  We introduce and investigate the space of \emph{equidistant}
$X$-\emph{cactuses}. These are rooted, arc weighted, phylogenetic networks with
leaf set $X$, where $X$ is a finite set of species, and all leaves have the
same distance from the root. The space contains as a subset the space of
ultrametric trees on $X$ that was introduced by Gavryushkin and Drummond. We
show that equidistant-cactus space is a CAT(0)-metric space which implies, for
example, that there are unique geodesic paths between points. As a key step to
proving this, we present a combinatorial result concerning \emph{ranked} rooted
$X$-cactuses. In particular, we show that such networks can be encoded in terms
of a pairwise compatibility condition arising from a poset of collections of
pairs of subsets of $X$ that satisfy certain set-theoretic properties. As a
corollary, we also obtain an encoding of ranked, rooted $X$-trees in terms of
partitions of $X$, which provides an alternative proof that the space of
ultrametric trees on $X$ is CAT(0). As with spaces of phylogenetic trees, we
expect that our results should provide the basis for and new directions in
performing statistical analyses for collections of phylogenetic networks with
arc lengths.
