Three explicit examples of Lie 2-groupoids

Abstract

This note constructs three explicit classes of Lie $2$-groupoids arising from double Lie groupoids via the bar construction of Artin and Mazur, following a result of Mehta and Tang: double Lie groupoids give rise to bisimplicial manifolds whose truncations define Lie $2$-groupoids. This construction is carried out for comma double Lie groupoids (Brown and Mackenzie), transitive double Lie groupoids (Brown and Mackenzie), and split VB-groupoids described by $2$-term representations up to homotopy (Gracia-Saz and Mehta).