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 ] : A → B is an equivalence class of rigid objects T in the cluster category of A so that B is the right hom-ext perpendicular category of the underlying object |T | ∈ A. Factorizations of a morphism [T ] are given by totally orderings of the components of T . This is equivalent to a “signed exceptional sequences.” For an algebra of finite representation type, the geometric realization of the cluster morphism category is the Eilenberg-MacLane space with fundamental group equal to the “picture group” introduced by the authors in [IOTW15b].