{\tau}-Cluster Morphism Categories and Picture Groups

Abstract

$\tau$-cluster morphism categories were introduced by Buan and Marsh as a generalization of cluster morphism categories (as defined by Igusa and Todorov) to $\tau$-tilting finite algebras. In this paper, we show that the classifying space of such a category is a cube complex, generalizing results of Igusa and Todorov and Igusa. We further show that the fundamental group of this space is isomorphic to a generalized version of the picture group of the algebra, as defined by Igusa, Todorov, and Weyman. We end this paper by showing that if the algebra is Nakayama, then this space is locally $\mathsf{CAT}(0)$, and hence a $K(\pi,1)$. We do this by constructing a combinatorial interpretation of the 2-simple minded collections for Nakayama algebras. A key step in the proof is to show that, for Nakayama algebras, 2-simple minded collections are characterized by pairwise compatibility conditions, a fact not true in general.Comment: 29 pages, 14 figures. v3: Numerous improvements have been made following suggestions of an anonymous referee. v2: Lemma 4.13 has been combined with Lemma 4.12 (now Lemma 4.11) and its proof has been changed. The introduction has been rewritten and minor typos have been fixe