}$ of preordered sets and monotone maps.

2. (2) Metric, ultrametric, and bounded metric spaces à la Reference Lawvere Lawvere (1973) can be seen as categories enriched in left continuous t-norms:

   1. a. With multiplication $*$ , $[0,1]_*\text{-}\mathsf{Cat}$ is equivalent to the category $\mathsf{Met}$ of (generalized) metric spaces and non-expansive maps.

   2. b. With infimum $\wedge$ , $[0,1]_\wedge\text{-}\mathsf{Cat}$ is equivalent to the category $\mathsf{UMet}$ of (generalized) ultrametric spaces and non-expansive maps.

   3. c. With the Łukasiewicz tensor $\odot$ , $[0,1]_{\odot}\text{-}\mathsf{Cat}$ is equivalent to the category $\mathsf{BMet}$ of (generalized) bounded-by-1 metric spaces and non-expansive maps.

3. (3) Categories enriched in a free quantale $\mathsf{P} M$ on a monoid M (such as $M=\Sigma^*$ for some alphabet $\Sigma$ ) can be interpreted as labeled transition systems with labels in M: in a $\mathsf{P} M$ -category (X,a), the objects represent the states of the system, and we can read $m\in a(x,y)$ as an m-labeled transition from x to y.

Definition 14. The dual of a $\mathcal{V}$ -category (X,a) is the $\mathcal{V}$ -category ${(X,a)}^\mathrm{op} = (X,a^\circ)$ .

Remark 15. The quantale $\mathcal{V}$ becomes a $\mathcal{V}$ -category under the canonical structure $\hom \colon \mathcal{V} \times \mathcal{V} \to \mathcal{V}$ . In fact, for every set S, we can form the S-power $\mathcal{V}^S$ of $\mathcal{V}$ which has as underlying set all functions $h \colon S \to \mathcal{V}$ and $\mathcal{V}$ -category structure $[-,-]$ given by:

\[ [h,l] = \bigwedge_{s \in S} \hom(h(s),l(s)), \]

for all $h,l \colon S \to \mathcal{V}$ . For instance, for the quantale $([0,1],\oplus,0)$ , where [0, 1] is equipped with the dual of the natural ordering (Example 4(2c)), this distance on $[0,1]\text{-}\mathsf{Rel}(X,Y)=[0,1]^{X\times Y}$ is given in terms of the natural order on [0, 1] by:

\[ [r,s] = \sup \big\{\max(s(x,y)-r(x,y),0) \mid (x,y) \in X \times Y\big\}. \]

Every $\mathcal{V}$ -category (X,a) carries a natural order defined by:

\[ x \le y \text{ if } k \le a(x,y),\]

which induces a faithful functor $\mathcal{V}\text{-}\mathsf{Cat} \to \mathsf{Ord}$ . A $\mathcal{V}$ -category (X,a) is separated if

\begin{equation} (k\le a(x,y)\quad\&\quad k\leq a(y,x))\implies x=y\end{equation}

for all $x,y\in X$ . That is, (X,a) is separated if the natural order defined above is anti-symmetric.

Remark 16. The natural order of the $\mathcal{V}$ -category $\mathcal{V}$ is just the order of the quantale $\mathcal{V}$ . The natural order of $\mathcal{V}^S$ is calculated pointwise, and as such is complete. Furthermore, the $\mathcal{V}$ -category $\mathcal{V}^{S}$ is complete in the sense of enriched category theory; in particular, $\mathcal{V}^{S}$ has tensors, which are given by $(u\otimes h)(s)=u\otimes h(s)$ , for $u\in\mathcal{V}$ and $h\in\mathcal{V}^{S}$ . Tensors are compatible with composition, that is, $(u \otimes f) \cdot g = u \otimes (f \cdot g)$ for all $u \in \mathcal{V}$ , $f \colon S \to \mathcal{V}$ , and $g \colon S' \to S$ . Here, we recall that a $\mathcal{V}$ -category (X,a) is tensored if, for every $x\in X$ , the $\mathcal{V}$ -functor $a(x,-)\colon X\to\mathcal{V}$ has a left adjoint $-\otimes x \colon\mathcal{V}\to X$ in $\mathcal{V}\text{-}\mathsf{Cat}$ . We also note that $\mathcal{V}$ -functors between tensored $\mathcal{V}$ -categories are characterized by a pleasant property: a map $f \colon X\to Y$ between tensored $\mathcal{V}$ -categories X and Y is a $\mathcal{V}$ -functor if and only if f is monotone and, for all $u\in\mathcal{V}$ and $x\in X$ , $u\otimes f(x)\le f(u\otimes x)$ (Stubbe Reference Stubbe2006).

2.2 Lax extensions

A lax extension of a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ to $\mathcal{V}\text{-}\mathsf{Rel}$ consists of a map:

such that

(L1) $r \le r' \implies \mathsf{L} r \le \mathsf{L} r'$ ,

(L2) $\mathsf{L} s\cdot\mathsf{L} r\le\mathsf{L} (s\cdot r)$ ,

(L3) $\mathsf{F} f \le \mathsf{L} f$ and ${(\mathsf{F} f)}^\circ\le\mathsf{L}(f^\circ)$

for all , , and $f \colon X \to Y$ . The first condition means precisely that $\mathsf{L}$ induces a monotone map:

(1) \begin{equation} \mathsf{L}_{X,Y} \colon \mathcal{V}\text{-}\mathsf{Rel}(X,Y) \longrightarrow \mathcal{V}\text{-}\mathsf{Rel}(\mathsf{F} X, \mathsf{F} Y),\end{equation}

for all sets X and Y. We say that a lax extension is $\mathcal{V}$ -enriched if this map satisfies the stronger condition of being a $\mathcal{V}$ -functor, for all sets X and Y (see Remark 15):

(L1’) $[r,r'] \leq [\mathsf{L} r, \mathsf{L} r']$

for all . A lax extension $\mathsf{L}$ is identity-preserving if $\mathsf{L} 1_X = 1_{\mathsf{F} X}$ for every set X.

Remark 17. It is common to refer to various forms of extensions of $\mathsf{Set}$ -functors to $\mathsf{Rel}$ as relators, relational liftings, or lax relational liftings.

Example 18. The prototypical example of a lax extension is the Barr extension to $\mathsf{Rel}$ of a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ that preserves weak pullbacks. Taking advantage of the fact that every relation can be described as a span:

the Barr extension of $\mathsf{F}$ sends r to the relation $\mathsf{F} p_2 \cdot {\mathsf{F} p_1}^\circ$ .

Kurz and Velebil (Reference Kurz and Velebil2016) provide a concise survey-oriented toward applications of lax extensions in coalgebra and logic that deals mostly with lax extensions to $\mathsf{Rel}$ . Regarding lax extensions to $\mathcal{V}\text{-}\mathsf{Rel}$ , work within the framework of monoidal topology (Clementino and Hofmann Reference Clementino and Hofmann2004; Schubert and Seal Reference Schubert and Seal2008; Seal Reference Seal2005) encompasses a substantial amount of results.

Our main motivation to study lax extensions stems from the fact that they provide a framework for the coalgebraic treatment of various notions of quantale-valued (bi)simulation (Gavazzo Reference Gavazzo2018; Goncharov et al. Reference Goncharov, Hofmann, Nora, Schröder, Wild, Kupferman and Sobocinski2023; Hughes and Jacobs Reference Hughes and Jacobs2004; Marti and Venema Reference Marti and Venema2015; Rutten Reference Rutten1998;Wild and Schröder Reference Wild, Schröder, Konnov and Kovács2020, Reference Wild, Schröder, Kiefer and Tasson2021). Recall that an $\mathsf{F}$ -coalgebra for a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ is a pair $(X,\alpha)$ consisting of a set X of states and a transition map $\alpha\colon X\to\mathsf{F} X$ ; such coalgebras are viewed as transition systems, with $\mathsf{F}$ determining the transition type (e.g., if $\mathsf{F}$ is just powerset, then $\mathsf{F}$ -coalgebras are relational transition systems in the usual sense). Now given a lax extension $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ of $\mathsf{F}$ , an $\mathsf{L}$ -simulation between $\mathsf{F}$ -coalgebras $(X,\alpha)$ and $(Y,\beta)$ is a $\mathcal{V}$ -relation such that

\[ \beta \cdot s \leq \mathsf{L} s \cdot \alpha.\]

If the lax extension $\mathsf{L}$ preserves converse, then $\mathsf{L}$ -simulations are more suitably called $\mathsf{L}$ -bisimulations. Since $\mathcal{V}\text{-}\mathsf{Rel}$ is a quantaloid, there is the largest $\mathsf{L}$ -(bi)simulation between two given coalgebras, which is termed $\mathsf{L}$ -(bi)similarity. In the two-valued case, it has been shown by Marti and Venema (Reference Marti and Venema2015) that the notion of $\mathsf{L}$ -bisimilarity arising from an identity-preserving lax extension that preserves converse coincides with the standard coalgebraic notion of behavioral equivalence. On the other hand, the notion of $\mathcal{V}$ -enriched lax extension has been instrumental in establishing quantitative Hennessy-Milner and van Benthem-type theorems (Wild and Schröder Reference Wild, Schröder, Konnov and Kovács2020, Reference Wild, Schröder, Kiefer and Tasson2021). It has been introduced with $\mathcal{V}$ -enrichment replaced with (L1) together with the condition:

(L4) For every set X and every $u\in\mathcal{V}$ , $u\otimes 1_{\mathsf{F} X}\le \mathsf{L}(u\otimes 1_{X})$ ,

where $u \otimes 1_A$ denotes the tensor of u and $1_A$ as defined in Remark 16 (and this condition is shown to be equivalent to a variant of $\mathcal{V}$ -enrichment where $\mathcal{V}$ is equipped with the symmetrized distance). The next result records the equivalence of these and other conditions.

Theorem 19. For a lax extension $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ , the following assertions are equivalent.

1. (i) $\mathsf{L}$ is $\mathcal{V}$ -enriched.

2. (ii) For every set X and every $u\in\mathcal{V}$ , $u\leq[1_{\mathsf{F} X},\mathsf{L}(u\otimes 1_{X})]$ .

3. (iii) For every set X and every $u\in\mathcal{V}$ , $u\otimes 1_{\mathsf{F} X}\le \mathsf{L}(u\otimes 1_{X})$ .

4. (iv) For every $\mathcal{V}$ -relation and every $u\in\mathcal{V}$ , $u\otimes\mathsf{L} r\le\mathsf{L}(u\otimes r)$ .

Proof. Note that a lax extension satisfies the monotonicity condition (L1); therefore, the statements (i) and (iv) are equivalent by general results about tensored $\mathcal{V}$ -categories recalled in Remark 16. Now assume (i). Then, for every $u\in\mathcal{V}$ ,

\begin{equation} u\le [1_{X},u\otimes 1_{X}]\le [\mathsf{L} 1_{X},\mathsf{L}(u\otimes 1_{X})]\le [1_{\mathsf{F} X},\mathsf{L}(u\otimes 1_{X})]. \end{equation}

The implication (ii) $\Rightarrow$ (iii) follows from the adjunction $-\otimes 1_{\mathsf{F} X}\dashv[1_{\mathsf{F} X},-]$ . Finally, assume (iii). Then, for every $u\in\mathcal{V}$ and ,

\begin{equation} \mathsf{L}(u\otimes r) =\mathsf{L}(u\otimes (r\cdot 1_{X})) =\mathsf{L}(r\cdot (u\otimes 1_{X})) \geq \mathsf{L} r\cdot \mathsf{L}(u\otimes 1_{X}) \geq \mathsf{L} r\cdot (u\otimes 1_{\mathsf{F} X}) =u\otimes \mathsf{L} r. \end{equation}

Remark 20. If $k=\top$ in $\mathcal{V}$ , then $u\otimes 1_{X}\le 1_{X}$ and therefore $[u\otimes 1_{X},1_{X}]=k=\top$ . Hence,

\begin{equation} [1_{X},u\otimes 1_{X}]=[1_{X},u\otimes 1_{X}]\wedge [u\otimes 1_{X},1_{X}], \end{equation}

and $\mathcal{V}$ -enrichment of a lax extension can be equivalently expressed using the symmetrization of the canonical structure on $\mathcal{V}$ .

Remark 21. For the quantale [0, 1] of Example 4(2c), Wild and Schröder (Reference Wild, Schröder, Konnov and Kovács2020) prove the equivalence (i) $\Leftrightarrow$ (iii) of Theorem 19, but with non-expansiveness of (1) defined with respect to the symmetric Euclidean metric on [0, 1] and with $\Delta_{\varepsilon,X}$ denoting $\varepsilon\otimes 1_{X}$ . Since this quantale is integral, Remark 20 ensures that this is equivalent to considering the asymmetric distance.

In the remainder of this subsection, we collect some useful properties of lax extensions. First, we note that they preserve certain composites of $\mathcal{V}$ -relations and functions strictly.

Proposition 22. Let $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ be a lax extension, and let $f \colon X \to Y$ , $g \colon W \to Z$ be functions and a $\mathcal{V}$ -relation. Then,

1. (1) $\mathsf{L} (s\cdot f)=\mathsf{L} s\cdot\mathsf{L} f=\mathsf{L} s\cdot \mathsf{F} f$ ,

2. (2) $\mathsf{L} (g^\circ\cdot s)= \mathsf{L}(g^{\circ})\cdot\mathsf{L} s={(\mathsf{F} g)}^\circ\cdot\mathsf{L} s$ .

Proof. See, for example, Hofmann et al. (Reference Hofmann, Seal, Tholen, Clementino, Colebunders, Hofmann, Lowen, Lucyshyn-Wright, Seal and Tholen2014, Proposition III.1.4.4).

Proposition 23. Let $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ be a lax functor. Then, items 5 and 5 of Proposition 5 are equivalent as conditions on $\mathsf{L}$ . If $\mathsf{L}$ satisfies them, then $\mathsf{L}$ is a lax extension of $\mathsf{F}$ .

Proof. See, for example, Hofmann et al. (Reference Hofmann, Seal, Tholen, Clementino, Colebunders, Hofmann, Lowen, Lucyshyn-Wright, Seal and Tholen2014, Proposition III.1.4.3).

A morphism of lax extensions $\alpha \colon (\mathsf{G}, \mathsf{L}) \to (\mathsf{F},\mathsf{L}_\mathsf{F})$ is a natural transformation $\alpha \colon \mathsf{G} \to \mathsf{F}$ that is oplax as a transformation $\alpha \colon \mathsf{L}_\mathsf{G} \to \mathsf{L}_\mathsf{F}$ ; that is, for every ,

\[ \alpha_Y \cdot \mathsf{L}_\mathsf{G} r \leq \mathsf{L}_\mathsf{F} r \cdot \alpha_X.\]

When no ambiguities arise, we simply write $\alpha \colon \mathsf{L}_\mathsf{G} \to \mathsf{L}_\mathsf{F}$ .

Disregarding size constraints, we have that lax extensions and their morphisms form a category which is topological over the category of endofunctors on $\mathsf{Set}$ (Schubert and Seal Reference Schubert and Seal2008, Remark 3.5). Given a family $\alpha = {(\alpha_i \colon \mathsf{F} \to \mathsf{F}_i)}_{i \in I}$ of natural transformations where each $\mathsf{F}_i$ carries a lax extension $\mathsf{L}_i$ to $\mathcal{V}\text{-}\mathsf{Rel}$ , the initial extension $\mathsf{L}_\alpha$ is defined by:

\[ \mathsf{L}_\alpha r = \bigwedge_{i \in I} \alpha_Y^\circ \cdot \mathsf{L}_i r \cdot \alpha_X,\]

for . In particular, the supremum and infimum over a class of lax extensions of a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ is a lax extension of $\mathsf{F}$ .

Every lax extension has a dual as introduced by Seal (Reference Seal2005). The dual lax extension $\mathsf{L}^\circ \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ of a lax extension $\mathsf{L}\colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ is the lax extension of $\mathsf{F} \colon \mathsf{Set} \to\mathsf{Set}$ that is defined by the assignment:

\[ r \longmapsto {\mathsf{L} (r^\circ)}^\circ.\]

Notably, this means that we can symmetrize lax extensions. The symmetrization ${\widehat{\mathsf{F}}}^s \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ of a lax extension $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ is the lax extension obtained as the meet of $\mathsf{L}$ and ${\mathsf{L}}^\circ$ .

Finally, an important application of lax extensions of a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ to $\mathcal{V}\text{-}\mathsf{Rel}$ is to construct liftings of $\mathsf{F}$ to $\mathcal{V}\text{-}\mathsf{Cat}$ ; that is, endofunctors on $\mathcal{V}\text{-}\mathsf{Cat}$ that make the following diagram commute, where the vertical arrows represent the forgetful functor:

The lifting $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Cat} \to \mathcal{V}\text{-}\mathsf{Cat}$ induced by a lax extension $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ sends a $\mathcal{V}$ -category (X,a) to the $\mathcal{V}$ -category $(\mathsf{F} X, \mathsf{L} a)$ .

2.3 Predicate liftings

Given a cardinal $\kappa$ , a $\kappa$ -ary predicate lifting for a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ is a natural transformation:

\[ \mu \colon \mathsf{Q}_{\mathcal{V}^\kappa} \longrightarrow \mathsf{Q}_\mathcal{V} \mathsf{F},\]

where, for a set Y, $\mathsf{Q}_Y \colon \mathsf{Set}^\mathrm{op} \to \mathsf{Set}$ denotes the functor:

\[ \mathsf{Set}(-,Y) \colon \mathsf{Set}^\mathrm{op} \to \mathsf{Set};\]

that is, we can think of elements of $\mathsf{Q}_{\mathcal{V}^\kappa} X$ as $\kappa$ -indexed families of $\mathcal{V}$ -valued predicates on X. We say that $\mu \colon \mathsf{Q}_{\mathcal{V}^\kappa}\to \mathsf{Q}_\mathcal{V} \mathsf{F}$ is monotone if for every set X the map:

\[ \mu_X \colon \mathsf{Set}(X,\mathcal{V}^\kappa) \to \mathsf{Set}(\mathsf{F} X,\mathcal{V})\]

is monotone w.r.t the pointwise orders induced by the corresponding powers of the complete lattice $\mathcal{V}$ , and $\mathcal{V}$ -enriched if this map is actually a $\mathcal{V}$ -functor w.r.t to the $\mathcal{V}$ -categorical structures induced by the corresponding powers of the $\mathcal{V}$ -category $\mathcal{V}$ (see Remark 15). Note that when talking about monotone or $\mathcal{V}$ -enriched predicate liftings, instead of $\mathsf{Q}_{\mathcal{V}^\kappa}\colon\mathsf{Set}^{\mathrm{op}}\to\mathsf{Set}$ we actually consider the functors $\mathsf{Q}_{\mathcal{V}^\kappa}\colon\mathsf{Set}^{\mathrm{op}}\to\mathsf{Pos}$ and $\mathsf{Q}_{\mathcal{V}^\kappa}\colon\mathsf{Set}^{\mathrm{op}}\to\mathcal{V}\text{-}\mathsf{Cat}$ , respectively. In any case, the functors $\mathsf{Q}_{\mathcal{V}^\kappa}$ are part of an adjunction: in general, for a category $\mathsf{A}$ and an object A with powers in $\mathsf{A}$ , we have

Moreover, for all functors $\mathsf{G} \colon \mathsf{Set}^{\mathrm{op}}\to\mathsf{A}$ and $\mathsf{F} \colon\mathsf{Set}\to\mathsf{Set}$ , the adjunction above induces a bijection between natural transformations $\mathsf{G}\to\mathsf{Q}_{A}\mathsf{F}$ and natural transformations $\mathsf{F}\to\mathsf{A}(\mathsf{G},A)$ . In particular, a $\kappa$ -ary predicate lifting $\mu \colon \mathsf{Q}_{\mathcal{V}^\kappa}\to\mathsf{Q}_\mathcal{V}\mathsf{F}$ corresponds to a natural transformation:

(2) \begin{equation} \overline{\mu}\colon\mathsf{F} \longrightarrow\mathsf{A}(\mathsf{Q}_{\mathcal{V}^\kappa},\mathcal{V}). \end{equation}

For $\mathsf{A}=\mathsf{Set}$ (respectively $\mathsf{A}=\mathsf{Pos}$ ) and $\mathcal{V}=\mathsf{2}$ , the codomain of $\overline{\mu}$ is the generalized (monotone) neighborhood functor (Marti and Venema Reference Marti and Venema2015; Schröder and Pattinson Reference Schröder and Pattinson2010). A collection M of predicate liftings is separating if the cone

\begin{equation} (\overline{\mu}_{X}\colon \mathsf{F} X\longrightarrow\mathsf{Set}(\mathsf{Q}_{\mathcal{V}^\kappa} X,\mathcal{V}))_{\mu\in M} \end{equation}

(with $\overline\mu$ as per (2)) is mono (i.e., jointy injective) for every set X.

Remark 24. Predicate liftings are instrumental in coalgebraic logic (Cîrstea et al. 2011; Pattinson Reference Pattinson2004; Schröder Reference Schröder2008). As a basic example, consider the Kripke semantics of modal logic. Coalgebras for the covariant powerset functor $\mathsf{P} \colon \mathsf{Set} \to \mathsf{Set}$ correspond precisely to Kripke frames. In coalgebraic modal logic, the Kripke semantics of the modal logic K is recovered by interpreting the modal operator $\Diamond$ as the predicate lifting $\Diamond \colon \mathsf{Q}_2 \to \mathsf{Q}_2\mathsf{P}$ whose X-component is defined by:

\[ \Diamond_X(A)= \{B \subseteq X \mid A \cap B \neq \varnothing\}, \]

or by interpreting the modal operator $\Box$ as the predicate lifting $\Box \colon \mathsf{Q}_2 \to \mathsf{Q}_2\mathsf{P}$ whose X-component is defined by:

\[ \Box_X(A)= \{B \subseteq X \mid B \subseteq A\}. \]

Both $\Diamond$ and $\Box$ are monotone (= $\mathsf{2}$ -enriched) predicate liftings, and both $\{\Diamond\}$ and $\{\Box\}$ are separating for $\mathsf{P}$ . Similarly, $\mathcal{V}$ -valued predicate liftings provide the semantical framework for modal operators in $\mathcal{V}$ -valued coalgebraic modal logic (Forster et al. Reference Forster, Goncharov, Hofmann, Nora, Schröder, Wild, Klin and Pimentel2023; Goncharov et al. Reference Goncharov, Hofmann, Nora, Schröder, Wild, Kupferman and Sobocinski2023; Wild and Schröder Reference Wild, Schröder, Konnov and Kovács2020, Reference Wild, Schröder, Kiefer and Tasson2021). In inductive definitions of the semantics of (coalgebraic) modal logics over a coalgebra $(X,\alpha)$ , formulae are typically interpreted as subsets of the state set X. If $\heartsuit$ is the predicate lifting interpreting a modality $\heartsuit$ , and $Y\subseteq X$ is the interpretation of a formula $\phi$ , then the formula $\heartsuit\phi$ is interpreted by the set $\alpha^{-1}[\heartsuit_X(Y)]$ . For instance, the interpretation of $\Diamond\phi$ in a $\mathsf{P}$ -coalgebra $(X,\alpha)$ , again with $Y\subseteq X$ being the interpretation of $\phi$ , consists of all $x\in X$ such that $\alpha(x)\cap Y\neq\emptyset$ , that is, of all states that have some successor satisfying $\phi$ .

Remark 25. Generalizing a corresponding observation for the $\mathsf{2}$ -valued case (Schröder Reference Schröder2008), we note that by the Yoneda lemma, a predicate lifting $\mu \colon \mathsf{Q}_{\mathcal{V}^\kappa} \to \mathsf{Q}_\mathcal{V} \mathsf{F}$ is equivalently given by a morphism of type $\mathsf{F} \mathcal{V}^\kappa \to \mathcal{V}$ , the image of the identity map on $\mathcal{V}^\kappa$ under $\mu_{\mathcal{V}^\kappa}$ . In particular, for given $\kappa$ , the collection of all $\kappa$ -ary predicate liftings is small. The X-component $\mu_X \colon \mathsf{Q}_{\mathcal{V}^\kappa} X \to \mathsf{Q}_\mathcal{V} \mathsf{F} X $ of the predicate lifting induced by a morphism $g\colon \mathsf{F} \mathcal{V}^\kappa \to \mathcal{V}$ is defined by:

\[ \mu_X(f)= g \cdot \mathsf{F} f. \]

In the two-valued case, it has been shown that separating sets of finitary predicate liftings for a finitary $\mathsf{Set}$ -functor give rise to expressive coalgebraic modal logics (Schröder Reference Schröder2008), and that a finitary $\mathsf{Set}$ -functor admits a separating set of finitary monotone predicate liftings if and only if the functor admits an identity-preserving lax extension to $\mathsf{Rel}$ (Marti and Venema Reference Marti and Venema2015). On the other hand, sets of $\mathcal{V}$ -enriched predicate liftings satisfying a quantitative analogue of separation feature in expressiveness results for quantitative coalgebraic modal logics for behavioral distances (Forster et al. Reference Forster, Goncharov, Hofmann, Nora, Schröder, Wild, Klin and Pimentel2023; König and Mika-Michalski Reference König, Mika-Michalski, Schewe and Zhang2018; Wild and Schröder Reference Wild, Schröder, Konnov and Kovács2020, Reference Wild, Schröder, Kiefer and Tasson2021, Reference Wild and Schröder2022). We conclude this subsection with a characterization of $\mathcal{V}$ -enriched predicate liftings, which corresponds to Theorem 19 and can be shown similarly.

Theorem 26. For a predicate lifting $\mu \colon \mathsf{Q}_{\mathcal{V}^\kappa} \to \mathsf{Q}_\mathcal{V}\mathsf{F}$ , the following are equivalent.

1. (i) $\mu$ is $\mathcal{V}$ -enriched.

2. (ii) $\mu$ is monotone and for every $u\in\mathcal{V}$ , $u\leq[\mu(1_{\mathcal{V}^\kappa}),\mu(u\otimes 1_{\mathcal{V}^\kappa})]$ .

3. (iii) $\mu$ is monotone and for every $u\in\mathcal{V}$ , $u\otimes \mu(1_{\mathcal{V}^\kappa}) \le \mu(u\otimes 1_{\mathcal{V}^\kappa})$ .

$\mu$ is monotone and for every function $f \colon X \to \mathcal{V}^\kappa$ and every $u\in\mathcal{V}$ , $u\otimes\mu(f) \le \mu(u\otimes f)$ .

Remark 27. Analogously to Remark 20, if $k = \top$ , then $\mathcal{V}$ -enrichment of predicate liftings can be equivalently expressed using the symmetrization of the canonical structure on $\mathcal{V}$ .

3. From Lax Extensions to Predicate Liftings

We proceed to investigate the relationship between lax extensions and predicate liftings, using $\mathcal{V}$ -relations as a natural language to express their interaction. We begin by expressing predicate liftings in terms of $\mathcal{V}$ -relations, with an view to constructing predicate liftings from lax extensions.

We recall that the category $\mathsf{Set}$ is Cartesian-closed. In particular, this means that for every set I, the evaluation map $\mathrm{ev}_I \colon \mathcal{V}^I \times I \to \mathcal{V}$ , which we think of as a $\mathcal{V}$ -relation defined by $\mathrm{ev}_I(f,i) = f(i)$ , exhibits the function space $\mathcal{V}^I$ as an exponential object. Therefore, for every set X, currying/uncurrying defines an isomorphism:

\[ \mathcal{V}\text{-}\mathsf{Rel}(X,I) \simeq \mathsf{Set}(X,\mathcal{V}^I).\]

We denote the right adjunct of a $\mathcal{V}$ -relation by $r^\sharp \colon X \to \mathcal{V}^I$ and the left adjunct of a function $f \colon X \to \mathcal{V}^I$ by .

Lemma 29. Let $\mathcal{V}$ be a quantale and I a set. For all functions $f \colon X \to Y$ and $g \colon Y \to \mathcal{V}^I$ ,

\[ (g \cdot f)^\flat = g^\flat \cdot f_\circ. \]

Proof. Let $x \in X$ and $i \in I$ . Then, by definition,

$$(g \cdot f)^\flat (x,i) = (g \cdot f)(x)(i) = g(f(x))(i) = g^\flat(f(x),i) = (g^\flat \cdot f_\circ) (x,i).$$

In the sequel, given a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ and a cardinal $\kappa$ , let $\mathcal{V}\text{-}\mathsf{Rel}_\circ(\mathsf{F} -, \kappa) \colon \mathsf{Set}^\mathrm{op} \to \mathsf{Set}$ denote the composite functor:

The above lemma says essentially that the uncurrying bijection $(-)^\flat$ is natural, so we immediately obtain the following:

Therefore, we mainly view predicate liftings as natural transformations $\mathcal{V}\text{-}\mathsf{Rel}_\circ(-,\kappa) \to \mathcal{V}\text{-}\mathsf{Rel}_\circ(\mathsf{F} -,1)$ from now on, although to keep the notation simple we still write $\mathsf{Q}_{\mathcal{V}^\kappa} \to \mathsf{Q}_\mathcal{V} \mathsf{F}$ . Explicitly, naturality of a transformation $\mu\colon\mathcal{V}\text{-}\mathsf{Rel}_\circ(-,\kappa) \to \mathcal{V}\text{-}\mathsf{Rel}_\circ(\mathsf{F} -,1)$ means that for $f\colon X\to Y$ and , we have

\begin{equation*} \mu_X(r\cdot f) = \mu_Y(r)\cdot \mathsf{F} f.\end{equation*}

Now, it is easy to see that every lax extension induces natural transformations with the desired domain:

This description suggests that to construct $\kappa$ -ary predicate liftings from a lax extension, we should construct natural transformations $\mathcal{V}\text{-}\mathsf{Rel}_\circ (\mathsf{F} -, F\kappa) \to \mathcal{V}\text{-}\mathsf{Rel}_\circ (\mathsf{F} -, 1)$ . The easiest way to do so is to select a $\mathcal{V}$ -relation . Then, each component of the resulting predicate lifting $\mathcal{V}\text{-}\mathsf{Rel}_\circ (-,\kappa) \to \mathcal{V}\text{-}\mathsf{Rel}_\circ (\mathsf{F} -, 1)$ is computed as:

\[ f \longmapsto \textbf{r} \cdot \mathsf{L} f.\]

This motivates our notion of predicate lifting induced by a lax extension, which, as we shall explain at the end of this section, generalizes the notion of Moss lifting (Kurz and Leal Reference Kurz, Leal, Abramsky, Mislove and Palamidessi2009; Leal Reference Leal, Adámek and Kupke2008; Marti and Venema Reference Marti and Venema2015;Wild and Schröder Reference Wild, Schröder, Kiefer and Tasson2021).

Immediately from the definition, we have:

Example 34. Consider the lax extension of the covariant powerset functor $\mathsf{P} \colon \mathsf{Set} \to \mathsf{Set}$ to $\mathsf{Rel}$ given by:

\[ B (\widehat{\mathsf{P}}r) C \iff \forall c \in C, \exists b \in B,\, b \mathrel{r} c. \]

The unary Moss lifting for $\widehat{\mathsf{P}}$ determined by the element $1 \in \mathsf{P} 1$ is the predicate lifting $\Diamond \colon \mathsf{Q}_\mathcal{V} \to \mathsf{Q}_\mathcal{V}\mathsf{P}$ whose X-component is defined by:

\[ A \longmapsto \{B \subseteq X \mid A \cap B \neq \varnothing\}. \]

The Moss lifting for the dual extension of $\widehat{\mathsf{P}}$ determined by the element $1 \in \mathsf{P} 1$ is the predicate lifting $\Box \colon \mathsf{Q}_\mathcal{V} \to \mathsf{Q}_\mathcal{V}\mathsf{P}$ whose X-component is computed as:

\[ A \longmapsto \{B \subseteq X \mid B \subseteq A\}. \]

On the other hand, the unary Moss lifting for the Barr extension $\bar{\mathsf{P}}$ of $\mathsf{P}$ (the symmetrization of $\widehat{\mathsf{P}}$ ) determined by the element $1 \in \mathsf{P} 1$ is the predicate lifting $\nabla \colon \mathsf{Q}_\mathcal{V} \to \mathsf{Q}_\mathcal{V}\mathsf{P}$ whose X-component is defined by:

\[ A \longmapsto \{B \subseteq X \mid B \neq\varnothing \wedge B \subseteq A\}. \]

As mentioned in Remark 25, by the Yoneda Lemma, $\kappa$ -ary predicate liftings for a functor are completely determined by their action on the identity function on $\mathcal{V}^\kappa$ and the action of the functor. In the relational point of view, this means that $\kappa$ -ary predicate liftings are completely determined by the image of the evaluation relation . In the following, we will see that $\kappa$ -ary predicate liftings induced by a lax extension are completely determined by their action on the identity $\mathcal{V}$ -relation $1_\kappa \colon \kappa \to \kappa$ and the action of the lax extension. This characterization makes it easy to construct, detect, and manipulate predicate liftings induced by lax extensions (see Corollary 37, Lemma 77, and Proposition 39).

Proof. Let be a $\mathcal{V}$ -relation.

(i)⇒(ii) By definition.

(ii)⇒(iii) $ \mu(\mathrm{ev}_\kappa) = \textbf{r} \cdot \mathsf{L}\,\mathrm{ev}_\kappa = \textbf{r} \cdot \mathsf{L}1_\kappa \cdot \mathsf{L}\,\mathrm{ev}_\kappa = \mu(1_\kappa) \cdot \mathsf{L}\,\mathrm{ev}_\kappa. $

(iii)⇒(iv) By adjointness, the assumption implies that so we obtain .

(iv)⇒(i) Put . Then, since $\mu$ is a natural transformation,

\begin{align*} \mu(f) = \mu(\mathrm{ev}_\kappa \cdot f^\sharp) & = \mu(\mathrm{ev}_\kappa) \cdot \mathsf{F} f^\sharp \\ & = \textbf{r} \cdot \mathsf{L}\,\mathrm{ev}_\kappa \cdot \mathsf{F} f^\sharp = \textbf{r} \cdot \mathsf{L} f \end{align*}

where the last equality is by Proposition 22.

We crystallize the above into a Yoneda-style characterization of predicate liftings that is analogous to the one mentioned in Remark 25 but uses the lax extension instead of the underlying set functor:

Theorem 36. A predicate lifting $\mu \colon \mathsf{Q}_{\mathcal{V}^\kappa} \to \mathsf{Q}_\mathcal{V}\mathsf{F}$ is induced by a lax extension $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ if and only if it is defined from $\mu(1_\kappa)$ (in the $\mathcal{V}$ -relational view) by:

\[ \mu(f)= \mu(1_\kappa) \cdot \mathsf{L}(f). \]

Proof. “If” is immediate by Condition (iii) in Lemma 35; we prove “only if.” Varying the calculation in the proof of (iv) $\implies$ (i) in Lemma 35 and using condition (iii) of the lemma, we have

\begin{equation*} \mu(f) = \mu(\mathrm{ev}_\kappa \cdot f^\sharp) = \mu(\mathrm{ev}_\kappa) \cdot \mathsf{F} f^\sharp \overset{\text(iii)}{=}\mu(1_\kappa) \cdot \mathsf{L}\,\mathrm{ev}_\kappa\cdot \mathsf{F} f^\sharp=\mu(1_\kappa)\cdot \mathsf{L} f. \end{equation*}

Corollary 37. A predicate lifting $\mu \colon \mathsf{Q}_{\mathcal{V}^\kappa} \to \mathsf{Q}_\mathcal{V}\mathsf{F}$ is induced by a lax extension $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ if and only if for every relation and every function $g \colon \kappa \to Y$ ,

\[ \mu(g^\circ \cdot r) = \mu(g^\circ) \cdot \mathsf{L} r. \]

Proof. For “if,” just apply the assumption to $g=\mathrm{id}_\kappa$ and use Theorem 36. For “only if,” apply Theorem 36 to $g^\circ \cdot r$ , obtaining

\begin{equation*} \mu(g^\circ \cdot r)= \mu(1_\kappa) \cdot \mathsf{L}(g^\circ \cdot r)=\mu(1_\kappa) \cdot \mathsf{L}(g^\circ) \cdot \mathsf{L} r = \mu(g^\circ)\cdot \mathsf{L} r \end{equation*}

where the second equality is by Proposition 22.

In Section 4, we will see that every lax extension is induced by its class of Moss liftings and turn now to the question of characterizing the Moss liftings induced by a lax extension in the usual point of view.

Lemma 39. Let $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ be a lax extension of a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ , and $\mu \colon \mathsf{Q}_{\mathcal{V}^\kappa} \to \mathsf{Q}_\mathcal{V}\mathsf{F}$ a predicate lifting induced by $\mathsf{L}$ . Let denote the structure of the $\mathcal{V}$ -category ${(\mathcal{V}^\kappa)}^\mathrm{op}$ . Then,

\begin{equation*} \mu(\mathrm{ev}_\kappa) = \mu(1_\kappa) \cdot {(\mathsf{F} 1_\kappa^\sharp)}^\circ \cdot \mathsf{L} h. \end{equation*}

Proof. Observe that $\mathrm{ev}_\kappa = {(1_\kappa^\sharp)}^\circ \cdot h$ . Thus, we have

\begin{equation*} \mu(\mathrm{ev}_\kappa)=\mu({(1_\kappa^\sharp)}^\circ \cdot h)=\mu({(1_\kappa^\sharp)}^\circ)\cdot\mathsf{L}(h)=\mu(1_\kappa) \cdot \mathsf{L} ((1_\kappa^\sharp)^\circ) \cdot \mathsf{L} h=\mu(1_\kappa) \cdot {(\mathsf{F} 1_\kappa^\sharp)}^\circ \cdot \mathsf{L} h, \end{equation*}

using Corollary 37, Theorem 36, and Proposition 22, in that order.

Proof. Let $\mu^\textbf{k} \colon \mathsf{Q}_{\mathcal{V}^\kappa} \to \mathsf{Q}_\mathcal{V}\mathsf{F}$ be a Moss lifting induced by $\mathsf{L}$ . Note that $\mu(1_\kappa) \cdot {(\mathsf{F} 1_\kappa^\sharp)}^\circ \cdot \mathsf{L} h = \textbf{k}^\circ \cdot {(\mathsf{F} 1_\kappa^\sharp)}^\circ \cdot \mathsf{L} h$ . Hence, from Lemma 39, we conclude that under the isomorphism between $\kappa$ -ary predicate liftings for $\mathsf{F}$ and maps of type $\mathsf{F}\mathcal{V}^\kappa \to \mathcal{V}$ (see Remark 25), the predicate lifting $\mu^\textbf{k}$ corresponds to the map $\mathsf{F}\mathcal{V}^\kappa \to \mathcal{V}$ defined by:

\[ \textbf{v} \longmapsto \mathsf{L} h(\textbf{v},\mathsf{F} 1_\kappa^\sharp(\textbf{k})). \]

That is, $\mu^\textbf{k}$ corresponds to the representable $\mathcal{V}$ -functor $(\mathsf{F}\mathcal{V}^\kappa, \mathsf{L} h) \to \mathcal{V}^\mathrm{op}$ with representing object $\mathsf{F}1_\kappa^\sharp(\textbf{k})$ . On the other hand, this also makes it clear that every representable $\mathcal{V}$ -functor $(\mathsf{F}\mathcal{V}^\kappa, \mathsf{L} h) \to \mathcal{V}^\mathrm{op}$ with representing object in the image of the map $\mathsf{F}1_\kappa^\sharp \colon \mathsf{F}\kappa \to \mathsf{F}\mathcal{V}^\kappa$ corresponds to a Moss lifting.

By Lemma 35, the predicate liftings induced by a lax extension $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ are determined by the fixed points of the monotone map:

which are precisely the $\mathcal{V}$ -relations that can be factorized as $\textbf{r} \cdot \mathsf{L}\,\mathrm{ev}_\kappa$ for some $\mathcal{V}$ -relation . Therefore, the least fixed point is obtained by composing the $\mathcal{V}$ -relation with the $\mathcal{V}$ -relation , the constant function into $\bot$ . The corresponding $\kappa$ -ary predicate lifting sends a $\mathcal{V}$ -relation to the $\mathcal{V}$ -relation , the constant function into $\bot$ . This is the smallest predicate lifting with respect to the pointwise order of $\kappa$ -ary predicate liftings, which is defined by:

Similarly, the greatest fixed point is obtained by composing the $\mathcal{V}$ -relation with the $\mathcal{V}$ -relation $\top_{\mathsf{F}\kappa,1}$ , the constant function into $\top$ . However, the greatest predicate lifting induced by a lax extension is not necessarily the greatest predicate lifting.

The definition of Moss lifting of a lax extension used in the literature is seemingly different from the one presented here (Kurz and Leal Reference Kurz, Leal, Abramsky, Mislove and Palamidessi2009; Leal Reference Leal, Adámek and Kupke2008; Marti and Venema Reference Marti and Venema2015; Wild and Schröder Reference Wild, Schröder, Konnov and Kovács2020). To conclude this section, we show that both definitions are computed in the same way.

We recall that every accessible $\mathsf{Set}$ -functor admits a presentation (e.g., Adámek et al. Reference Adámek, Gumm and Trnková2010, Proposition 3.9 and Theorem 68) as follows. A $\lambda$ -ary presentation of a $\lambda$ -accessible functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ consists of a $\lambda$ -ary signature $\Sigma$ , that is, a set of operations of arity less than $\lambda$ , and for each operation $\sigma \in \Sigma$ of arity $\kappa$ , a natural transformation $\sigma \colon {(-)}^\kappa \to \mathsf{F}$ such that, for every $X \in \mathsf{Set}$ , the cocone ${(\sigma_X \colon X^\kappa \to \mathsf{F} X)}_{\sigma \in \Sigma}$ is epi. Every functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ has a $\lambda$ -accessible subfunctor $\mathsf{F}_\lambda \colon \mathsf{Set} \to \mathsf{Set}$ (Adámek et al. Reference Adámek, Gumm and Trnková2010) that maps a set X to the set:

The notion of Moss lifting of a lax extension $\mathsf{L} \colon\mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ was introduced by Marti and Venema (Reference Marti and Venema2015). They think of predicate liftings for a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ as natural transformations:

\[ {(\mathsf{Q}_\mathcal{V})}^n \longrightarrow \mathsf{Q}_\mathcal{V} \mathsf{F},\]

where n is a natural number and ${(\mathsf{Q}_\mathcal{V}})^n$ denotes the n-fold product of $\mathsf{Q}_\mathcal{V}$ . Then, given a lax extension $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ and a finitary presentation of $\mathsf{F}_\omega$ with signature $\Sigma$ , each n-ary $\sigma \in \Sigma$ induces a n-ary predicate lifting for $\mathsf{F}$ — a Moss lifting — whose X-component is defined by the assignment:

\[ (f_1, \dots, f_n ) \longmapsto (\textbf{x} \mapsto \mathsf{L}({\mathrm{ev}_X}^\circ)(\textbf{x}, \sigma_{\mathsf{Q}_\mathcal{V} X}(f_1, \dots, f_n)).\]

Of course, the functors $\mathsf{Q}_{\mathcal{V}^\kappa}$ and ${(\mathsf{Q}_\mathcal{V})}^\kappa$ are isomorphic, and the next proposition shows that both definitions of Moss lifting agree.

Proposition 42. Let $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ be a lax extension of a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ , $\Sigma$ the signature of a $\lambda$ -ary presentation of $\mathsf{F}_\lambda$ , and $\kappa$ a cardinal. Let $\sigma \colon {(-)}^\kappa \to \mathsf{F}_\lambda \to \mathsf{F}$ be the natural transformation induced by a $\kappa$ -ary operation symbol $\sigma \in \Sigma$ . Then, for every $\mathcal{V}$ -relation ,

\[ \mu^{\sigma_\kappa(1_\kappa)} (f) = {\sigma_{P_\mathcal{V} X} (\,^\ulcorner\! f^\urcorner)}^\circ \cdot \mathsf{L}({\mathrm{ev}}^\circ), \]

where $\,^\ulcorner\! f^\urcorner$ represents the $\mathcal{V}$ -relation f as a function of type $\kappa \to \mathsf{Q}_\mathcal{V} X$ .

Proof. Note that by definition of ${\mathrm{ev}_X}^\circ$ , $f = {\,^\ulcorner\! f^\urcorner}^\circ \cdot {\mathrm{ev}_X}^\circ$ . Hence, $\mathsf{L} f = {\mathsf{F} \,^\ulcorner\! f^\urcorner}^\circ \cdot \mathsf{L}(\mathrm{ev}^\circ)$ . Moreover, ${\,^\ulcorner\! f^\urcorner}^\kappa (1_\kappa) = \,^\ulcorner\! f^\urcorner$ by definition of the functor ${(-)}^\kappa \colon \mathsf{Set} \to \mathsf{Set}$ ; or with $\mathrm{id}_\kappa \colon 1 \to \kappa^\kappa$ denoting the function that selects the identity morphism on $\kappa$ , ${\,^\ulcorner\! f^\urcorner}^\kappa \cdot \mathrm{id}_\kappa = \,^\ulcorner\! f^\urcorner$ . Therefore, since $\sigma \colon {(-)}^\kappa \to \mathsf{F}$ is a natural transformation,

\begin{align*} \mu^{\sigma_\kappa(1_\kappa)} (f) & = {\mathrm{id}_\kappa}^\circ \cdot \sigma_\kappa^\circ \cdot \mathsf{L} f \\ & = {\mathrm{id}_\kappa}^\circ \cdot \sigma_\kappa^\circ \cdot {\mathsf{F} \,^\ulcorner\! f^\urcorner}^\circ \cdot \mathsf{L} ({\mathrm{ev}_X}^\circ) \\ & = {\mathrm{id}_\kappa}^\circ \cdot {{\,^\ulcorner\! f^\urcorner}^\kappa}^\circ \cdot \sigma_{\mathsf{Q}_\mathcal{V} X}^\circ \cdot \mathsf{L} ({\mathrm{ev}_X}^\circ) \\ & = {\sigma_{\mathsf{Q}_\mathcal{V} X} (\,^\ulcorner\! f^\urcorner)}^\circ \cdot \mathsf{L}({\mathrm{ev}_X}^\circ). \end{align*}

Therefore, $\kappa$ -ary Moss liftings are constructed in a very simple way. From the usual point of view, a Moss lifting $\mu^\textbf{k} \colon \mathsf{Q}_{\mathcal{V}^\kappa} \to \mathsf{Q}_\mathcal{V} \mathsf{F}$ of a lax extension $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ , for $\textbf{k} \colon 1 \to \mathsf{F} \kappa$ , is just the map that sends a function $f \colon X \to \mathcal{V}^\kappa$ (with left adjunct to the function $\mathsf{L} f^\flat (-,\textbf{k}) \colon \mathsf{F} X \to \mathcal{V}$ .

In terms of maps of type $\mathsf{F} \mathcal{V}^\kappa \to \mathcal{V}$ , Proposition 39 tells us that $\mu^\textbf{k}$ is determined by the map $\mu(1_{\mathcal{V}^\kappa}) \colon \mathsf{F}\mathcal{V}^\kappa \to \mathcal{V}$ defined by:

\[ \textbf{x} \longmapsto \mathsf{L} h(\textbf{x},\mathsf{F} 1_\kappa^\sharp(\textbf{k})),\]

where is the structure of the $\mathcal{V}$ -category ${(\mathcal{V}^\kappa)}^\mathrm{op}$ (see Remark 15). In other words, $\kappa$ -ary Moss liftings correspond precisely to the representable $\mathcal{V}$ -functors $(\mathsf{F}\mathcal{V}^\kappa, \mathsf{L} h) \to \mathcal{V}^\mathrm{op}$ with representing objects in the image of the map $\mathsf{F}1_\kappa^\sharp \colon \mathsf{F}\kappa \to\mathsf{F}\mathcal{V}^\kappa$ .

4. From Predicate Liftings to Lax Extensions

The main goal of this section is to provide a way to construct lax extensions to $\mathcal{V}\text{-}\mathsf{Rel}$ from predicate liftings and to understand which lax extensions arise in such way. The first task has already been discharged in earlier work (Wild and Schröder Reference Wild, Schröder, Konnov and Kovács2020, Reference Wild, Schröder, Kiefer and Tasson2021). In the following, we describe the Kantorovich extension of a collection of monotone predicate liftings (Wild and Schröder Reference Wild, Schröder, Konnov and Kovács2020, Reference Wild, Schröder, Kiefer and Tasson2021), but completely from the point of view of $\mathcal{V}$ -relations. In Theorem 49, we show that every Kantorovich extension arises as an initial extension w.r.t to canonical extensions of generalized monotone neighborhood functors.

Given a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ and a predicate lifting $\mu \colon \mathsf{Q}_{\mathcal{V}^\kappa} \to \mathsf{Q}_\mathcal{V} \mathsf{F}$ , this perspective makes it intuitive to construct a $\mathcal{V}$ -relation from a $\mathcal{V}$ -relation :

Theorem 44. Let $\mu \colon \mathsf{Q}_{\mathcal{V}^\kappa} \to \mathsf{Q}_\mathcal{V} \mathsf{F} $ be a $\kappa$ -ary predicate lifting. For every $\mathcal{V}$ -relation , consider the $\mathcal{V}$ -relation given by:

(3)

If $\mu$ is monotone, then the assignment $r \mapsto \mathsf{L}^\mu r $ defines a lax extension $\mathsf{L}^\mu \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel} $ , which is $\mathcal{V}$ -enriched whenever $\mu$ is $\mathcal{V}$ -enriched.

Proof. (L1)/(L1’) Monotonicity is immediate from the fact that

\begin{equation} \mathsf{L}^{\mu}(-)\colon\mathcal{V}\text{-}\mathsf{Rel}(X,Y) \longrightarrow\mathcal{V}\text{-}\mathsf{Rel}(\mathsf{F} X,\mathsf{F} Y) \end{equation}

is a composite of monotone maps; similarly for $\mathcal{V}$ -enrichment.

(L2) Let and be $\mathcal{V}$ -relations. We have to show that $\mathsf{L} s\cdot\mathsf{L} r\le\mathsf{L} (s\cdot r)$ . Let $h \colon Z \to \kappa$ . By definition, we have

Therefore, by Proposition 6(3)

which implies the claim.

(L3) Let $f\colon X\to Y$ . First, observe that since $\mu$ is a natural transformation, we have $\mu(g)\cdot \mathsf{F} f=\mu(g\cdot f)$ , and therefore

for all . Hence, $\mathsf{F} f \leq \mathsf{L}^\mu f$ . Second, note that because $\mu$ is monotone and natural, we have

for all . Therefore, by Proposition 11, $\mu(i)\cdot(\mathsf{F} f)^\circ\le \mu(i \cdot f^\circ)$ , and hence

Thus, $(\mathsf{F} f^\circ)\le \mathsf{L}^\mu (f^\circ)$ .

The formula (3) of Theorem 44 is entailed by the view of predicate liftings as natural transformations of type ${\mathcal{V}\text{-}\mathsf{Rel}}_\circ(-,\kappa) \to {\mathcal{V}\text{-}\mathsf{Rel}}_\circ(\mathsf{F} -,1)$ . By applying the involution on $\mathcal{V}\text{-}\mathsf{Rel}$ , we could also think of predicate liftings as natural transformations ${\mathcal{V}\text{-}\mathsf{Rel}}^\circ(\kappa,-) \to {\mathcal{V}\text{-}\mathsf{Rel}}^\circ(1, \mathsf{F} -)$ between functors defined according to the schema:

This point of view would lead us to the dual extension of (3).

Proposition 45. Let $\mu \colon \mathcal{V}\text{-}\mathsf{Rel}_\circ(-,\kappa) \to \mathcal{V}\text{-}\mathsf{Rel}_\circ(\mathsf{F} -, 1)$ be a predicate lifting and $\bar{\mu} \colon \mathcal{V}\text{-}\mathsf{Rel}^\circ(\kappa,-) \to \mathcal{V}\text{-}\mathsf{Rel}^\circ(1,\mathsf{F} -)$ be the natural transformation defined by:

\[ r \longmapsto {\mu(r^\circ)}^\circ. \]

Then,

Proof. Taking into account Proposition 6(1) and the fact that $(-)^\circ$ preserves infima, we have

Definition 46. Let $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ be a functor and M a class of monotone predicate liftings. The Kantorovich lax extension of $\mathsf{F}$ with respect to M is the lax extension:

\[ \mathsf{L}^M = \bigwedge_{\mu \in M} \mathsf{L}^\mu. \]

Examples: 47. Let $\mathcal{V}$ be a quantale.

1. (1) The identity functor on $\mathcal{V}\text{-}\mathsf{Rel}$ is the Kantorovich extension of the identity functor on $\mathsf{Set}$ with respect to the identity natural transformation $\mathsf{Q}_\mathcal{V} \to \mathsf{Q}_\mathcal{V}$ .

2. (2) The largest extension of a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ to $\mathcal{V}\text{-}\mathsf{Rel}$ arises as the Kantorovich extension of $\mathsf{F}$ with respect to the natural transformation $\top \colon \mathsf{Q}_\mathcal{V} \to \mathsf{Q}_\mathcal{V} \mathsf{F}$ that sends every map to the constant map $\top$ and also as the Kantorovich extension with respect to the natural transformation $\bot \colon \mathsf{Q}_\mathcal{V} \to \mathsf{Q}_\mathcal{V} \mathsf{F}$ that sends every map to the constant map $\bot$ .

3. (3) For a subquantale $\mathcal{W}$ of $\mathcal{V}$ (i.e., $\mathcal{W}$ is a submonoid of $\mathcal{V}$ closed under suprema), it is easy to construct a unary predicate lifting of the covariant $\mathcal{W}$ -powerset functor $\mathsf{P} \colon \mathsf{Set} \to \mathsf{Set}$ , which, in terms of $\mathcal{V}$ -relations, is defined by:

   \begin{align*} \mathsf{P} X & =\mathcal{W}\text{-}\mathsf{Rel}(X, 1) \subseteq \mathcal{V}\text{-}\mathsf{Rel}(X,1), \\ \mathsf{P} f & = (-) \cdot f^\circ. \end{align*}
   A straightforward calculation shows that “evaluating” induces a predicate lifting $\Diamond \colon \mathsf{Q}_\mathcal{V} \to \mathsf{Q}_\mathcal{V} \mathsf{P}$ whose X-component is defined by:
   \begin{align*} \Diamond_X \colon \mathcal{V}\text{-}\mathsf{Rel}(X,1) & \longrightarrow \mathcal{V}\text{-}\mathsf{Rel}(\mathsf{P} X, 1). \\ \phi & \longmapsto \phi \cdot \mathrm{ev}_{\mathcal{W},X} \end{align*}

   Then, by Proposition 6(4) and Corollary 9,

   

   for every $\mathcal{V}$ -relation . Therefore, for $\mathcal{W}$ -relations and ,

   \[ \widehat{\mathsf{P}}^\Diamond r (\phi, \psi) = \bigwedge_{y \in Y} \hom(\psi(y), \bigvee_{x \in X} r(x,y) \otimes \phi(x)). \]
   Furthermore, for $\mathcal{W} = \mathsf{2}$ this formula simplifies to
   \[ \widehat{\mathsf{P}}^\Diamond r (A,B) = \bigwedge_{b \in B} \bigvee_{a \in A} r(a,b). \]

   1. a. For $\mathcal{W} = \mathcal{V} = \mathsf{2}$ we obtain a generalization of the upper half of the Egli–Milner order: for every , and all $A \in \mathsf{P} X$ and $B \in \mathsf{P} Y$ ,

      \[ A (\widehat{\mathsf{P}}^\Diamond r) B \iff \forall b \in B, \exists a \in A, a\, r\, b. \]

   2. b. For $\mathcal{W} = 2$ and $\mathcal{V}$ a left continuous t-norm, we obtain a generalization of the upper half of the Hausdorff metric: for every , and all $A \in \mathsf{P} X$ and $B \in \mathsf{P} Y$ ,

      \[ \widehat{\mathsf{P}}^\Diamond r (A,B) = \bigwedge_{b \in B} \bigvee_{a \in A} r(a,b). \]

4. (4) The dual lax extensions of the extensions 47(3a) and 47(3b) are generalizations of the lower half of the Egli–Milner order and of the lower half of the Hausdorff metric, respectively. Therefore, the symmetrization of these lax extensions are generalizations of the Egli–Milner order and the Hausdorff metric.

5. (5) Let us now consider a faithful functor ${\mathsf{|}{-}\mathsf{|}} \colon\mathsf{A}\to\mathsf{Pos}$ with some $\mathsf{A}$ -object $\mathcal{V}$ over the partially ordered set $\mathcal{V}$ and the functor ${{}^{\kappa}\mathsf{U}}=\mathsf{A}(\mathsf{Q}_{\mathcal{V}^\kappa},\mathcal{V})\colon\mathsf{Set}\to\mathsf{Set}$ . Some typical examples are $\mathsf{A}=\mathsf{Pos}$ and $\mathsf{A}=\mathcal{V}\text{-}\mathsf{Cat}$ ; for instance, for $\mathsf{A}=\mathsf{Pos}$ and $\mathcal{V}=\mathsf{2}$ , we obtain the generalized monotone neighborhood functor ${{}^{\kappa}\mathsf{U}}\colon\mathsf{Set}\to\mathsf{Set}$ . There is a canonical predicate lifting corresponding to the identity transformation $1 \colon {{}^{\kappa}\mathsf{U}}\to{{}^{\kappa}\mathsf{U}}$ , and we denote the induced extension as $\widehat{{{}^{\kappa}\mathsf{U}}}$ . Then, for a $\mathcal{V}$ -relation and for $\Phi \colon \mathsf{Q}_{\mathcal{V}^\kappa} X\to\mathcal{V}$ and $\Psi \colon \mathsf{Q}_{\mathcal{V}^\kappa} Y\to\mathcal{V}$ ,

This extension coincides with the one considered by Reference Schubert and Seal Schubert and Seal (2008) for the classical monotone neighborhood functor. In particular, it follows that, for the identity , the $\mathcal{V}$ -category $({{}^{\kappa}\mathsf{U}} X,\widehat{{{}^{\kappa}\mathsf{U}}} 1_{X})$ is separated.

Remark 48. To see that the Kantorovich extension defined by Wild and Schröder (Reference Wild, Schröder, Konnov and Kovács2020) coincides with the one presented here, note that Theorem 44 requires $\mu$ to be monotone; hence, we can define $\mathsf{F}^\mu$ with respect to $\mathcal{V}$ -relations of type with the same result, that is,

Moreover, in the language of Wild and Schröder (Reference Wild, Schröder, Konnov and Kovács2020), a pair (f,g) of $\kappa$ -indexed families of maps of type $X \to \mathcal{V}$ is r-non-expansive precisely when , when interpreting f and g as $\mathcal{V}$ -relations.

The following result explains the distinguished role of the canonical extensions of generalized monotone neighborhood functors in the process of constructing lax extensions.

Theorem 49. Let M be a collection of predicate liftings for a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ . The Kantorovich extension $\mathsf{L}^{M}$ is the initial extension of $\mathsf{F}$ with respect to the cone:

\begin{equation} (\overline{\mu}\colon\mathsf{F} \longrightarrow{{}^{\kappa}\mathsf{U}})_{\mu\in M} \end{equation}

(with $\overline\mu$ as per (2)) and the lax extensions $\widehat{{{}^{\kappa}\mathsf{U}}}$ of ${{}^{\kappa}\mathsf{U}}$ described in Examples 47(5). Also, note that if all predicate liftings in M are even $\mathcal{V}$ -enriched, then so is $\mathsf{L}^{M}$ .

Proof. Let $\mu \colon \mathsf{Q}_{\mathcal{V}^\kappa} \to \mathsf{Q}_\mathcal{V} \mathsf{F} $ be a $\kappa$ -ary predicate lifting in M. Then, for a $\mathcal{V}$ -relation and $\textbf{x}\in\mathsf{F} X$ and $\textbf{y}\in\mathsf{F} Y$ ,

That is, $\mathsf{L}^{\mu}$ is the initial extension with respect to $\overline{\mu}\colon\mathsf{F}\to{{}^{\kappa}\mathsf{U}}$ and the lax extension $\widehat{{{}^{\kappa}\mathsf{U}}}$ of ${{}^{\kappa}\mathsf{U}}$ .

We proceed to collect some properties of Kantorovich extensions. We begin by observing that Kantorovich extensions are compatible with initial extensions along a natural transformation. This property will be particularly useful in Section 5 to generalize Theorem 1.

Proposition 50. Let $\mathsf{L}_\mathsf{F} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ be a lax extension of a functor $\mathsf{F} \colon\mathsf{Set}\to\mathsf{Set}$ , and let $i \colon \mathsf{G}\to \mathsf{F}$ be a natural transformation. Consider the initial lax extension $\mathsf{L}_i \colon\mathcal{V}\text{-}\mathsf{Rel}\to\mathcal{V}\text{-}\mathsf{Rel}$ of $\mathsf{G}$ with respect to $i \colon \mathsf{G}\to \mathsf{F}$ . If $\mathsf{L}_\mathsf{F}$ is Kantorovich w.r.t a class M of monotone predicate liftings, then $\mathsf{L}_i$ is Kantorovich w.r.t to the class of monotone predicate liftings:

\[ M_i = \{ (\mathsf{Q}_\mathcal{V} i) \cdot \mu \mid \mu \in M \}. \]

Proof. Clearly, every predicate lifting in $M_i$ is monotone. Now, let be a $\mathcal{V}$ -relation. Then,

Therefore, by Corollary 8,

Example 51. We recall that an endofunctor on $\mathsf{Set}$ is called taut if it preserves inverse images (Manes Reference Manes2002). Every taut functor $\mathsf{F} \colon\mathsf{Set}\to\mathsf{Set}$ admits a natural transformation:

\begin{equation} \mathrm{supp} \colon \mathsf{F} \longrightarrow\mathsf{U} \end{equation}

into the monotone neighborhood functor with X-component:

\begin{equation} \mathrm{supp}_{X}\colon \mathsf{F} X \longrightarrow\mathsf{U} X,\quad \textbf{x} \longmapsto \{A\subseteq X\mid \textbf{x}\in \mathsf{F} A\}. \end{equation}

We note that every $\mathrm{supp}_{X}(\textbf{x})$ is actually a filter (Gumm Reference Gumm, Fiadeiro, Harman, Roggenbach and Rutten2005; Manes Reference Manes2002). As observed by Schubert and Seal (Reference Schubert and Seal2008), the op-canonical extension (Seal Reference Seal2005) of a taut functor is the initial lift with respect to supp of the extension $\widehat{\mathsf{U}}$ of the monotone neighborhood functor $\mathsf{U} \colon\mathsf{Set}\to\mathsf{Set}$ of Example 47(5), that is, the extension induced by the predicate lifting $\mu$ corresponding to the identity transformation $1 \colon\mathsf{U}\to\mathsf{U}$ . Hence, by Proposition 50, the op-canonical extension of a taut functor $\mathsf{F}$ is induced by the predicate lifting:

\begin{equation} \mathsf{Q}_\mathcal{V}\mathrm{supp} \cdot \mu, \end{equation}

which, by adjunction, corresponds to the natural transformation:

\begin{equation} \mathrm{supp} \colon\mathsf{F} \longrightarrow\mathsf{U}. \end{equation}

The next result entails that we can use the Kantorovich extension to major a lax extension by extracting all predicate liftings induced by the lax extension. This is the first step toward representing lax extensions by collections of predicate liftings.

Proof. Let be a $\mathcal{V}$ -relation. Then, by (L2),

Notably, as a consequence of the previous results, we can use the Kantorovich extension to detect predicate liftings induced by lax extensions.

Proof. According to Lemma 35, it suffices to show $\mu(\mathrm{ev}_\kappa) = \mu(1_\kappa) \cdot \mathsf{L}^\mu\mathrm{ev}_\kappa$ . First, observe that by Proposition 52 and Lemma 35((iii)) we have

\[ \mu(\mathrm{ev}_\kappa) = \mu(1_\kappa) \cdot \mathsf{L}(\mathrm{ev}_\kappa) \leq \mu(1_\kappa) \cdot \mathsf{L}^\mu(\mathrm{ev}_\kappa). \]

Second, note that by definition of $\mathsf{L}^\mu$ , we obtain

Example 55. The predicate lifting $\Diamond \colon \mathsf{Q}_\mathcal{V} \to \mathsf{Q}_\mathcal{V}\mathsf{P}$ of Example 47(3) is induced by the lax extension $\widehat{\mathsf{P}}^\Diamond \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ . With $k \colon 1 \to \mathcal{W}$ denoting the function that selects the element $k \in \mathcal{W}$ , we have

\[ \Diamond(f) = k^\circ \cdot \widehat{\mathsf{P}}^\Diamond f \]

for every $\mathcal{V}$ -relation .

We already know from Proposition 33 that every predicate lifting induced by a lax extension is monotone. The next example shows that the converse statement does not hold.

Example 56. Let [0, 1] denote the quantale consisting of the unit interval equipped with the usual order and multiplication. Consider the unary monotone predicate lifting for the identity functor $\mu \colon \mathsf{P}_{[0,1]} \to \mathsf{P}_{[0,1]}$ determined by the map $\mu(\mathrm{ev}_1) \colon [0,1] \to [0,1]$ defined by:

\[ \mu(\mathrm{ev}_1)(v) = \begin{cases} 0\;\;\;\; \text{ if } v \leq \frac{1}{2}; \\ 1\;\;\;\; \text{ otherwise}. \end{cases} \]

Given that [0, 1] is an integral quantale and $\mu$ is a unary predicate lifting in order to be induced by $\mathsf{L}^\mu$ , $\mu$ would need to satisfy the condition:

\[ \mu(\mathrm{ev}_1) \leq \mathsf{L}^\mu(\mathrm{ev}_1); \]

that is, for every ,

However, with $g = \frac{2}{3}$ and $v = \frac{3}{4}$ ,

Finally, we tackle the problem of recovering a lax extension to $\mathcal{V}\text{-}\mathsf{Rel}$ as the Kantorovich extension w.r.t. some class of predicate liftings.

Proof. Note that $\textbf{k} = \mathsf{F} i(\textbf{y})$ means $\textbf{k} = \mathsf{F} i \cdot \textbf{y}$ , when considering elements as functions.

1. (1) $ \mu^\textbf{k} (s) = \textbf{k}^\circ \cdot \mathsf{L} s = \textbf{y}^\circ \cdot {\mathsf{F} i}^\circ \cdot \mathsf{L} s = \textbf{y}^\circ \cdot \mathsf{L} (i^\circ \cdot s) $ .

2. (2)

   1. a. Since i is a monomorphism, $i^\circ \cdot i = 1_Y$ . Therefore, the claim follows by applying 1 with $s = i \cdot r$ .

   2. b. Applying 2a, and recalling Proposition 6(2) and Condition (L3), yields

Lemma 58 allows approximating a lax extension with respect to the cardinality of the codomain of $\mathcal{V}$ -relations.

Therefore, as the collection of all Moss liftings of a lax extension is not larger than the class of all sets,

From Theorem 49, we obtain

From Proposition 33, we obtain

The next result is a generalization of Theorem 2 mentioned in the introduction. Note that, for $\mathcal{V}=\mathsf{2}$ and a lax extension $\mathsf{L}\colon \mathcal{V}\text{-}\mathsf{Rel}\to \mathcal{V}\text{-}\mathsf{Rel}$ that preserves converses, $\mathsf{L} 1_{X}$ is an equivalence relation on X. Hence, $\mathsf{L}$ is identity-preserving if and only if the ordered set $(\mathsf{F} X,\mathsf{L}1_{X})$ is anti-symmetric.

Proof. Suppose that there is such a lax extension $\mathsf{L}\colon \mathcal{V}\text{-}\mathsf{Rel}\to \mathcal{V}\text{-}\mathsf{Rel}$ . Let X be a set and $\textbf{x},\textbf{y}\in\mathsf{F} X$ with $\textbf{x}\neq\textbf{y}$ . By assumption, then have w.l.o.g. that $k\nleq\mathsf{L} 1_{X}(\textbf{x},\textbf{y})$ . Hence, by Theorem 61, there is a predicate lifting $\mu \colon \mathsf{Q}_{\mathcal{V}^\kappa}\to\mathsf{Q}_\mathcal{V}\mathsf{F}$ such that

\begin{equation} k\nleq{{}^{\kappa}\mathsf{U}}(\overline{\mu}(\textbf{x}),\overline{\mu}(\textbf{y})), \end{equation}

and therefore $\overline{\mu}(\textbf{x})\neq\overline{\mu}(\textbf{y})$ (see Example 47(5)). On the other hand, if $\mathsf{F}$ has a separating class of monotone predicate liftings, then the Kantorovich extension of $\mathsf{F}$ with respect to this class has the desired property by Example 47(5).

To conclude this section, we note that quantale-valued lax extensions that preserve converses and satisfy the condition of Corollary 64 lead to quantale-valued notions of bisimilarity that extend the canonical coalgebraic notion of behavioral equivalence.

Proof. Consider the lax homomorphisms of quantales $\phi \colon \mathsf{2} \to \mathcal{V}$ defined by $\phi(0) = \bot$ and $\phi(1) = k$ , and $\psi \colon \mathcal{V} \to \mathsf{2}$ defined by $\psi(v) = 1$ if $k \leq v$ and $\psi(v) = 0$ , otherwise. It is well known that lax homomorphisms of quantales give rive to lax functors between the corresponding categories of $\mathcal{V}$ -relations (Hofmann et al. Reference Hofmann, Seal, Tholen, Clementino, Colebunders, Hofmann, Lowen, Lucyshyn-Wright, Seal and Tholen2014). Hence, we obtain lax functors:

\[ \phi \colon \mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel} \text{ and } \psi \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathsf{Rel} \]

that act identically on sets and postcompose the lax homomorphisms of quantales with relations (interpreted as maps into the quantales). It is easy to see that if we start with a lax extension $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ of $\mathsf{F}$ that satisfies the conditions of the proposition, then, as $\mathcal{V}$ is nontrivial, we obtain an identity-preserving lax extension $\mathsf{L}_2 \colon \mathsf{Rel} \to \mathsf{Rel}$ of $\mathsf{F}$ that preserves converses as the composite:

Therefore, by Marti and Venema (Reference Marti and Venema2015, Theorem 14), $\mathsf{L}_2$ -bisimilarity coincides with behavioural equivalence. Moreover, as $\mathsf{L}$ -bisimilarity is itself an $\mathsf{L}$ -bisimulation, it follows that two states in an $\mathsf{F}$ -coalgebra are $\mathsf{L}_2$ -bisimilar if and only if their $\mathsf{L}$ -bisimilarity is greater or equal than k.

5. Small Lax Extensions

We next discuss the possibility of recovering a lax extension from a set of predicate liftings.

We will see next that every lax extension of an accessible functor is small. We recall that, for a regular cardinal $\lambda$ , a functor $\mathsf{F}\colon \mathsf{Set}\to \mathsf{Set} $ is called $\lambda$ -accessible if $\mathsf{F}$ preserves $\lambda$ -directed colimits. Furthermore, a functor $\mathsf{F}\colon \mathsf{Set}\to \mathsf{Set} $ is called accessible if $\mathsf{F}$ is $\lambda$ -accessible for some regular cardinal $\lambda$ .

Clearly, every $\lambda$ -accessible functor $\mathsf{F}\colon \mathsf{Set}\to \mathsf{Set} $ is $\lambda$ -bounded, that is, for every set X and every $\textbf{x}\in \mathsf{F} X$ , there exists a subset $m \colon A\to X$ with $|A|<\lambda$ and $\textbf{x}$ is in the image of $\mathsf{F} m$ . This property is in fact equivalent to accessibility:

An immediate consequence of the result above is an algebraic presentation of accessible functors.

Remark 69. The implication (i) $\implies$ (ii) above can be justified as follows. First, recall (MacLane Reference MacLane1998) that $\mathsf{F} \colon\mathsf{Set}\to\mathsf{Set}$ is a colimit of the (large) diagram

\begin{equation} {(\text{elements $(X,\textbf{x})$ of $\mathsf{F}$})}^{\mathrm{op}} \longrightarrow [\mathsf{Set},\mathsf{Set}]. \end{equation}

By (i),

\begin{equation} {(\text{elements $(X,\textbf{x})$ of $\mathsf{F}$ where $|X|<\lambda$})}^{\mathrm{op}} \longrightarrow {(\text{elements $(X,\textbf{x})$ of $\mathsf{F}$})}^{\mathrm{op}} \end{equation}

is cofinal.

Proposition 70. Every lax extension of an $\lambda$ -accessible functor is $\lambda$ -small. In fact, $\mathsf{L}=\mathsf{L}^{M_{\lambda}}$ for

\begin{equation} M_{\lambda}=\{\text{all $\alpha$-ary Moss-liftings, $\alpha<\lambda$}\}. \end{equation}

Corollary 72. Let $\mathsf{F}\colon \mathsf{Set}\to \mathsf{Set} $ be a $\lambda$ -accessible functor with a lax extension $\mathsf{L}\colon \mathcal{V}\text{-}\mathsf{Rel}\to \mathcal{V}\text{-}\mathsf{Rel}$ such that, for all sets X with $|X|<\lambda$ , the $\mathcal{V}$ -category $(\mathsf{F} X,\mathsf{L}1_{X})$ is separated. Then the set

\begin{equation} M_{\lambda}=\{\text{all $\alpha$-ary Moss liftings, $\alpha<\lambda$}\} \end{equation}

is separating.

We can relax slightly the condition of Proposition 70 by taking advantage of the structure of $\mathcal{V}$ -category.

Remark 75. A map $i \colon A \to X$ is dense in (X,a) if and only if the image of A is dense in (X,a) with respect to the closure operator $\overline{(-)}$ on $\mathcal{V}\text{-}\mathsf{Cat}$ introduced by Hofmann and Tholen (Reference Hofmann and Tholen2010). We recall that for every $x \in X$ and $M \subseteq X$ ,

\begin{equation} x \in \overline{M} \iff k \le \bigvee_{z\in M} a(x,z) \otimes a(z,x). \end{equation}

This closure operator generalizes the one induced by the usual topology associated with a metric space. In fact, one can show that for categories enriched in a value quantale, such as metric spaces, this closure operator coincides with the one induced by the symmetric topology considered by Flagg (Reference Flagg and Seely1992).

The following lemma records that dense maps are compatible with Moss liftings (see Theorem 36); in the sense that to determine the action of a $\kappa$ -ary Moss lifting, it suffices to focus on a dense subset of $(\mathsf{F} \kappa, \mathsf{L} 1_\kappa)$ .

Lemma 77. Let $\mathsf{L} \colon \mathcal{V}\text{-}\mathsf{Rel} \to \mathcal{V}\text{-}\mathsf{Rel}$ be a lax extension of a functor $\mathsf{F} \colon \mathsf{Set} \to \mathsf{Set}$ . Furthermore, let be a $\mathcal{V}$ -relation, $\textbf{k}$ be an element of $\mathsf{F}\kappa$ , and $i \colon A \to \mathsf{F}\kappa$ be a dense map in $(\mathsf{F}\kappa,\mathsf{L} 1_{\kappa})$ . Then,

\begin{equation} \mu^{\textbf{k}}(\varphi)= \mu^\textbf{k}(1_\kappa) \cdot i \cdot i^\circ \cdot \mathsf{L}(\varphi). \end{equation}

Proof. Let be a $\mathcal{V}$ -relation, $\textbf{k}\in\mathsf{F}\kappa$ , and consider $i_{\kappa} \colon \mathsf{G} \kappa \to \mathsf{F} \kappa$ . Then, by Lemma 77, for all ,

Hence, by Propositions 6(4) and 9,

Now, the claim follows from the fact that $\mathsf{G}$ is $\lambda$ -accessible.

Finally, we obtain a generalization of Theorem 1 mentioned in the introduction.