  $\tau$-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.
