nLab
prometric space

Definition

A prometric on a set $X$ is a family $G$ of functions $d:X\times X\to [0,\mathrm{\infty })$ such that

1. $G$ is a $\ge$-filter, i.e.

   1. there exists an element $d\in G$,
   2. if $d_1,d_2\in G$ there is a $d_3\in G$ with $d_3\ge d_1$ and $d_3\ge d_2$ pointwise, and
   3. if $d_1\in G$ and $d_2\le d_1$ pointwise, then $d_2\in G$.

2. For every $d\in G$ and $x\in X$, we have $d(x,x)=0$.

3. For any $d\in G$ there exists an $e\in G$ such that for all $x,y,z\in X$ we have

   $$d(x,z)\le e(x,y)+e(y,z).$$d(x,z) \le e(x,y)+e(y,z).

4. For every $d\in G$, there is an $e\in G$ with $d(x,y)\le e(y,x)$ for all $x,y\in X$.

If we drop the final condition, we obtain a quasi-prometric. We can also consider extended prometrics in which the $d$ can take the value $\mathrm{\infty }$.

A base for a prometric is a collection satisfying all these axioms except $\le$-closure; the $\le$-closure of a prometric base is a prometric. Any single pseudometric, and in fact any gauge, constitutes a base for a prometric.

Of course a prometric space is a space equipped with a prometric, and likewise for a quasi-prometric space.

Morphisms

A short map between prometric spaces $X$ and $Y$ is a function $f:X\to Y$ such that for every $d\in G_Y$, we have $d\circ (f\times f)\in G_X$. We write $\mathrm{ProMet}$ for the category of prometric spaces and short maps.

If the prometrics of $X$ and $Y$ are presented by bases, then this is equivalent to saying that for any basic $d$ on $Y$, there is a basic $e$ on $X$ such that $d(f(x),f(x\prime ))\le e(x,x\prime )$ for all $x,x\prime \in X$. Thus, for metric spaces and gauge spaces considered as prometric spaces, this reduces to the usual notion of short map (i.e. distance-decreasing map). Hence the category $\mathrm{Gau}$ of gauge spaces and short maps is included as a full subcategory of $\mathrm{ProMet}$.

Subcategories

Since $\mathrm{Gau}$ includes the categories of metric spaces and uniform spaces (disjointly), so does $\mathrm{ProMet}$. Likewise, since $\mathrm{QGau}$ includes the category of topological spaces (disjointly from metric and uniform spaces), so does $\mathrm{QProMet}$.

There is also another embedding of $\mathrm{Unif}$ into $\mathrm{ProMet}$, however, which is notably simpler than its embedding into $\mathrm{Gau}$. Given a uniform space $X$, we define for each entourage $U\subseteq X\times X$ a distance function

$$d_U(x,y)=\{\begin{array}{ll}0& (x,y)\in U\\ 1& (x,y)\notin U.\end{array}$$d_U(x,y) = \begin{cases} 0 & (x,y)\in U\\ 1 & (x,y)\notin U. \end{cases}

The collection of such $d_U$ is a base for a prometric on $X$. The short maps between such prometric spaces are precisely the uniformly continuous ones, so this defines another embedding of $\mathrm{Unif}$ into $\mathrm{ProMet}$. The full image of this embedding consists precisely of those prometric spaces generated by a base of $\{0,1\}$-valued functions. Note that replacing $1$ by any other positive real number defines a different embedding of $\mathrm{Unif}$ into $\mathrm{ProMet}$.

Conversely, every prometric induces a uniformity, where the entourages are the sets

$$U_{d,ϵ}=\{(x,y)\mid d(x,y)<ϵ\}.$$U_{d,\epsilon} = \{(x,y) | d(x,y)\lt\epsilon\}.

In this way every short map induces a uniformly continuous map as well. This operation is compatible with the above inclusion of $\mathrm{Unif}$, as well as with the inclusion of $\mathrm{Gau}$.

Categorical interpretation

As observed by Lawvere, an (extended quasi pseudo) metric space is a category enriched over $([0,\mathrm{\infty }],\ge ,+,0)$. In other words, it is a monoid (or monad) in the bicategory $[0,\mathrm{\infty }]\mathrm{Mat}$ of matrices with values in this monoidal category. Analogously, an (extended quasi) prometric space is a monoid in the bicategory $\mathrm{Pro}[0,\mathrm{\infty }]\mathrm{Mat}$ whose hom-categories are the categories of pro-objects in the hom-categories of $[0,\mathrm{\infty }]\mathrm{Mat}$.

Note that if $\mathrm{Rel}=\{0,1\}\mathrm{Mat}$ denotes the bicategory of relations in $\mathrm{Set}$, then a monoid in $\mathrm{Rel}$ is a preorder, while a monoid in $\mathrm{ProSet}$ is a (quasi) uniform space.

In all these cases, in order to recover the correct notion of morphism abstractly, we must consider monoids in a double category or equipment rather than merely a bicategory.

References

• Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen. “One Setting for All: Metric, Topology, Uniformity, Approach Structure.”