Connections, Local Subgroupoids, and a Holonomy Lie Groupoid of a Line Bundle Gerbe

Our main aim is to associate a holonomy Lie groupoid to the connective structure of an abelian gerbe. The construction has analogies with a procedure for the holonomy Lie groupoid of a foliation, in using a locally Lie groupoid and a globalisation procedure. We show that path connections and 2–holonomy on line bundles may be formulated using the notion of a connection pair on a double category, due to Brown–Spencer, but now formulated in terms of double groupoids using the thin fundamental groupoids introduced by Caetano–Mackaay–Picken. To obtain a locally Lie groupoid to which globalisation applies, we use methods of local subgroupoids as developed by Brown–İçen–Mucuk.