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
- ^ namely, the property that distinct elements have nonzero distance between them and the property that the distance between two elements is always finite.
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" (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.
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
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