In mathematics, specifically in category theory, a generalized metric space is a metric space but without some properties, such as symmetry.Namely, the property that distinct elements have non-zero distance between them and the property that the distance between two elements is always finite. Precisely, it is a category enriched over , the one-point compactification of . The notion was introduced in 1973 by William Lawvere who noticed that a metric space can be viewed as a particular kind of 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.
Discussion We can view as a symmetric monoidal category as follows: objects are points in , the set of morphisms between objects is
and composition is given by the 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 and with . Then 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 its well-definedness is exactly the triangle inequality.
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