We show how the Hopf algebra known as the Drinfeld double arises in this context. Such cocycles in particular represent higher principal bundles, gerbes, - possibly equivariant, possibly with connection - as well as the corresponding associated higher vector bundles. 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. 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. 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. For instance quantum groups, that is Hopf algebras, have been familiar to theoretical physicists for a while now. Quantum field theory allows more general symmetries than groups and Lie algebras.
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