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 , 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.
We can view as a symmetric monoidal category as follows. An object there is a point in , the hom set between objects
and the composition given by sum
The tensor operation is . This category structure is equivalent to one obtained by viewing the poset as a category in the usual way. The above definition is analogous to the following example: let be the Boolean algebra generated by some subsets of a finite set and with to mean and with , is a symmetric monoidal category.
Now, let be a metric space. Then it can be viewed as a category enriched over as follows. The objects are the points of and we let . The composition for is a morphism in
and that that is well-defined is exactly the triangular inequality.