$\infty$-Bundles

TL;DR

This paper reviews the theory of $ abla$-bundles, emphasizing principal $ abla$-bundles, and explores their applications in geometry, topology, physics, and higher gauge theories through an $ abla$-categorical perspective.

Contribution

It provides a comprehensive review of $ abla$-bundles, especially principal $ abla$-bundles, and discusses their applications using an $ abla$-categorical framework.

Findings

Higher bundles generalize classical fibre bundles with homotopy coherence.

$ abla$-categorical formulation enables applications beyond smooth manifolds.

Survey of applications in geometry, topology, and physics.

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.