Generalized metric space explained

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

[ 0 , inf ty] , 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

• 10.1007/BF02924844 . Metric spaces, generalized logic, and closed categories . 1973 . Lawvere . F. William . Rendiconti del Seminario Matematico e Fisico di Milano . 43 . 135–166 .
  • Lawvere . F. W. . Metric spaces, generalized logic and closed categories . Reprints in Theory and Applications of Categories . 2002 . 1 . 1–37 .
• Cauchy completion in category theory . Cahiers de Topologie et Géométrie Différentielle Catégoriques . 1986 . 27 . 2 . 133–146 . Borceux . Francis . Dejean . Dominique .
• 10.1016/S0304-3975(97)00042-X . Generalized metric spaces: Completion, topology, and powerdomains via the Yoneda embedding . 1998 . Bonsangue . M.M. . Van Breugel . F. . Rutten . J.J.M.M. . Theoretical Computer Science . 193 . 1–2 . 1–51 .

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

Notes and References

1. namely, the property that distinct elements have nonzero distance between them and the property that the distance between two elements is always finite.