Generalized metric space

In mathematics, specifically in category theory, a generalized metric space is a metric space but without the symmetry property and some other properties.^[1] Precisely, it is a category enriched over ${\displaystyle [0,\infty ]}$, the one-point compactification of ${\displaystyle \mathbb {R} }$. 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.

We can view ${\displaystyle {\overline {\mathbb {R} _{+}}}=[0,\infty ]}$ as a symmetric monoidal category as follows.^[2] An object there is a point in ${\displaystyle {\overline {\mathbb {R} _{+}}}}$, the hom set between objects ${\displaystyle a,b}$

${\displaystyle \operatorname {Hom} (a,b)={\begin{cases}\{b-a\},&{\text{if }}b\geq a\\\emptyset ,&{\text{else}}\end{cases}}.}$

and the composition given by sum

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

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

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

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

and that is well-defined is exactly the triangular inequality.

• 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" (PDF). Reprints in Theory and Applications of Categories (1): 1–37.
• Borceux, Francis; Dejean, Dominique (1986). "Cauchy completion in category theory". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 27 (2): 133–146.
• 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.

• 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