Up-to techniques for behavioural metrics via fibrations

5.1 ${\mathcal{V}}$ -predicate liftings

Liftings of $\mathsf{Set}$ -functors to the category $\mathsf{Pred}$ (for ${\mathcal{V}} = 2$ ) of predicates have been widely studied in the context of coalgebraic modal logic, as they correspond to modal operators (see e.g. Schröder Reference Schröder2008). For $\mathcal{V}\textrm{-}\mathsf{Pred}$ , we proceed in a similar way. Let us analyse what it means to have a fibred lifting ${\widehat{F}}$ to $\mathcal{V}\textrm{-}\mathsf{Pred}$ of an endofunctor F on $\mathsf{Set}$ . First, recall that the fibre ${\mathcal{V}\textrm{-}\mathsf{Pred}_{X}}$ is just the poset ${\mathcal{V}}^X$ . So the restriction ${\widehat{F}_{X}}$ to such a fibre corresponds to a monotone map ${\mathcal{V}}^X\to{\mathcal{V}}^{FX}$ . The fact that ${\widehat{F}}$ is a fibred lifting essentially means that the maps $({\mathcal{V}}^X\to{\mathcal{V}}^{FX})_{X}$ form a natural transformation between the contravariant functors ${\mathcal{V}}^{-}$ and ${\mathcal{V}}^{F-}$ . Furthermore, by Yoneda lemma we know that natural transformations ${\mathcal{V}}^-\Rightarrow{\mathcal{V}}^{F-}$ are in one-to-one correspondence with maps $\mathit{ev}:F{\mathcal{V}}\to{\mathcal{V}}$ , which we will call hereafter evaluation maps.

One can characterize the evaluation maps which correspond to the monotone natural transformations. In Proposition 14, we show that these are the monotone evaluation maps $\mathit{ev}:(F{\mathcal{V}},{\ll})\to({\mathcal{V}},\le)$ with respect to the usual order $\le$ on ${\mathcal{V}}$ and an order ${\ll}$ on $F{\mathcal{V}}$ defined below and obtained by applying the standard canonical relation lifting of F – in the sense of Barr (Reference Barr1970) – to the relation $\le$ . Explicitly, we apply the functor F to the relation $\le$ seen as the span below in order to obtain a relation on $F{\mathcal{V}}$ . Note that $[{\le}] = \{(v_1,v_2)\in {\mathcal{V}}\times{\mathcal{V}}\mid v_1 \le v_2)\}$ and o is the embedding of $[\le]$ in ${\mathcal{V}}\times{\mathcal{V}}$ .

Definition 13 (Relation ${\ll}$ on $F{\mathcal{V}}$ ). We define a relation ${{\ll}}$ on $F{\mathcal{V}}$ : let $v_1,v_2\in F{\mathcal{V}}$ . We define $v_1{\ll} v_2$ whenever

\begin{equation*} \exists r\in F[\le] \mbox{ s.t. } F(\pi_1\circ o)(r) = v_1\mbox{ and } F(\pi_2\circ o)(r) = v_2 \end{equation*}

The relation ${\ll}$ will also be denoted by $\le^F$ (i.e. the order $\le$ lifted under F via the standard relation lifting Barr Reference Barr1970).

According to Balan and Kurz (Reference Balan and Kurz2011) relation lifting transforms preorders into preorders whenever F preserves weak pullbacks (but not necessarily orders into orders).

${\widehat{F}}$ is a lifting of F to $\mathcal{V}\textrm{-}\mathsf{Pred}$ if and only if the following two conditions are met for all sets X and functions $f\colon X\to Y$ :

1. (1) ${\widehat{F}_{X}}\colon{\mathcal{V}\textrm{-}\mathsf{Pred}_{X}}\to{\mathcal{V}\textrm{-}\mathsf{Pred}_{FX}}$ is monotone, and,

2. (2) the inequality ${\widehat{F}_{X}}\circ {f^*}(R)\le{({Ff})^*}\circ{\widehat{F}_{Y}}$ holds.

These two conditions alone are equivalent to the laxness of the following square

However, ${\widehat{F}}$ is a fibred lifting of F if and only if item 1 holds and the inequality in item 2 above is in fact an equality. Hence, ${\widehat{F}}$ is a fibred lifting if and only if the above square is actually commutative, which amounts to the existence of a natural transformation $\gamma\colon{\mathcal{V}}^{-}\to{\mathcal{V}}^{F-}$ with each component $\gamma_X$ being monotone.

We have thus proved the equivalence of the first two conditions. Now, let us turn to the equivalence between the first and third one. By Yoneda lemma, we know that natural transformations ${\mathcal{V}}^{-}\to{\mathcal{V}}^{F-}$ are in one-to-one correspondence with evaluation maps $\mathit{ev}\colon F({\mathcal{V}})\to{\mathcal{V}}$ . It remains to characterize the monotonicity condition. We show that this is equivalent to requiring that $\mathit{ev}_F\colon F{\mathcal{V}}\to {\mathcal{V}}$ is monotone for the order ${\ll}$ on $F{\mathcal{V}}$ and $\le$ on ${\mathcal{V}}$ .

‘ $\Leftarrow$ ’ Assume that $\mathit{ev}_F$ is monotone and take $f_1,f_2\colon X\to {\mathcal{V}}$ such that $f_1\le f_2$ . This means that $\langle{f_1},{f_2}\rangle$ factors through o as depicted below, where $u\colon X\to [\le]$ is defined as $u(x) = (f_1(x),f_2(x))$ .

If we apply F to the diagram above and post-compose with $F\pi_1,F\pi_2,\mathit{ev}_F$ , we obtain the following diagram.

Let $t\in FX$ . Our aim is to show ${\widehat{F}} f_1(t) \le {\widehat{F}} f_2(t)$ , which implies ${\widehat{F}} f_1 \le {\widehat{F}} f_2$ .

First, define $r = Fu(t) \in F[\le]$ . Now observe that $F(\pi_1\circ o)(r) = F(\pi_1\circ o\circ u)(t) = F(\pi_1\circ \langle{f_1},{f_2}\rangle)(t) = Ff_1(t)$ . Analogously, $F(\pi_2\circ o)(r) = Ff_2(t)$ . Hence $Ff_1(t) {\ll} Ff_2(t)$ .

Using the monotonicity of $\mathit{ev}_F,$ we can conclude that

\begin{equation*}{\widehat{F}} f_1(t) = \mathit{ev}_F(Ff_1(t)) \le \mathit{ev}_F(Ff_2(t)) = {\widehat{F}} f_2(t)\,.\end{equation*}

‘ $\Rightarrow$ ’ Assume that ${\widehat{F}}$ is monotone. In order to show monotonicity of $\mathit{ev}_F,$ take $v_1,v_2\in F{\mathcal{V}}$ such that $v_1{\ll} v_2$ . This means that there exists $r\in F[\le]$ such that $F(\pi_1\circ o)(r) = v_1$ , $F(\pi_2\circ o)(r) = v_2$ .

Now consider $\pi_1\circ o$ , $\pi_2\circ o\colon [\le]\to {\mathcal{V}}$ . It holds that $\pi_1\circ o\le \pi_2\circ o$ and with monotonicity of ${\widehat{F}}$ we can conclude ${\widehat{F}}(\pi_1\circ o) \le {\widehat{F}}(\pi_2\circ o)$ . Hence,

that is, we have shown that $\mathit{ev}_F$ is monotone.

Notice that the correspondence between fibred liftings and monotone evaluation maps is given in one direction by $\mathit{ev}={\widehat{F}}(\mathit{id}_{\mathcal{V}})$ , and conversely, by ${\widehat{F}}(p\colon X\to{\mathcal{V}})=\mathit{ev}\circ F(p)$ .

Evaluation maps as Eilenberg-Moore algebras. Evaluation maps have also been extensively considered in the coalgebraic approach to modal logics (Schröder Reference Schröder2008). A special kind of evaluation map arises when the truth values ${\mathcal{V}}$ have an algebraic structure for a given monad $(T,\mu,\eta)$ , that is, we have ${\mathcal{V}}=T{\Omega}$ for some object ${\Omega}$ and the evaluation map $T{\mathcal{V}}\to{\mathcal{V}}$ is an Eilenberg-Moore algebra for T. This notion of monadic modality has been studied in Hasuo (Reference Hasuo2015) where the category of free algebras for T was assumed to be order enriched. In Lemma 15 below we show that under reasonable assumptions, the evaluation map obtained as the free Eilenberg-Moore algebra on ${\Omega}$ (i.e. $\mathit{ev}\colon T{\mathcal{V}}\to{\mathcal{V}}$ is just $\mu_{{\Omega}}\colon T^2{\Omega}\to T{\Omega}$ ) is a monotone evaluation map, and hence gives rise to a fibred lifting of T.

Since $\le$ is obtained by lifting $\sqsubseteq$ under T we can infer that there exists a witness function $w\colon \le\ \to T(\sqsubseteq)$ that assigns to every pair of elements $t_1,t_2\in {\mathcal{V}}$ with $t_1\le t_2$ a witness $w(t_1,t_2)\in T(\sqsubseteq)$ with $T\pi_i(w(t_1,t_2)) = t_i$ . Hence $T\pi_i\circ w = \pi'_i$ , where $\pi_i\colon \sqsubseteq\ \to {{\Omega}}$ and $\pi'_i\colon \le\ \to {\mathcal{V}}$ are the usual projections.

Since $t'_1\le^T t'_2$ , there exists a witness $t'\in T(\le)$ with $T\pi'_i(t') = t'_i$ .

We show that $t = \mu_\sqsubseteq(Tw(t'))$ is a witness for $\mu_{{\Omega}}(t'_1) \le \mu_{{\Omega}}(t'_2)$ . It holds that $T\pi_i\circ \mu_\sqsubseteq \circ Tw = \mu_{{\Omega}}\circ TT\pi_i \circ Tw = \mu_{{\Omega}}\circ T(T\pi_i\circ w) = \mu_{{\Omega}}\circ T\pi'_i$ , where the first equality holds since $\mu$ is a natural transformation. This implies $T\pi_i(t) = (T\pi_i\circ \mu_\sqsubseteq \circ Tw)(t') = (\mu_{{\Omega}}\circ T\pi'_i)(t') = \mu_{{\Omega}}(t'_i)$ .

We provide next several examples of monotone evaluation maps which arise in this fashion.

The canonical evaluation map. In the case ${\mathcal{V}}=2$ , there exists a simple way of lifting a functor $F \colon \mathsf{Set}\to\mathsf{Set}$ : given a predicate $p\colon U \rightarrowtail X$ , one defines the canonical predicate lifting ${\widehat{F}_{\mathsf{can}}}(U)$ of F as the epi-mono factorization of $Fp\colon FU \to FX$ . This lifting corresponds to a canonical evaluation map ${\mathsf{true}} \colon 1 \to 2$ which maps the unique element of 1 into the element 1 of the quantale 2. For ${\mathcal{V}}$ -relations, a generalized notion of canonical evaluation map was introduced in Hofmann (Reference Hofmann2007). For $r\in{\mathcal{V,}}$ consider the subset ${\uparrow r}=\{v\in{\mathcal{V}}\mid v\ge r\}$ and write ${\mathsf{true}_{r}}\colon{\uparrow r}\hookrightarrow{\mathcal{V}}$ for the inclusion. Given $u\in F{\mathcal{V,}}$ we write $u\in F({\uparrow r})$ when u is in the image of the injective function $F({\mathsf{true}_{r}})$ . Following Hofmann (Reference Hofmann2007), we define ${\mathit{ev}_{\mathsf{can}}}:F{\mathcal{V}}\to{\mathcal{V}}$ as follows:

\begin{equation*} {\mathit{ev}_{\mathsf{can}}}(u)=\bigvee\{ r\mid u\in F({\uparrow r})\}.\end{equation*}

Example 18. Assume F is the powerset functor ${\mathcal{P}}$ and let $u\in{\mathcal{P}}({\mathcal{V}})$ . We obtain that

\begin{equation*} {\mathit{ev}_{\mathsf{can}}}(u)=\bigvee\{ r\mid u\subseteq\,{\uparrow r}\} \text{, or equivalently, } {\mathit{ev}_{\mathsf{can}}}(u)=\bigwedge u\,. \end{equation*}

When ${\mathcal{V}}=2,$ we obtain ${\mathit{ev}_{\mathsf{can}}}\colon{\mathcal{P}}2\to 2$ given by ${\mathit{ev}_{\mathsf{can}}}(u)= 1$ iff $u=\emptyset$ or $u=\{1\}$ . This corresponds to the $\Box$ operator from modal logic. If ${\mathcal{V}} = [0,\infty]$ we have ${\mathit{ev}_{\mathsf{can}}}(u)=\sup u$ .

The canonical evaluation map ${\mathit{ev}_{\mathsf{can}}}$ is monotone whenever the functor F preserves weak pullbacks (see Lemma 52 in Appendix A). For such functors, by Proposition 14, the map ${\mathit{ev}_{\mathsf{can}}}$ induces a fibred lifting ${\widehat{F}_{\mathsf{can}}}$ of F, called the canonical $\mathcal{V}\textrm{-}\mathsf{Pred}$ -lifting of F and defined by

\begin{equation*} {\widehat{F}_{\mathsf{can}}}(p)(u) = \bigvee\{r\mid F(p)(u)\in F({\uparrow r})\}\ \text{ for } p\in\mathcal{V}\textrm{-}\mathsf{Pred}_X\text{ and } u\in FX\,.\end{equation*}

The canonical $\mathcal{V}\textrm{-}\mathsf{Pred}$ -lifting of $T_\Sigma$ is defined for all $p \in \mathcal{V}\textrm{-}\mathsf{Pred}$ and $t\in T_\Sigma X$ by

\begin{equation*} {\widehat{T}_{\Sigma \mathsf{can}}}(p)(t) = \bigwedge_{x\in Var(t)} p(x)\,.\end{equation*}

5.2 From predicates to relations via Wasserstein

We describe next how functor liftings to $\mathcal{V}\textrm{-}\mathsf{Rel}$ can be systematically obtained using the change-of-base situation described above. In particular, we will show how the Wasserstein metric between probability distributions (defined in terms of couplings of distributions) can be naturally modelled in the fibrational setting.

Consider a ${\mathcal{V}}$ -predicate lifting ${\widehat{F}}$ of a $\mathsf{Set}$ -functor F. A natural way to lift F to ${\mathcal{V}}$ -relations using ${\widehat{F}}$ is to regard a ${\mathcal{V}}$ -relation $r\colon X\times X\to{\mathcal{V}}$ as a ${\mathcal{V}}$ -predicate on the product $X\times X$ . Formally, we will use the isomorphism ${{\iota_{X}}}$ described in Section 4. We can apply the functor ${\widehat{F}}$ to the predicate ${{\iota_{X}}}(r)$ in order to obtain the predicate ${\widehat{F}}\circ {{\iota_{X}}}(r)$ on the set $F(X\times X)$ . Ideally, we would want to transform this predicate into a relation on FX. So first, we have to transform it into a predicate on $FX\times FX$ . To this end, we use the natural transformation

(7)

We drop the superscript and simply write ${\lambda}$ when the functor F is clear from the context. Additionally, the bifibrational structure of $\mathcal{V}\textrm{-}\mathsf{Rel}$ plays a crucial role, as we can use the direct image functor ${\Sigma_{{\lambda}_{X}}}$ to transform ${\widehat{F}}\circ {{\iota_{X}}}(r)$ into a predicate on $FX\times FX$ . Putting all the pieces together, we define a lifting of F on the fibre ${{\mathcal{V}\textrm{-}\mathsf{Rel}_{X}}}$ as the composite ${W({\widehat{F}})}_X$ given by:

(8)

We still have to verify that ${W({\widehat{F}})}$ , as explained in Remark 21 hereafter denoted by ${\overline{F}}$ , is indeed a lifting of F to $\mathcal{V}\textrm{-}\mathsf{Rel}$ . The above construction provides the definition of ${\overline{F}}$ on the fibres and, in particular, on the objects of $\mathcal{V}\textrm{-}\mathsf{Rel}$ . For a morphism between ${\mathcal{V}}$ -relations $p\in{{\mathcal{V}\textrm{-}\mathsf{Rel}_{X}}}$ and $q\in{{\mathcal{V}\textrm{-}\mathsf{Rel}_{Y}}}$ , that is, a map $f\colon X\to Y$ such that $p\le {f^*}(q)$ , we define ${\overline{F}}(f)$ as the map $Ff\colon FX\to FY$ . To see that this is well defined, it remains to show that ${\overline{F}} p\le {({Ff})^*}({\overline{F}} q)$ . This is the first part of the next proposition.

This follows from the sequence of (in)equalities (9)–(14), where on each line we underlined the sub-expression that was rewritten and which we will explain in turn.

(9) \begin{equation*} {\overline{F}}\circ {f^*}(q) = {{\iota_{FX}}}^{-1} \circ {\Sigma_{{\lambda}_{X}}}\circ{\widehat{F}_{{\Delta} X}}\circ \underline{ {{\iota_{X}}}\circ{f^*}} (q)\end{equation*}

(10) \begin{equation*} ={{\iota_{FX}}}^{-1} \circ {\Sigma_{{\lambda}_{X}}}\circ\underline{{\widehat{F}_{{\Delta} X}}\circ{({{\Delta} f})^*}}\circ {{\iota_{Y}}}(q)\end{equation*}

(11) \begin{equation*}\le {{\iota_{FX}}}^{-1} \circ \underline{{\Sigma_{{\lambda}_{X}}}\circ{({F{\Delta} f})^*}}\circ {\widehat{F}_{{\Delta} Y}}\circ {{\iota_{Y}}}(q)\end{equation*}

(12) \begin{equation*}\le \underline{{{\iota_{FX}}}^{-1} \circ{({{\Delta} F f})^*}}\circ {\Sigma_{{\lambda}_{Y}}}\circ {\widehat{F}_{{\Delta} Y}}\circ {{\iota_{Y}}}(q)\end{equation*}

(13)

(14) \begin{equation*}= {({F f})^*}\circ{\overline{F}}(q) \end{equation*}

We obtained (9) and (14) using the definition of ${\overline{F}}$ . To derive the equalities in (10) and (13), we used the fact that ${{\iota}}$ is a fibred lifting of ${\Delta}$ . The inequality (11) follows from the fact that ${\widehat{F}}$ is a lifting of F, and hence, we have the inequality

(15) \begin{equation*} {\widehat{F}_{{\Delta} X}}\circ{({{\Delta} f})^*}\le {({F{\Delta} f})^*}\circ {\widehat{F}_{{\Delta} Y}}\,. \end{equation*}

Finally, the inequality (12) follows from the commutativity of the naturality squares of ${\lambda}$ as an instance of (5).

(16) \begin{equation*} {\Sigma_{{\lambda}_{X}}}\circ{({F{\Delta} f})^*}\le {({{\Delta} Ff})^*}\circ {\Sigma_{{\lambda}_{Y}}}\,. \end{equation*}

Now let us focus on the second part of the proof. Since ${\widehat{F}}$ is a fibred lifting by assumption, then the inequality (15) becomes an equality. When the functor F preserves weak pullbacks, then by Lemma 51 in Appendix A we know that the naturality squares of ${\lambda}$ are weak pullbacks. Hence, since the fibration $\mathcal{V}\textrm{-}\mathsf{Rel}$ has the Beck-Chevalley property for weak pullback squares, it follows that (16) is also an equality. We obtain that all the inequalities (9)–(14) are in fact equalities. This amounts to the fact that ${\overline{F}}$ is a fibred lifting.

Spelling out the concrete description of the direct image functor and of ${\lambda}_X$ , we obtain for a relation $p\in\mathcal{V}\textrm{-}\mathsf{Rel}_{X}$ and $t_1,t_2\in FX$ , that

(17) \begin{equation*} {\overline{F}}(p)(t_1,t_2)=\bigvee\{{\widehat{F}}(p)(t)\mid t\in F(X\times X), F\pi_i(t)=t_i\}\end{equation*}

Unravelling the definition of ${\widehat{F}}(p)(t)=\mathit{ev}\circ F(p)$ , we obtain for ${\overline{F}}(p)$ the same formula as for the extension of F on ${\mathcal{V}}$ -matrices, as given in Hofmann (Reference Hofmann2007, Definition 3.4). This definition in Hofmann (Reference Hofmann2007) is obtained by a direct generalization of the Barr extensions of $\mathsf{Set}$ -functors to the bicategory of relations. In contrast, we obtained (17) by exploiting the fibrational change-of-base situation and by first considering a $\mathcal{V}\textrm{-}\mathsf{Pred}$ -lifting.

We call a lifting of the form ${\overline{F}}$ the Wasserstein lifting of F corresponding to ${\widehat{F}}$ . This terminology is motivated by the next example.

Example 23. When $F = {\mathcal{D}}$ (the distribution functor), ${\mathcal{V}} = [0,1]$ and $ev_F$ is as in Example 17, then ${\overline{F}}$ is the original Wasserstein metric from transportation theory (Villani Reference Villani2009), which – by the Kantorovich-Rubinstein duality – is the same as the Kantorovich metric. Here we compare two probability distributions $t_1,t_2\in {\mathcal{D}} X$ and obtain as a result the coupling $t\in {\mathcal{D}}(X\times X)$ with marginal distributions $t_1,t_2$ , giving us the optimal plan to transport the ‘supply’ $t_1$ to the ‘demand’ $t_2$ . More concretely, given a metric $d\colon X\times X\to {\mathcal{V}}$ , the (discrete) Wasserstein metric is defined as

\begin{equation*} d^W(t_1,t_2) = \inf \{ \sum_{x,y\in X} d(x,y)\cdot t(x,y) \mid \sum_y t(x,y) = t_1(x), \sum_x t(x,y) = t_2(y) \}. \end{equation*}

On the other hand, when $\mathit{ev}_F$ is the canonical evaluation map of Example 19 the corresponding $\mathcal{V}\textrm{-}\mathsf{Rel}$ -lifting ${\overline{F}_{\mathsf{can}}}$ minimizes the longest distance (and hence the required time) rather than the total cost of transport.

Then, given a metric $d\colon X\times X\to [0,1]$ and $X_1,X_2\subseteq X$ , the lifted metric is defined as follows (remember that the order is reversed on [0,1]):

\begin{equation*} {\overline{F}}(d)(X_1,X_2) = \inf \{ \sup d[Y] \mid Y \subseteq X\times X, \pi_i[Y] = X_i \} \end{equation*}

As explained in Reference Baldan, Bonchi, Kerstan and König Baldan et al. (2018) , Reference Mémoli Mémoli (2011) , this is the same as the Hausdorff metric $d^H$ defined by:

\begin{equation*} d^H(X_1,X_2) = \sup \{\sup_{x_1\in X_1} \inf_{x_2\in X_2} d(x_1,x_2), \sup_{x_2\in X_2} \inf_{x_1\in X_1} d(x_1,x_2)\} \end{equation*}

Example 25. Recall the ${\widehat{T}_{\Sigma \mathsf{can}}}$ , the canonical $\mathcal{V}\textrm{-}\mathsf{Pred}$ -lifting of the term monad $T_\Sigma$ , from Example 20. We now illustrate $\overline{T_{\Sigma}}_\mathsf{can}$ , the Wasserstein lifting corresponding to ${\widehat{T}_{\Sigma \mathsf{can}}}$ . By (17) we have that for all $d\in \mathcal{V}\textrm{-}\mathsf{Rel}$ and $t_1,t_2\in T_\Sigma X$ , it holds that

\begin{equation*}\overline{T_{\Sigma}}_\mathsf{can}(d)(t_1,t_2) = \bigvee\{{\widehat{T}_{\Sigma \mathsf{can}}}(d)(t)\mid t\in T_\Sigma(X\times X), T_\Sigma\pi_i(t)=t_i\}\end{equation*}

Assume that for $t_1,t_2\in T_\Sigma X,$ there exists a $\Sigma$ -context $C(-_1, \dots, -_n)$ such that $t_1 =C(x_1, \dots, x_n)$ and $t_2=C(y_1,\dots,y_j)$ for $j\in \{1,\dots,n\}$ and variables $x_j,y_j \in X$ . ^{Footnote 1} Then, such a context C is unique and the above set contains exactly one $t \in T_\Sigma (X \times X)$ that is $t = C((x_1,y_1),\dots,(x_n,y_n))$ . Thus, $\overline{T_{\Sigma}}_\mathsf{can}(d)(t_1,t_2) = {\widehat{T}_{\Sigma \mathsf{can}}}(d)( C((x_1,y_1),\dots,(x_n,y_n)))$ that, by definition of ${\widehat{T}_{\Sigma \mathsf{can}}}$ , is $\bigwedge_{j\in \{1,\dots, n\}} d(x_j,y_j)$ . Instead, if such a context C does not exist, then the above set is empty and $\overline{T_{\Sigma}}_\mathsf{can}(d)(t_1,t_2)=\bot$ . In a nutshell,

(18) \begin{equation*} \overline{T_{\Sigma}}_\mathsf{can}(d)(t_1,t_2)= \begin{cases} \bigwedge_{j\in\{1,\dots,n\}} d(x_j,y_j) & \exists C(-_1, \dots, -_n) \text{ such that } \\ & \qquad t_1= C(x_1,\dots,x_n) \text{ and } t_2=C(y_1,\dots,y_n)\\ \bot & \text{otherwise}\\ \end{cases} \end{equation*}

As an example take as terms $t_1=f(g(x),h(y))$ and $t_2=f(g(z),h(x))$ , then the context is $C(-_1, -_2)= f(g(-_1), h(-_2))$ and the term $t\in T_{\Sigma}(X\times X)$ is $t=f(g( (x,z)), h((y,x)))$ . Thus, $\overline{T_{\Sigma}}_\mathsf{can}(d)(t_1,t_2)= d(x,z) \wedge d(y,x)$ . Instead, if one takes $t_1=f(g(x),h(y))$ and $t_2=f(g(z),y)$ , then $\overline{T_{\Sigma}}_\mathsf{can}(d)(t_1,t_2)=\bot$ .

The next lemma establishes that this construction is functorial: liftings of natural transformations to $\mathcal{V}\textrm{-}\mathsf{Pred}$ can be converted into liftings of natural transformations between the corresponding Wasserstein liftings on $\mathcal{V}\textrm{-}\mathsf{Rel}$ .

We now prove the following lemma:

Proof. The existence (and in this case uniqueness) of the lifting ${\widehat{\zeta}}$ is equivalent to the fact that ${\widehat{F}_{X}}\le{({{\zeta}_X})^*}\circ{\widehat{G}_{X}}$ for all X. This is fairly standard, but we include here an explanation for the sake of completeness. If ${\widehat{\zeta}}$ exists, then for all $p\in{\mathcal{V}\textrm{-}\mathsf{Pred}_{X}}$ we have the next diagram, where the dashed arrow exists and is unique by the universal property in Definition 4.

Since the fibres in $\mathcal{V}\textrm{-}\mathsf{Pred}$ are posets, this means that ${\widehat{F}}(p)\le{({{\zeta}_X})^*}\circ{\widehat{G}}(p)$ , since there is a unique morphism in the fibre from ${\widehat{F}}(p)$ to ${\widehat{G}}(p)$ . For the same reason, any two liftings of ${\zeta}$ must coincide. Conversely, if the inequality ${\widehat{F}}(p)\le{({{\zeta}_X})^*}\circ{\widehat{G}}(p)$ holds, we compose with ${\widetilde{{\zeta}_X}_{{\widehat{G}}(p)}}$ in order to obtain ${\widehat{\zeta}}_{p}$ .

We have to show that ${\overline{F}_{X}}\le{({{\zeta}_X})^*}\circ{\overline{G}_{X}}$ . We obtain:

To show the inequality on the third line, we notice that ${\Sigma_{{\lambda}_X^F}}\circ{({{\zeta}_{X\times X}})^*} \le {({{\zeta}_X\times {\zeta}_X})^*}\circ {\Sigma_{{\lambda}_X^G}}$ is the mate, as in (5), of the following commutative square, which in turn commutes by the naturality of ${\zeta}$ and the uniqueness of mediating morphisms into the product.

To summarize, the proof of the first part of the lemma follows from the next lax diagram, by composing with the isomorphisms ${{\iota_{X}}}$ and ${{\iota_{FX}}}^{-1}$ .

It remains to prove that ${{\widehat{F}}} \le {\zeta}^*_X\circ {{\widehat{G}}}$ is equivalent to $\mathit{ev}_F\le \mathit{ev}_G\circ {\zeta}_{\mathcal{V}}$ . The implication from left to right is obtained by setting $X = {\mathcal{V}}$ and applying the functors on both sides to $\mathit{id}_{\mathcal{V}}$ . We get the other direction by taking $p\colon X\to {\mathcal{V}}$ and computing $({({{\zeta}_X})^*}\circ {{\widehat{G}}})(p) = \mathit{ev}_G\circ Gp\circ {\zeta}_X = \mathit{ev}_G \circ {\zeta}_V\circ Fp \ge \mathit{ev}_F\circ Fp = {{\widehat{F}}}(p)$ . Note that this uses the naturality of ${\zeta}$ .

5.3 Preservation of reflexivity, symmetry and transitivity

For ${\mathcal{V}}=[0,\infty]$ , one is also interested in lifting functors to the category of (generalized) pseudo-metric spaces, not just of $[0,\infty]$ -valued relations. This motivates the next question: when does the lifting ${\overline{F}}$ restrict to a functor on $\mathcal{V}\textrm{-}\mathsf{Cat}$ and $\mathcal{V}\textrm{-}\mathsf{Cat}_\mathsf{sym}$ ? We have the following characterization theorem, where ${\kappa_{X}}\colon X\to {\mathcal{V}}$ is the constant function $x\mapsto 1$ and $u\otimes v\colon X\to {\mathcal{V}}$ denotes the pointwise tensor of two predicates $u,v\colon X\to {\mathcal{V}}$ , that is, $(u\otimes v)(x)=u(x)\otimes v(x)$ .

Furthermore, let ${{\delta_{X}}}:X\to X\times X$ be the diagonal function on a set X. A relation $r:X\times X\to{\mathcal{V}}$ is reflexive if and only if

(19) \begin{equation*} {{{{\delta_{X}}}}^*}\circ {{\iota_{X}}}(r)\ge {\kappa_{X}}\,.\end{equation*}

Consequently, when all the above hypotheses are satisfied, then the corresponding $\mathcal{V}\textrm{-}\mathsf{Rel}$ Wasserstein lifting ${\overline{F}}$ restricts to a lifting of F to both $\mathcal{V}\textrm{-}\mathsf{Cat}$ and $\mathcal{V}\textrm{-}\mathsf{Cat}_\mathsf{sym}$ .

For ${\mathcal{V}}=[0,\infty]$ , the first condition of Theorem 27 is a relaxed version of a condition in Baldan et al. (Reference Baldan, Bonchi, Kerstan and König2018, Definition 5.14) used to guarantee reflexivity. The second condition (for transitivity) is equivalent to a non-symmetric variant of a condition in Baldan et al. (Reference Baldan, Bonchi, Kerstan and König2018) (see Lemma 54 in Appendix A).

The proof is immediate from Lemmas 28, 30 and 31 which we prove next.

Lemma 28. Assume ${{\widehat{F}}}$ is a lifting of F such that

\begin{equation*} {{\widehat{F}}}({\kappa_{X}})\ge{\kappa_{FX}}\,. \end{equation*}

Then, ${\overline{F}}({\mathit{diag}_{X}}) \ge {\mathit{diag}_{FX}}$ ; hence, ${\overline{F}}$ preserves reflexive relations.

Proof. Notice that

\begin{equation*}{\mathit{diag}_{X}}={{\iota_{X}}}^{-1}{\Sigma_{{{\delta_{X}}}}}({\kappa_{X}})\,.\end{equation*}

Using this observation, we obtain that

(20)

(21) \begin{equation*} \ge {{\iota_{FX}}}^{-1}\circ{\Sigma_{{\lambda}_{X}}}\circ{\Sigma_{F{{\delta_{X}}}}}\circ {\widehat{F}} ({\kappa_{X}})\end{equation*}

(22) \begin{equation*} \ge {{\iota_{FX}}}^{-1}\circ{\Sigma_{{\lambda}_{X}}}\circ{\Sigma_{F{{\delta_{X}}}}} ({\kappa_{FX}})\end{equation*}

(23) \begin{equation*} = {{\iota_{FX}}}^{-1}\circ{\Sigma_{{{\delta_{FX}}}}} ({\kappa_{FX}})\end{equation*}

(24) \begin{equation*} = {\mathit{diag}_{FX}}\end{equation*}

In (20), we used the definition of ${\overline{F}}$ . For the inequality (21), we used that ${\widehat{F}}$ is a lifting of F and the mate of (3), that is, ${\widehat{F}}\circ {\Sigma_{{{\delta_{X}}}}}\ge {\Sigma_{F{{\delta_{X}}}}}\circ {\widehat{F}}$ . The inequality (22) is the hypothesis, while in (23) we used that ${\lambda}_{X}\circ F{{\delta_{X}}}={{\delta_{FX}}}$ .

Preservation of reflexive relations is now immediate. For $r\in\mathcal{V}\textrm{-}\mathsf{Rel}_{X}$ is reflexive when $r\ge{\mathit{diag}_{X}}$ . Hence, ${\overline{F}}(r)\ge{\overline{F}}({\mathit{diag}_{X}})\ge{\mathit{diag}_{FX}}$ , which entails that ${\overline{F}}$ is reflexive.

We now turn our attention to the preservation of composition of relations and of the transitivity property.

We will use the notations $\pi_i:X\times X\times X\to X$ to denote the $i\textrm{th}$ projection on $X^3$ and $\tau_i:FX\times FX\times FX\to FX$ to denote the $i\textrm{th}$ projection on $(FX)^3$ .

We will use the fact that the composition $p{\boldsymbol{\ \cdot\ }} q$ of two relations $p,q\colon X\times X\to {\mathcal{V}}$ can be written as the composite

(25) \begin{equation*} p{\boldsymbol{\ \cdot\ }} q={{\iota_{X}}}^{-1}{\Sigma_{\langle{\pi_1},{\pi_3}\rangle}}({{\langle{\pi_2},{\pi_3}\rangle}^*}({{\iota_{X}}}q) \otimes {{\langle{\pi_1},{\pi_2}\rangle}^*}({{\iota_{X}}}p))\end{equation*}

Proof. It is easy to verify that the square below is a pullback.

By applying F to the diagram, we obtain the diagram below where the square is a weak pullback (since F preserves weak pullbacks).

Using this diagram, we can show that the square below is a weak pullback as well.

Assume that $t_1,t_2\in F(X^2)$ , $(s_1,s_2,s_3)\in (FX)^3$ are given such that ${\lambda}_X(t_1) = (s_2,s_3)$ (which means $F\pi_1(t_1) = s_2$ , $F\pi_2(t_1) = s_3$ ) and ${\lambda}_X(t_2) = (s_1,s_2)$ (which means $F\pi_1(t_2) = s_1$ , $F\pi_2(t_2) = s_2$ ). That is, $t_1,t_2$ live on the middle level and $s_3,s_2,s_1$ on the lower level (in that order) in the diagram above. Since the square is a weak pullback, there exists $t\in F(X^3)$ such that $F\langle{\pi_2},{\pi_3}\rangle(t) = t_1$ and $F\langle{\pi_1},{\pi_2}\rangle(t) = t_2$ . It remains to verify that $\nu_X(t) = (s_1,s_2,s_3)$ : for instance $F\pi_1(t) = (F\pi_1\circ F\langle{\pi_1},{\pi_2}\rangle)(t) = F\pi_1(t_2) = s_1$ . (Analogously for $s_2,s_3$ .)

Since the Beck-Chevalley condition holds, we obtain

\begin{equation*} {\Sigma_{\nu_X}}{{\langle{F\langle{\pi_2},{\pi_3}\rangle},{}\rangle {F\langle{\pi_1},{\pi_2}\rangle}}^*} = {{\langle{\langle{\tau_2},{\tau_3}\rangle},{\langle{\tau_1},{\tau_2}\rangle}\rangle}^*}{\Sigma_{{\lambda}_X\times {\lambda}_X}}. \end{equation*}

Then, we will apply this to a predicate of the form $\otimes \circ (u\times w)$ and using the facts

• ${{\langle{h_1},{h_2}\rangle}^*}(\otimes\circ(u\times w))={{h_1}^*}(u) \otimes {{h_2}^*}(w)$ .

• ${\Sigma_{f\times f}}(\otimes\circ (u\times w)) = \otimes \circ({\Sigma_{f}}(u)\times {\Sigma_{f}}(w))$ .

we derive the desired equality.

While the first item above is straightforward, the second has to be further explained. Whenever $f\colon X\to Y$ , $p,p'\colon X\to {\mathcal{V}}$ , $y,y'\in Y$ , we have (using distributivity):

Lemma 30. Assume F preserves weak pullbacks and ${{\widehat{F}}}$ is a fibred lifting of F such that

(26) \begin{equation*} {{\widehat{F}}}(u\otimes v)\ge{{\widehat{F}}}(u)\otimes{{\widehat{F}}}(v) \end{equation*}

Then, ${\overline{F}}(p{\boldsymbol{\ \cdot\ }} q) \ge {\overline{F}}(p){\boldsymbol{\ \cdot\ }} {\overline{F}}(q);$ hence, ${\overline{F}}$ preserves transitive relations.

Proof. We denote by $\nu_X\colon F(X^3)\to (FX)^3$ the map defined as $\nu_X = \langle F\pi_1,F\pi_2,F\pi_3\rangle$ .

(27)

(28) \begin{equation*} \ge {{\iota_{X}}}^{-1}{\Sigma_{{\lambda}_X}}{\Sigma_{F\langle{\pi_1},{\pi_3}\rangle}}{\widehat{F}} ({{\langle{\pi_2},{\pi_3}\rangle}^*}({{\iota_{X}}}q)\otimes{{\langle{\pi_1},{\pi_2}\rangle}^*}({{\iota_{X}}}p))\end{equation*}

(29) \begin{equation*} \ge {{\iota_{X}}}^{-1}{\Sigma_{{\lambda}_X}}{\Sigma_{F\langle{\pi_1},{\pi_3}\rangle}}{\widehat{F}}({{\langle{\pi_2},{\pi_3}\rangle}^*}({{\iota_{X}}}q))\otimes{\widehat{F}}({{\langle{\pi_1},{\pi_2}\rangle}^*}({{\iota_{X}}}p)) \end{equation*}

(30) \begin{equation*} = {{\iota_{X}}}^{-1}{\Sigma_{{\lambda}_X}}{\Sigma_{F\langle{\pi_1},{\pi_3}\rangle}}({{F\langle{\pi_2},{\pi_3}\rangle}^*}{\widehat{F}}({{\iota_{X}}}q)\otimes{{F\langle{\pi_1},{\pi_2}\rangle}^*}{\widehat{F}}({{\iota_{X}}}p)) \end{equation*}

(31) \begin{equation*} = {{\iota_{X}}}^{-1}{\Sigma_{\langle{\tau_1},{\tau_3}\rangle}}{\Sigma_{\nu_X}}({{F\langle{\pi_2},{\pi_3}\rangle}^*}{\widehat{F}}({{\iota_{X}}}q)\otimes{{F\langle{\pi_1},{\pi_2}\rangle}^*}{\widehat{F}}({{\iota_{X}}}p))\end{equation*}

(32) \begin{equation*} ={{\iota_{X}}}^{-1}{\Sigma_{\langle{\tau_1},{\tau_3}\rangle}}({{\langle{\tau_2},{\tau_3}\rangle}^*}{\Sigma_{{\lambda}_X}}({\widehat{F}}({{\iota_{X}}}q))\otimes{{\langle{\tau_1},{\tau_2}\rangle}^*}{\Sigma_{{\lambda}_X}}({\widehat{F}}({{\iota_{X}}}p)))\end{equation*}

(33) \begin{equation*} ={{\iota_{X}}}^{-1}{\Sigma_{\langle{\tau_1},{\tau_3}\rangle}}({{\langle{\tau_2},{\tau_3}\rangle}^*}({{\iota_{X}}}{\overline{F}} q)\otimes{{\langle{\tau_1},{\tau_2}\rangle}^*}({{\iota_{X}}}{\overline{F}} p))\end{equation*}

(34) \begin{equation*} ={\overline{F}} p{\boldsymbol{\ \cdot\ }}{\overline{F}} q \end{equation*}

The equalities (27), (33) and (34) follow by unravelling the definition of ${\overline{F}}$ and from (25). The inequality (28) follows using by the mate of (3). The inequality (29) follows from the hypothesis on ${\widehat{F}}$ . The equality (30) is obtained using the ${\overline{F}}$ is a fibred lifting. To prove the equality in (31), we use that ${\lambda}_X\circ F\langle{\pi_1},{\pi_3}\rangle=\langle{\tau_1},{\tau_3}\rangle\circ\nu_X$ . Finally, (32) follows from Lemma 29.

Assume $r\in\mathcal{V}\textrm{-}\mathsf{Rel}_X$ is transitive, that is, $r{\boldsymbol{\ \cdot\ }} r\le r$ . Then, we have ${\overline{F}} r{\boldsymbol{\ \cdot\ }} {\overline{F}} r\le{\overline{F}}(r{\boldsymbol{\ \cdot\ }} r)\le {\overline{F}} r$ , hence ${\overline{F}} r$ is transitive.

Proof. We first observe that the square below commutes.

Knowing that ${\lambda}_X = \langle{F\pi_1^X},{F\pi_2^X}\rangle$ and that ${\mathsf{sym}_{X}} = \langle{\pi_2},{\pi_1^X}\rangle$ , where $\pi_i^X\colon X\times X\to X$ , we can easily show that the square commutes:

Recall that $p\in{{\mathcal{V}\textrm{-}\mathsf{Rel}_{Y}}}$ is symmetric when $p=p\circ {\mathsf{sym}_{Y}}$ . We cannot perform a reindexing along ${\mathsf{sym}_{Y}}$ in the fibration $\mathcal{V}\textrm{-}\mathsf{Rel}$ , since ${\mathsf{sym}_{Y}}$ is not a morphism on Y, but on $Y\times Y$ . Instead, we have that p is symmetric if and only if

\begin{equation*}{{\iota_{Y}}}p={({{\mathsf{sym}_{Y}}})^*}({{\iota_{Y}}}p)\end{equation*}

in $\mathcal{V}\textrm{-}\mathsf{Pred}$ . Hence, we want to show that for any $r\in{{\mathcal{V}\textrm{-}\mathsf{Rel}_{X}}}$ the implication holds

\begin{equation*}{{\iota_{X}}}r={({{\mathsf{sym}_{X}}})^*}({{\iota_{X}}}r)\Rightarrow {{\iota_{FX}}}{\overline{F}} r={({{\mathsf{sym}_{FX}}})^*}({{\iota_{FX}}}{\overline{F}} r)\end{equation*}

We have the following inequalities:

However, using the idempotency of ${\mathsf{sym}_{FX}}$ and the monotonicity of ${({{\mathsf{sym}_{FX}}})^*}$ from the inequality

\begin{equation*}{{\iota_{FX}}}{\overline{F}} r\le {({{\mathsf{sym}_{FX}}})^*} ({{\iota_{FX}}}{\overline{F}} r)\end{equation*}

that we have just proved above we can infer that the equality also holds.

We can establish generic sufficient conditions on a monotone evaluation map $\mathit{ev}$ so that the corresponding $\mathcal{V}\textrm{-}\mathsf{Pred}$ -lifting ${\widehat{F}}$ satisfies the conditions of Theorem 27. In Proposition 53 in Appendix A, we show that ${\widehat{F}}(p\otimes q)\ge{\widehat{F}}(p)\otimes{\widehat{F}}(q)$ holds whenever the map $\otimes\colon{\mathcal{V}}\times{\mathcal{V}}\to{\mathcal{V}}$ is the carrier of a lax morphism in the category of F-algebras between $({\mathcal{V}},\mathit{ev})^2\to({\mathcal{V}},\mathit{ev})$ , that is, $\otimes \circ(\mathit{ev}\times\mathit{ev}) \circ{\lambda}_{\mathcal{V}} \le \mathit{ev} \circ F(\otimes)$ . Furthermore, ${\widehat{F}}({\kappa_{X}})\ge{\kappa_{X}}$ holds whenever the map ${\kappa_{\mathbb{1}}}\colon\mathbb{1}\to{\mathcal{V}}$ is the carrier of a lax morphism from the one-element F-algebra $!\colon F\mathbb{1}\to\mathbb{1}$ to $({\mathcal{V}},\mathit{ev})$ , that is, ${\kappa_{\mathbb{1}}}\circ !\le\mathit{ev}\circ F{\kappa_{\mathbb{1}}}$ . These two requirements correspond to the conditions $(Q_\otimes)$ , respectively $(Q_{k})$ satisfied by a topological theory in the sense of Hofmann (Reference Hofmann2007, Definition 3.1). Since these two are satisfied by the canonical evaluation map ${\mathit{ev}_{\mathsf{can}}}$ , ^{Footnote 2} we immediately obtain

Proof. (1) Given $t\in FX$ , we have on one hand that

and on the other, that

Hence, in order to show the desired inequality it is sufficient to show that

\begin{equation*} Fp(t)\in F({\uparrow r}),\ Fq(t)\in F({\uparrow s})\textrm{ imply } F(p\otimes q)(t) \in F({\uparrow (r\otimes s)})\,. \end{equation*}

Let $r,s\in{\mathcal{V}}$ so that $Fp(t)\in F({\uparrow r})$ and $Fq(t)\in F({\uparrow s})$ . Note that $p\otimes q\colon X\to {\mathcal{V}}$ is the composite:

Hence, $F(p\otimes q)$ is the composite of the arrows on the top line of the diagram below:

Note that the triangle and the square above are commutative. Using the abbreviation $\theta = F((p\times q)\circ {{\delta_{X}}})(t),$ we have that:

(35) \begin{eqnarray*} F(p\otimes q)(t) & = & F(\otimes)(\theta) \end{eqnarray*}

(36) \begin{eqnarray*} ((Fp)(t),(Fq)(t)) & = & {\lambda}_{\mathcal{V}}(\theta)\end{eqnarray*}

From Lemma 51 in Appendix A, we know that the square in the diagram below is a weak pullback.

(37)

By hypothesis, we know that there exists $u\in F({\uparrow r})$ and $v\in F({\uparrow s})$ such that $Fp(t) = F{\mathsf{true}_{r}}(u)$ and $Fq(t) = F{\mathsf{true}_{s}}(v)$ . Hence,

\begin{equation*} (F{\mathsf{true}_{r}}\times F{\mathsf{true}_{s}})(u,v)={\lambda}_{\mathcal{V}}(\theta)\,. \end{equation*}

Using the fact that the square (37) is a weak pullback, there exists $w\in F(({\uparrow r})\times ({\uparrow s}))$ such that $F({\mathsf{true}_{r}}\times {\mathsf{true}_{s}})(w) = \theta$ , $F\pi_1(w) = u$ and $F\pi_2(w) = v$ .

Thus far, we have shown that

\begin{equation*} F(p\otimes q)(t)=F(\otimes)(\theta) =F(\otimes)\circ F({\mathsf{true}_{r}}\times {\mathsf{true}_{s}})(w) \end{equation*}

for some $w\in F(({\uparrow r})\times ({\uparrow s}))$ . To finish the proof of the first item, we will prove that $F(\otimes)\circ F({\mathsf{true}_{r}}\times {\mathsf{true}_{s}})$ factors through $F{\mathsf{true}_{r\otimes s}}\colon F({\uparrow (r\otimes s)})\to F{\mathcal{V}}$ .

To this end, notice that due to monotonicity of the tensor product, we know that $({\uparrow r})\otimes ({\uparrow s}) \subseteq \,{\uparrow (r\otimes s)}$ . Hence, $\otimes\colon{\mathcal{V}}\times{\mathcal{V}}\to{\mathcal{V}}$ restricts to a function ${\otimes_{|{\uparrow r},{\uparrow s}}}$ on ${\uparrow r}\times{\uparrow s}$ so that the square below commutes.

Now we put $z:=F(\otimes_{|{\uparrow r},{\uparrow s}})(w)$ and observe that

We conclude that $F(p\otimes q)(t)\in F({\uparrow (r\otimes s)})$ .

(2) Now let us prove the second item. Given $t\in FX$ , we know that

In order to show that ${\widehat{F}_{\mathsf{can}}}{\kappa_{X}}(t) \ge 1$ it is sufficient to prove that $F{\kappa_{X}}(t) \in F({\uparrow {1}})$ .

Let $e\colon X\to {\uparrow {1}}$ a constant mapping with $e(x) = 1$ . Then, the diagram to the left below commutes and by applying the functor F we obtain the diagram below.

Now $F~{\kappa_{X}}(t)=F{\mathsf{true}_{1}}(Fe(t))$ , hence $F~{\kappa_{X}}(t)\in F({\uparrow {1}})$ .

An immediate consequence of Proposition 32 and of Theorem 27 is that the Wasserstein lifting ${\overline{F}_{\mathsf{can}}}$ that corresponds to ${\widehat{F}_{\mathsf{can}}}$ restricts to a lifting of F to both $\mathcal{V}\textrm{-}\mathsf{Cat}$ and $\mathcal{V}\textrm{-}\mathsf{Cat}_\mathsf{sym}$ .