In Igusa, Todorov and Weyman (Picture groups of finite type and cohomology in type An 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 (Contemp Math 265: 99–127, 2000) where he called it the “Stasheff group”. In this paper, we give an elementary combinatorial interpretation of the “cluster morphism category” constructed in as reported by Igusa and Todorov, (in: Signed exceptional sequences and the cluster morphism category, arXiv:1706.02041) 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 certain “cubical categories” are locally CAT(0). The objects of our category are the classical noncrossing partitions introduced by Kreweras (Discrete Math 1: 333–350, 1972) . The morphisms are binary forests. This paper is independent of as reported by Igusa and Todorov (in: Signed exceptional sequences and the cluster morphism category, arXiv:1706.02041)and as reported by Igusa, Todorov and Weyman (in: Picture groups of finite type and cohomology in type An arXiv:1609.02636)except in the last section where we use as reported by Igusa and Todorov (in: Signed exceptional sequences and the cluster morphism category, arXiv:1706.02041) to compare our category with the category with the same name given by Hubery and Krause (J Eur Math Soc 18: 2273–2313, 2016).
