  The three main objects in noncrossing Catalan combinatorics associated to a
finite Coxeter system are noncrossing partitions, clusters, and sortable
elements. The first two of these have known Fuss-Catalan generalizations. We
provide new viewpoints for both and introduce the missing generalization of
sortable elements by lifting the theory from the Coxeter system to the
associated positive Artin monoid. We show how this new perspective ties
together all three generalizations, providing a uniform framework for
noncrossing Fuss-Catalan combinatorics. Having developed the combinatorial
theory, we provide an interpretation of our generalizations in the language of
the representation theory of hereditary Artin algebras.
