τ-Cluster Morphism Categories and Picture Groups

Eric J Hanson, Kiyoshi Igusa

Published 2018-09-24Version 1

{\tau}-cluster morphism categories were introduced in [BM18a] as a generalization of cluster morphism categories to {\tau}-tilting finite algebras. In this paper, we show that the classifying space of such a category is a cube complex, generalizing a result of [Igu14] and [IT17]. 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 in [IOTW16]. We end this paper by showing that if our algebra is Nakayama, then this space is locally CAT(0), and hence a K({\pi},1). We do this by constructing a combinatorial interpretation of the 2-simple minded collections of Nakayama algebras.