  In [arXiv:1609.02636], we introduced "picture groups" and computed the
cohomology of the picture group of type $A_n$. This is the same group what was
introduced by Loday [21] where he called it the "Stasheff group". In this
paper, we give an elementary combinatorial interpretation of the category
associated to $A_n$ by the general construction given in [arXiv.1706.02041] and
prove that the classifying space of this category is locally $CAT(0)$ and thus
a $K(\pi,1)$. We prove a more general statement that classifying spaces of
certain "cubical categories" are locally $CAT(0)$. The objects of our category
are the classical noncrossing partitions introduced by Kreweras [20]. The
morphisms are binary forests. This paper is independent of [14] and [18] except
in the last section where we use [14] to compare our category with the category
with the same name given by Hubery and Krause [9].
