nLab
prometric space

Definition

A prometric on a set XX is a family GG of functions d:X×X[0,)d:X\times X\to [0,\infty) such that

  1. GG is a \ge-filter, i.e.

    1. there exists an element dGd\in G,
    2. if d 1,d 2Gd_1,d_2\in G there is a d 3Gd_3\in G with d 3d 1d_3\ge d_1 and d 3d 2d_3\ge d_2 pointwise, and
    3. if d 1Gd_1\in G and d 2d 1d_2\le d_1 pointwise, then d 2Gd_2\in G.
  2. For every dGd\in G and xXx\in X, we have d(x,x)=0d(x,x)=0.

  3. For any dGd\in G there exists an eGe\in G such that for all x,y,zXx,y,z\in X we have

    d(x,z)e(x,y)+e(y,z).d(x,z) \le e(x,y)+e(y,z).
  4. For every dGd\in G, there is an eGe\in G with d(x,y)e(y,x)d(x,y)\le e(y,x) for all x,yXx,y\in X.

If we drop the final condition, we obtain a quasi-prometric. We can also consider extended prometrics in which the dd can take the value \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 XX and YY is a function f:XYf:X\to Y such that for every dG Yd\in G_Y, we have d(f×f)G Xd\circ (f\times f) \in G_X. We write ProMetProMet for the category of prometric spaces and short maps.

If the prometrics of XX and YY are presented by bases, then this is equivalent to saying that for any basic dd on YY, there is a basic ee on XX such that d(f(x),f(x))e(x,x)d(f(x),f(x'))\le e(x,x') for all x,xXx,x'\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 GauGau of gauge spaces and short maps is included as a full subcategory of ProMetProMet.

Subcategories

Since GauGau includes the categories of metric spaces and uniform spaces (disjointly), so does ProMetProMet. Likewise, since QGauQGau includes the category of topological spaces (disjointly from metric and uniform spaces), so does QProMetQProMet.

There is also another embedding of UnifUnif into ProMetProMet, however, which is notably simpler than its embedding into GauGau. Given a uniform space XX, we define for each entourage UX×XU\subseteq X\times X a distance function

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

The collection of such d Ud_U is a base for a prometric on XX. The short maps between such prometric spaces are precisely the uniformly continuous ones, so this defines another embedding of UnifUnif into ProMetProMet. The full image of this embedding consists precisely of those prometric spaces generated by a base of {0,1}\{0,1\}-valued functions. Note that replacing 11 by any other positive real number defines a different embedding of UnifUnif into ProMetProMet.

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

U d,ϵ={(x,y)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 UnifUnif, as well as with the inclusion of GauGau.

Categorical interpretation

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

Note that if Rel={0,1}MatRel = \{0,1\} Mat denotes the bicategory of relations in SetSet, then a monoid in RelRel is a preorder, while a monoid in ProSetProSet 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.” (pdf)

Revised on July 21, 2013 05:57:18 by Todd Trimble (67.80.8.75)