The category of noncrossing partitions

Kiyoshi Igusa

Published 2014-11-02Version 1

In [13], picture groups are introduced and the cohomology of the picture group of type $A_n$ with straight orientation is computed. In this paper, we give an elementary combinatorial interpretation of the category associated to $A_n$ and prove that the classifying space of this category is a $K(\pi,1)$. The objects of the category are the classical noncrossing partitions introduced in [19]. The morphisms are binary forests. This paper is independent of the later papers in this series except for the last section in which we compare our category with the one in [arXiv:1310.1907].