Bass-Serre theory for groupoids

TL;DR

This paper extends Bass-Serre theory to the setting of groupoids, establishing a correspondence between groupoid actions on forests and graphs of groupoids, with a structure theorem confirming their inverse relationship.

Contribution

It develops a Bass-Serre theory for groupoids, introducing fundamental groupoids and forests, and proving a structure theorem for their mutual correspondence.

Findings

Established a structure theorem for groupoid actions on forests.

Defined the fundamental groupoid associated with a graph of groupoids.

Proved the mutual inverse relationship between groupoid actions and graphs of groupoids.

Abstract

In this paper a Bass-Serre theory in the groupoid setting is developed and a structure theorem is established. Any groupoid action without inversion of edges on a forest induces a graph of groupoids, while any graph of groupoids satisfying certain hypothesis admits a canonical associated groupoid, called the fundamental groupoid, and a forest, called the Bass-Serre forest, such that the fundamental groupoid acts on the Bass-Serre forest. The structure theorem states that these processes are mutually inverse.