From double Lie groupoids to local Lie 2-groupoids

TL;DR

This paper develops a method to construct local Lie 2-groupoids from double Lie groupoids using the bar construction, connecting to fundamental groupoids and symplectic structures.

Contribution

It introduces a novel application of the bar construction to double Lie groupoids, leading to the concept of symplectic 2-groupoids.

Findings

Recovered Haefliger's fundamental groupoid from double groupoids

Defined symplectic 2-groupoids via induced 2-forms

Established a new framework linking double groupoids and higher Lie groupoids

Abstract

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated local Lie 2-groupoid, which leads us to propose a definition of a symplectic 2-groupoid.