A Geometric Interpretation of Categories of Type à and of Morphisms in the Infinite Radical

We give a geometric realization of the module category of a hereditary algebra of type A~A~. We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type A~A~. To get a description of the module category, we introduce long moves between arcs. These allow us to include the infinite radical in the geometric description. Finally, our results can also be used to describe the corresponding cluster categories by taking unoriented arcs instead.