20 May 2026. Talk by Jaime Pedregal at the KU Leuven Poisson seminar.
Abstract: A connection on a vector bundle is equivalently given by its parallel transport functor from the thin fundamental groupoid of the base to the general linear groupoid of the bundle. The isotropy of the image of such functor is the holonomy group of the connection. For higher connections of a Lie algebroid on a graded vector bundle, the expectation is that “higher parallel transport” should be a functor that integrates the different parts of the higher connection along surfaces (or simplices) of the appropriate dimension. In this talk I will introduce the notion of the higher general linear groupoid of a graded vector bundle, such that parallel transport genuinely becomes a higher functor from the homotopy groupoid of the algebroid into it. The isotropy of the image of such functor will then be the higher holonomy group of the higher connection. From this perspective, the holonomy group of a classical Lie algebroid connection becomes a 2-group, a point of view that explains Fernandes’s version of the Ambrose–Singer theorem for Lie algebroid connections. Depending on time, I will also touch upon smoothness of higher holonomy groups and the extension of the framework to higher Lie algebroids.