Generalized metric space

Template:Distinguish In mathematics, specifically in category theory, a generalized metric space is a metric space but without some properties, such as symmetry.^[1] Precisely, it is a category enriched over $[0, \infty]$ , the one-point compactification of $ℝ$ . The notion was introduced in 1973 by William Lawvere who noticed that a metric space can be viewed as a particular kind of category.

The categorical point of view is useful since by Yoneda's lemma, a generalized metric space can be embedded into a much larger category in which, for instance, one can construct the Cauchy completion of the space.

Discussion

We can view ${\overline{ℝ}}_{+} = [0, \infty]$ as a symmetric monoidal category as follows:Template:Sfnp objects are points in ${\overline{ℝ}}_{+}$ , the set of morphisms between objects $a, b$ is

Hom (a, b) = {\begin{matrix} {b - a}, & if b \geq a, \\ \emptyset, & otherwise, \end{matrix}

and composition $\circ : Hom (b, c) \times Hom (a, b) \to Hom (c, a)$ is given by the sum

({c - b}, {b - a}) \mapsto {c - b + b - a} .

The tensor operation is $a \otimes b = a + b$ .

This category structure is equivalent to one obtained by viewing the poset $({\overline{ℝ}}_{+}, \geq)$ as a category in the usual way. The above definition is analogous to the following example: let $M$ be the Boolean algebra generated by some subsets of a finite set and with $a \to b := b \supset a$ and with $a \otimes b := a \cup b$ . Then $M$ is a symmetric monoidal category.

Now, let $(X, d)$ be a metric space. Then it can be viewed as a category enriched over ${\overline{ℝ}}_{+}$ as follows. The objects are the points of $X$ and we let $Hom (x, y) = d (x, y)$ . The composition for $x, y, z$ is a morphism in ${\overline{ℝ}}_{+}$