The Hausdorff metric
The Hausdorff metric is a metric on the power set of a given metric space.
Let be a metric space, regarded as a category enriched over (a Lawvere metric space). The enriched functor category is, concretely, the set of short maps with the supremum metric?, and the contravariant Yoneda embedding sends to .
Now, for any subset , each point gives rise to the representable functor , and we can define the functor to be the coproduct of these representables over all . Concretely, this means
Note that if is closed, then it can be recovered from as the set of points such that . If is not closed, then in this way we recover its closure.
Finally, since is also a Lawvere metric space, we obtain an induced metric on the set of subspaces of :
This metric is not symmetric (so a quasimetric); its symmetrization? is the Hausdorff metric. Equivalently, we could start out by considering instead the functor category where is the symmetrization of .
- Taking categories seriously, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. (pdf)