Principal ∞-bundles: general theory

The theory of principal bundles makes sense in any -topos, such as the -topos of topological, of smooth, or of otherwise geometric -groupoids/-stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure -group these -principal -bundles reproduce various higher structures that have been considered in the literature and further generalize these to a full geometric model for twisted higher nonabelian sheaf cohomology. We discuss here this general abstract theory of principal -bundles, observing that it is intimately related to the axioms that characterize -toposes. A central result is a natural equivalence between principal -bundles and intrinsic nonabelian cocycles, implying the classification of principal -bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber -bundles associated to principal -bundles subsumes a theory of -gerbes and of twisted -bundles, with twists deriving from local coefficient -bundles, which we define, relate to extensions of principal -bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice infinity-topos.