  Quantum field theory allows more general symmetries than groups and Lie
algebras. For instance quantum groups, that is Hopf algebras, have been
familiar to theoretical physicists for a while now. Nowdays many examples of
symmetries of categorical flavor -- categorical groups, groupoids, Lie
algebroids and their higher analogues -- appear in physically motivated
constructions and faciliate constructions of geometrically sound models and
quantization of field theories. Here we consider two flavours of categorified
symmetries: one coming from noncommutative algebraic geometry where varieties
themselves are replaced by suitable categories of sheaves; another in which the
gauge groups are categorified to higher groupoids. Together with their gauge
groups, also the fiber bundles themselves become categorified, and their gluing
(or descent data) is given by nonabelian cocycles, generalizing group
cohomology, where infinity-groupoids appear in the role both of the domain and
the coefficient object. Such cocycles in particular represent higher principal
bundles, gerbes, -- possibly equivariant, possibly with connection -- as well
as the corresponding associated higher vector bundles. We show how the Hopf
algebra known as the Drinfeld double arises in this context. This article is an
expansion of a talk that the second author gave at the 5th Summer School of
Modern Mathematical Physics in 2008.
