Generalized metric space

Not to be confused with Generalised metric.

In mathematics, specifically in category theory, a generalized metric space is a metric space but without some properties, such as symmetry. Precisely, it is a category enriched over $[0,\mathrm{\infty }]$, 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 ${\overline{\mathbb{ℝ}}}_{+}=[0,\mathrm{\infty }]$ as a symmetric monoidal category as follows: objects are points in ${\overline{\mathbb{ℝ}}}_{+}$, the set of morphisms between objects $a,b$ is

$\mathrm{Hom}(a,b)=\{\begin{array}{ll}\{b-a\},& \text{if }b\ge a,\\ \varnothing ,& \text{otherwise,}\end{array}$

and composition $\circ :\mathrm{Hom}(b,c)\times \mathrm{Hom}(a,b)\to \mathrm{Hom}(c,a)$ 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 $({\overline{\mathbb{ℝ}}}_{+},\ge )$ 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 $\mathrm{Hom}(x,y)=d(x,y)$. The composition for $x,y,z$ is a morphism in ${\overline{\mathbb{ℝ}}}_{+}$

$\circ :\mathrm{Hom}(y,z)\otimes \mathrm{Hom}(x,y)\to \mathrm{Hom}(x,z)$

and its well-definedness is exactly the triangle inequality.

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. 

• Borceux, Francis; Dejean, Dominique (1986). "Cauchy completion in category theory". Cahiers de Topologie et Géométrie Différentielle Catégoriques 27 (2): 133–146. http://eudml.org/doc/91378. 

• 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. 

Further reading

• Avery, Tom (2014). "Metric Spaces, Generalized Logic, and Closed Categories". https://golem.ph.utexas.edu/category/2014/02/metric_spaces_generalized_logi.html#more. 

• "Metric spaces". https://ncatlab.org/nlab/show/metric+space#LawvereMetricSpace. 

• Willerton, Simon (2022). "Optimal Transport and Enriched Categories IV: Examples of Kan-type Centres". https://golem.ph.utexas.edu/category/2022/01/optimal_transport_and_enriched_2.html#more. 

• Willerton, Simon (2023). "Metric Spaces as Enriched Categories II". https://golem.ph.utexas.edu/category/2023/05/metric_spaces_as_enriched_categories_ii.html#more.