{\tau}-Cluster Morphism Categories and Picture Groups

TL;DR

This paper explores the structure of $ au$-cluster morphism categories, showing their classifying spaces are cube complexes with fundamental groups called picture groups, and proves for Nakayama algebras these spaces are $K( ext{pi},1)$ spaces.

Contribution

It generalizes previous results by showing classifying spaces are cube complexes and identifies their fundamental groups as picture groups, especially for Nakayama algebras.

Findings

Classifying space of $ au$-cluster morphism category is a cube complex.

Fundamental group of this space is the picture group of the algebra.

For Nakayama algebras, the space is a $K( ext{pi},1)$ space.

Abstract

$\tau$-cluster morphism categories, introduced by Buan and Marsh, are a generalization of cluster morphism categories (defined by Igusa and Todorov). We show the classifying space of such a category is a cube complex, generalizing results of Igusa and Todorov and Igusa. Furthermore, the fundamental group of this space is the picture group of the algebra, first defined by Igusa, Todorov, and Weyman. Finally, we show that for Nakayama algebras, this space is a $K(\pi,1)$. The key step is a combinatorial proof that, for Nakayama algebras, 2-simple minded collections are characterized by pairwise compatibility conditions, a fact not true in general.