Signed exceptional sequences and the cluster morphism category

We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism category. The objects of this category are wide subcategories of the module category of a hereditary algebra. A morphism $[T]:\mathcal A\to \mathcal B$ is the equivalence class of a rigid object $T$ in the cluster category of $\mathcal A$ so that $\mathcal B$ is the right hom-ext perpendicular category of the underlying object $|T|\in \mathcal A$. Factorizations of a morphism $[T]$ are given by total orderings of the components of $T$. This is equivalent to a "signed exceptional sequence." For an algebra of finite representation type, the geometric realization of the cluster morphism category is an Eilenberg-MacLane space with fundamental group equal to the "picture group" introduced by the authors in [IOTW4].