  The cluster morphism category of an hereditary algebra was introduced in [5]
to show that the picture space of an hereditary algebra of finite
representation type is a $K(\pi,1)$ for the associated picture group, thereby
allowing for the computation of the homology of picture groups of finite type
as carried out in [7] for the case of $A_n$.
  In this paper we show that the cluster morphism category is a
$CAT(0)$-category for hereditary algebras of finite or tame type with only
small tubes. As a consequence, we get that the classifying space of the cluster
morphism category is a locally $CAT(0)$ space and, as a consequence of that, we
get that this classifying space is a $K(\pi,1)$.
