Generalized metric space

Template:Distinguish 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 $ℝ$. 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.

References

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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