nLab Hausdorff metric

Idea

The Hausdorff metric is a metric on the power set of a given metric space.

Definition

Let $A$ be a metric space, regarded as a category enriched over $V = [0,\infty]$ (a Lawvere metric space). The enriched functor category $[A,V]$ is, concretely, the set of short maps $A \to [0,\infty]$ with the supremum metric?, and the contravariant Yoneda embedding $A^{op} \to [A,V]$ sends $a$ to $d(a,-)$.

Now, for any subset $X \subseteq A$, each point $x \in X$ gives rise to the representable functor $d(x,-)$, and we can define the functor $d(X,-)\colon A \to V$ to be the coproduct of these representables over all $x \in X$. Concretely, this means

$$d(X,a)=inf_{x\in X} d(x,a) .$$

Note that if $X$ is closed, then it can be recovered from $d(X,-)$ as the set of points $x$ such that $d(X,x)=0$. If $X$ is not closed, then in this way we recover its closure.

Finally, since $[A,V]$ is also a Lawvere metric space, we obtain an induced metric on the set of subspaces of $A$:

$$d(X,Y) = d(d(X,-), d(Y,-)) .$$

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 $[A, V_{sym}]$ where $V_{sym}$ is the symmetrization of $V = [0,\infty]$.

References

• Taking categories seriously, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. (pdf)