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.[1] Precisely, it is a category enriched over , the one-point compactification of . 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.
Notes
References
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
- ↑ namely, the property that distinct elements have nonzero distance between them and the property that the distance between two elements is always finite.