Pedagogical Introduction to Higher Principal Bundles

From the Standard Model of particle physics to condensed matter systems, gauge theories form a powerful framework to understand Nature. Mathematically, gauge fields correspond to connections on principal bundles, which are described by Lie algebra valued 1-forms. String theory considerations motivate the lift of this picture to a categorified setting. Principal bundles are lifted to higher principal bundles (or gerbes) and higher connections are then described by higher degree forms valued in some L-infinity algebra. In this lecture, I will introduce these notions underpinning the geometric framework of higher gauge theory, reviewing elements of higher category theory along the way. In particular, the emphasis in this lecture will be on the cocycle description of gerbes with connection.