From double Lie groupoids to local Lie 2-groupoids (Q420562)

This paper is a study of the behavior of the bar construction with respect to simplicial manifolds arising from double Lie groupoids. Let us recall that a double Lie groupoid is essentially a groupoid object in the category of Lie groupoids. In this work, the authors apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. The authors start with some review of the definition of simplicial and bisimplicial manifolds and the bar construction. They apply the bar construction to the nerve of a double Lie groupoid and study the Kan properties of the resulting simplicial manifold. The case of a symplectic double groupoid is also studied. This case enables the authors to propose a definition of a symplectic groupoid.