  By analogy with methods of Spivak, there is a realization functor which takes
a persistence diagram $Y$ in simplicial sets to an extended pseudo-metric space
(or ep-metric space) $Re(Y)$. The functor $Re$ has a right adjoint, called the
singular functor, which takes an ep-metric space $Z$ to a persistence diagram
$S(Z)$. We give an explicit description of $Re(Y)$, and show that it depends
only on the $1$-skeleton $sk_{1}Y$ of $Y$. If $X$ is a totally ordered
ep-metric space, then there is an isomorphism $Re(V_{\ast}(X)) \cong X$,
between the realization of the Vietoris-Rips diagram $V_{\ast}(X)$ and the
ep-metric space $X$. The persistence diagrams $V_{\ast}(X)$ and $S(X)$ are
sectionwise equivalent for all such $X$.
