Infinitesimal higher symmetries and connections on higher bundles

Every principal bundle on a manifold has a universal symmetry group. It controls equivariant structures, and its tangent Lie algebra controls connections on the bundle.
In this talk we extend these concepts to higher, or categorified bundles. We will use a family‐version of the Lurie-Pridham Theorem from derived deformation theory to compute the associated L_∞-algebras, or rather L_∞-algebroids. That allows us to provide a unified definition of connections on higher bundles and an algebraic formulation of differential cohomology. We elaborate in particular on the case of higher U(1)‐bundles, or n‐gerbes.
This is joint work with Lukas Müller (Perimeter Institute), Joost Nuiten (Toulouse) and Richard Szabo (Heriot-Watt).