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 the ``Stasheff 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 certain ``cubical 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].

37 pages, 15 figures, presented at Workshop on "Hall and cluster algebras" May 8-12, 2014, CRM, Universite de Montreal, and at Conference on "Geometric Methods in Representation Theory" Nov 22-24, 2014, University of Iowa. v2: minor corrections, figures redrawn, v3: cubical category more clearly defined. v4: minor corrections, figures added. v5: lots of changes (in blue)

References in corpus (1)

• Periodic trees and semi-invariants

Cited by in corpus (6)

• Signed exceptional sequences and the cluster morphism category
• Pairwise Compatibility for 2-Simple Minded Collections
• Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank
• τ-Cluster Morphism Categories and Picture Groups
• Exceptional sequences and rooted labeled forests
• Horizontal and vertical mutation fans