The category of noncrossing partitions

Abstract In [18], we introduced “picture groups” and computed the cohomology of the picture group of type An. 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 An by the general construction given in [14] and prove that the classifying space of this category is locally CAT(0) and thus a K(π,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]. 2010 Mathematics Subject Classification. 16G20; 20F55.