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>>)?

withoutcategory theory. I'd like the metric geometry community to tell me what is known about PM spaces, and let's see if there's some more category theory to throw at the target. $\endgroup$