Seminarium Gamma

Double Lie groupoids generalize Lie groupoids by the addition of two distinct manifolds of arrows and a manifold of 2-arrows that span them. The classical Ambrose-Singer theorem states that the curvature of a principal bundle connection generates the holonomy. This was later adapted to the setting of Lie groupoids by Mackenzie, who established that the Lie algebroid of a holonomy groupoid is the curvature reduction of the original Lie algebroid. In this talk, we introduce the constructions of a path connection and a path curving on a double Lie groupoid and those of a connection and a curving on the tangent LA-groupoid. We further discuss the relation between the holonomy double groupoid of a path curving and the 3-curvature of the associated LA-groupoid curving. This is joint work in progress with Derek Krepski.