$\infty$-Bundles

Abstract

Higher bundles are homotopy coherent generalisations of classical fibre bundles. They appear in numerous contexts in geometry, topology and physics. In particular, higher principal bundles provide the geometric framework for higher-group gauge theories with higher-form gauge potentials and their higher-dimensional holonomies. An $\infty$-categorical formulation of higher bundles further allows one to identify these objects in contexts outside the worlds of smooth manifolds or topological spaces. This article reviews the theory of $\infty$-bundles, focussing on principal $\infty$-bundles, and surveys several of their applications. It is an invited contribution to the Topology section in the second edition of the Encyclopedia of Mathematical Physics.