  We determine regularity results for energy minimizing maps from an
$n$-dimensional Riemannian polyhedral complex $X$ into a CAT(1) space. Provided
that the metric on $X$ is Lipschitz regular, we prove H\"older regularity with
H\"older constant and exponent dependent on the total energy of the map and the
metric on the domain. Moreover, at points away from the $(n-2)$-skeleton, we
improve the regularity to locally Lipschitz. Finally, for points $x \in
X^{(k)}$ with $k \leq n-2$, we demonstrate that the H\"older exponent depends
on geometric and combinatorial data of the link of $x \in X$.
