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 $V=[0,\mathrm{\infty }]$ (a Lawvere metric space). The enriched functor category $[A,V]$ is, concretely, the set of short maps $A\to [0,\mathrm{\infty }]$ with the supremum metric?, and the contravariant Yoneda embedding $A^{\mathrm{op}}\to [A,V]$ 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 $d(X,-):A\to V$ to be the coproduct of these representables over all $x\in X$. Concretely, this means

$$d(X,a)={\inf }_{x\in X}d(x,a).$$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 $A$:

$$d(X,Y)=d(d(X,-),d(Y,-)).$$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 $[A,V_{\mathrm{sym}}]$ where $V_{\mathrm{sym}}$ is the symmetrization of $V=[0,\mathrm{\infty }]$.

References

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