Bass-Serre theory for groupoids

In this paper, Bass-Serre theory is developed in the groupoid framework, and a structure theorem is established. We show that, when a groupoid action on a forest is without inversion of edges, it induces a graph of groupoids. Conversely, a graph of groupoids that satisfies certain conditions admits a canonical associated groupoid, which we call the fundamental groupoid, and a forest, which we call the Bass-Serre forest. The fundamental groupoid acts on the Bass-Serre forest. The structure theorem asserts that these processes are mutually inverse.