Generalized metric space

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 $\mathbb {R}$ . 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 ${\overline {\mathbb {R} }}_{+}=[0,\infty ]$ as a symmetric monoidal category as follows:^[2] objects are points in ${\overline {\mathbb {R} }}_{+}$ , the set of morphisms between objects $a,b$ is

\operatorname {Hom} (a,b)={\begin{cases}\{b-a\},&{\text{if }}b\geq a,\\\quad \;\varnothing ,&{\text{otherwise,}}\end{cases}}

and composition ${\displaystyle \circ$ 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 $({\overline {\mathbb {R} }}_{+},\geq )$ 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 ${\overline {\mathbb {R} }}_{+}$ as follows. The objects are the points of $X$ and we let $\operatorname {Hom} (x,y)=d(x,y)$ . The composition for $x,y,z$ is a morphism in ${\overline {\mathbb {R} }}_{+}$

{\displaystyle \circ

and its well-definedness is exactly the triangle inequality.

Notes

↑ Namely, the property that distinct elements have non-zero distance between them and the property that the distance between two elements is always finite.
↑ Lawvere (1973), p. 145, § 1.

References

Bonsangue, M. M.; Van Breugel, F.; Rutten, J. J. M. M. (1998). "Generalized metric spaces: Completion, topology, and powerdomains via the Yoneda embedding". Theoretical Computer Science. 193 (1–2): 1–51. doi:10.1016/S0304-3975(97)00042-X. hdl:1887/4083537.

Borceux, Francis; Dejean, Dominique (1986). "Cauchy completion in category theory". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 27 (2): 133–146. Zbl 0595.18001.

Lawvere, F. William (1973). "Metric spaces, generalized logic, and closed categories". Rendiconti del Seminario Matematico e Fisico di Milano. 43: 135–166. doi:10.1007/BF02924844.

Discussion

Notes

References

Further reading