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 the symmetry property and some other properties. Precisely, it is a category enriched over $[0,\mathrm{\infty }]$, the one-point compactification of $\mathbb{ℝ}$. The notion was introduced in 1973 by Lawvere who noticed that a metric space can be viewed as a particular kind of a 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. An object there is a point in $\overline{{\mathbb{ℝ}}_{+}}$, the hom set between objects $a,b$

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

and the composition given by sum

$\circ :\mathrm{Hom}(b,c)\times \mathrm{Hom}(a,b)\to \mathrm{Hom}(c,a),(\{c-b\},\{b-a\})\mapsto \{c-b+b-a\}$

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 $a\to b$ to mean $b\supset a$ and with $a\otimes b=a\cup b$, $M$ 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 that that is well-defined is exactly the triangular inequality.

References

• 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. 
  • Lawvere, F. W. (2002). "Metric spaces, generalized logic and closed categories". Reprints in Theory and Applications of Categories (1): 1–37. http://www.tac.mta.ca/tac/reprints/articles/1/tr1.pdf. 
• 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. 
• 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. 

Further reading

• https://golem.ph.utexas.edu/category/2023/05/metric_spaces_as_enriched_categories_ii.html#more
• https://golem.ph.utexas.edu/category/2022/01/optimal_transport_and_enriched_2.html#more
• https://ncatlab.org/nlab/show/metric+space#LawvereMetricSpace
• https://golem.ph.utexas.edu/category/2014/02/metric_spaces_generalized_logi.html#more