A category of noncrossing partitions

Abstract: In [17], 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 [20] where he called it theStasheff group''. In this paper, we give an elementary combinatorial interpretation of the {\color{blue}cluster morphism category'' constructed in [13] in the special case of the linearly oriented quiver of type $A_n$.} We 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 certaincubical categories'' are locally $CAT(0)$. The objects of our category are the classical noncrossing partitions introduced by Kreweras [19]. The morphisms are binary forests. This paper is independent of [13] and [17] except in the last section where we use [13] to compare our category with the category with the same name given by Hubery and Krause [9].