What do category theorists know about "probabilistic metric spaces"? I recently stumbled upon the notion of probabilistic metric space as a generalization of Lawvere's metric spaces, and I am very interested in understanding it deeper.
In short, instead of a distance $d(x,y)$ between two elements of a set, one considers the probability that, if the distance between $x,y$ was measured, it would be $<t$, for $0\le t\le \infty$. More formally, instead of a set $X$ and a function $d : X\times X \to [0,\infty]$ + axioms, one specifies a set $X$ and a function $\gamma : X\times X \to D$, where $D$ is a space of "probability distributions", i.e. some variation of the function space $[0,1]^{[0,\infty]}$.
Originally, this notion was proposed by Menger (yes, Karl "sponge" Menger) who, well... didn't get it completely right, because although there's a set of metric space axioms which is very easy to translate in distribution terms (for example, each distribution $F_{xy}$ sends 0 to 0 and $F_{xy} = F_{yx}$ as distributions), it is not exactly clear how to handle the triangle inequality.
Various solutions have been proposed to estimate the value $F_{pq}(x+y)$ in terms of $F_{pq}(x), F_{qr}(y)$ for points $p,q,r\in X$ and real numbers $x,y\in[0,\infty]$.

*

*Menger proposed the definition of a "statistical metric space"; fix a function $T : [0,1]\times[0,1]\to [0,1]$ with suitable properties and ask that the triangle inequality holds as
$$ F_{pr}(x+y) \ge T(F_{pq}(x), F_{qr}(y))$$
This is certainly a suboptimal approach to the matter: the role of $T$ is not clear, not many examples are given by Menger, it is not clear what requests on $T$ are necessary and why, it is not clear how to gain at least some geometric intuition about a tuple $(X,F_{pq},T)$ of this sort (for example, it is unclear for which choices of $T$, if any, the statistical metric defines a topology?)...


*Almost immediately after, Abraham Wald (yes, the guy of the bullets!) had a better idea: distributions have a natural convolution operation, so why don't we ask the $F_{pq}$'s to satisfy the condition
$$\tag{conv} F_{pr} \ge F_{pq} * F_{qr}$$
pointwise as distributions?
It turns out that, if Menger's condition is too weak to produce meaningful examples, Wald's condition is too rigid: there's no intuition for what a "Menger space" is, and by contrast a "Wald space", a set $X$ equipped with a family of distributions $F : X\times X \to D$ such that (conv) is true, is very much like a metric space.
To get out of this cul-de-sac, the consensus in the following years has been to concentrate on asking Menger's $T$ functions to satisfy a certain amount of properties meant to rule out pathological examples, while allowing the generation of meaningful ones.
Building on Menger's $T$-functions idea, in the 50's, Schweizer and Sklar started to focus on a notion of $T$-probabilistic space where $T : [0,1]\times[0,1]\to[0,1] $ is one of, or akin to one of, the functions

*

*$(a,b)\mapsto \max(a+b-1,0)$

*$(a,b)\mapsto ab$

*$(a,b)\mapsto\min(a,b)$

*$(a,b)\mapsto \max(a,b)$

*$(a,b)\mapsto a+b-ab$

*$(a,b)\mapsto \min(a+b,1)$
and this ultimately led them to the following
Definition. A probabilistic $T$-space, or $T$-space for short, is a set $X$ equipped with a function $F : X\times X \to ([0,\infty]\to [0,1])$ and $T : [0,1]\times [0,1] \to [0,1]$ are subject to the following conditions:

*

*$F_{pq}(0)=0$;

*If $p=q$, then $F_{pq}(x)=1$ for all $x > 0$;

*If $p\ne q$, then there exists a $x>0$ such that $F_{pq}(x) < 1$;

*The $F_{pq}$'s are symmetric;

*$T$ is symmetric (i.e., $T(a,b)=T(b,a)$ for all $a,b\in[0,1]$), associative (in the obvious sense) and monotone when $[0,1]\times [0,1]$ has the product order;

*$T$ satisfies the triangular norm boundary condition $T(a,1)=a$ for all $a\in [0,1]$.

Curiously enough, the associativity request for these operations has a geometric interpretation: if $p, q, r, s$ are four points in $S$, and if $F_{pq} (x), F_{qr} (y)$, and $F_{rs} (z)$ are given, then $F_{ps} (x + y + z)$ can be estimated in two ways: either by estimating $F_{pr} (x + y)$ and combining this estimate with $F_{rs} (z)$, or by combining $F_{pq} (x)$ with the estimate of $F_{qs} (y + z)$. Requiring that these estimates be consistent leads naturally to the associativity property.
(Note also that the last three $T$ functions are not $t$-norms!)

But the important part (=category theory) is yet to come! To a trained eye, it's rather clear what we're doing:


*Consider the complete lattice $[0,1$];

*Consider a quantale structure $\otimes$ on $[0,1]$;

*Use the complete lattice anti-isomorphism $[0,\infty]\cong [0,1]$ given by the exponential function to "change scalars" and turn $[0,\infty]$ into a $[0,1]$-category;

*Consider the Day convolution monoidal structure on the set of all $[0,1]$-enriched presheaves $[0,\infty]\to [0,1]$.

*...

*Profit.

Let's be more precise: all what follows comes from the book "Monoidal Topology"
Fact. There is a quantale isomorphism
$ ([0,\infty]^{op},+,0)\cong ([0,1],\cdot,1) $
given by the functions
$$ \exp(-t) : [0,\infty]^{op} \leftrightarrows [0,1] : -\log t   $$
allowing to regard $[0,\infty]^{op}$ as just the $[0,1]$-category $[0,1]$, with its self-enrichment given by the closed structure.
Definition. A distance distribution is a function $\varphi : [0,\infty] \to [0,1]$ with the property that
$$ \tag{$\star$}\varphi (x) = \bigvee_{w<x} \varphi (w) $$
for all $x\in[0,\infty]$.
A distance distribution $\varphi$ is, of course, just a $[0,1]$-enriched presheaf.
From here we can define the convolution of two distance distributions as follows:
$$ \varphi * \psi : u\mapsto \bigvee_{v+w\le u} \varphi(v)\otimes\psi(w) \tag{$\heartsuit$}$$
According to the above interpretation of distributions as presheaves, this is just the Day convolution product on ${\sf PDist}=[0,1]\textsf{-Cat}([0,\infty],[0,1])$, that uses the quantale operation $\otimes$ on $[0,1]$, and $+$ on $[0,\infty]$, because $(\heartsuit)$ is just the coend $\int^{vw} \hom_{[0,\infty]}(v+w,u)\cdot \varphi(v)\otimes\psi(w)$.
The convolution operation $\_*\_ : {\sf PDist}\times{\sf PDist} \to {\sf PDist}$ is a monoidal structure on $\sf PDist$ and the monoidal unit is the distance distribution
$$  \kappa : x\mapsto \begin{cases} 0 & x=0 \\ 1 & x>0\end{cases} $$
In fact, $({\sf PDist}, *,\kappa)$ is a quantale (because the convolution product gives $[0,1]\textsf{-Cat}([0,\infty],[0,1])$ the structure of a symmetric monoidal closed, complete and cocomplete $[0,1]$-category), and thus we can form the bicategory of ${\sf PDist}$-relations and the 2-category of ${\sf PDist}$-categories, where by the latter we mean those $\sf PDist$-endorelations $a : X\times X \to {\sf PDist}$ that are reflexive and transitive: in simple terms a $\sf PDist$-category consists of a function $a : X\times X \to \sf PDist$ on a set $X$ such that
$$  \forall x\in X.a(x,x) = \kappa \qquad\forall xyz\in X. a(x,y) * a(y,z) \le a(x,z) $$
Proposition. There is a full embedding of the category of metric spaces in $\sf ProbMet$, (obtained by change of base and more precisely)
$$ B_{\pmb\delta} : {\sf Met} \to {\sf ProbMet} $$
induced by the Dirac map
$$  {\pmb\delta} : [0,\infty]^{op} \to {\sf PDist} : w\mapsto {\pmb\delta}_w $$
where ${\pmb\delta}_w$ sends $u\in [0,\infty]^{op}$ to the characteristic function of $(w,\infty]$. It is easy to see that ${\pmb\delta}$ is a quantale homomorphism, and that $B_{\pmb\delta}$ sends a metric space $(X,d)$ to the probabilistic metric space with the same underlying set, and where $\alpha : X\times X \times [0,\infty]^{op} \to [0,1]$ is the function
$$ (x,y;v)\mapsto \begin{cases} 0 & d(x,y) \ge v \\ 1 & d(x,y) < v\end{cases} $$
The Dirac map has a left adjoint ${\pmb\lambda} : {\sf PDist}\to [0,\infty]^{op}$ sending a distribution $\varphi$ to $\sup\{v\mid \varphi(v)\le 0\}$. This defines a quantale homomorphism, i.e.

*

*${\pmb\lambda}$ commutes with arbitrary suprema;

*${\pmb\lambda}$ preserves the quantale structure.

The Dirac map also has a right adjoint, ${\pmb\varrho} : {\sf PDist} \to [0,\infty]^{op}$, sending $\varphi$ to $\inf\{v\mid 1\le \varphi(v)\}$. Observe that

*

*${\pmb\varrho}(\kappa)=0$;

*${\pmb\varrho}(\varphi * \psi) = {\pmb\varrho}(\varphi) + {\pmb\varrho}(\psi)$,

but ${\pmb\varrho}$ fails to preserve arbitrary suprema, so it is only a lax quantale homomorphism. From this we obtain a triple of adjoints
$$ B_{\pmb\lambda} \dashv B_{\pmb\delta} \dashv B_{\pmb\varrho} $$
with obvious meaning of each symbol.

Now for the question: all this is a rephrasing of known facts in categorical terms. I want to go further. How to go further?

*

*What is known, in this area of research (that admittedly goes a bit over my head if it is not phrased in category-theoretic terms...) that can be categorized with profit?

*Is there some other question that the category-theoretic approach elegantly elucidates (on the lines of <<"just" take Day convolution on a presheaf category>>)?

 A: The idea of probabilistic metric spaces sounds cool but I don't believe it is the proper way to model non-deterministic problems.  You can phrase the question

What is the probability that $d(x,y)\lt t$ ?

quite satisfactorily within the framework of using the Kleisi category of the Giry monad $\mathcal{G}$. All you need is a measurement model, which amounts to a kernel $k: X \times X \rightarrow \mathcal{G}([0,\infty])$, where for a fixed pair $(x,y) \in X \times X$ the function $k_{(x,y)}$, evaluated on the measurable set $[0,t)$ yields the probability value to your question.  The kernel itself can be defined (chosen) using $d(x,y)$ as a parameter. (Of course I am implying $X\times X$ is the measurable space determined using the Borel-$\sigma$-algebra...)  Because $[0,\infty]$ is a super convex space you can even calculate the expected value which of course could be $\infty$ (depending upon your kernel).
I always presumed no one pursued the problem because they realized you don't need probabilistic metric spaces to model non-deterministic problems arising with metric spaces.
