Equivariant principal infinity-bundles

In this book we prove the classification theorem for equivariant principal bundles in the case that the topological structure group is truncated. The result is proven in a conceptually transparent manner as a consequence of a smooth Oka principle, which becomes available after faithfully embedding traditional equivariant topology into the singular-cohesive homotopy theory of globally equivariant higher smooth stacks. This works for discrete equivariance groups acting properly on smooth manifolds with resolvable singularities, whence we are equivalently describing principal bundles on good orbifolds. In setting up this proof, we re-develop the theory of equivariant principal bundles from scratch by systematic use of Grothendieck's internalization method. In particular we prove that all the intricate equivariant local triviality conditions considered in the literature are implied by regarding G-equivariant principal bundles as principal bundles internal to the BG-slice of the ambient cohesive infinity-topos. Generally we find that the characteristic subtle phenomena of equivariant classifying theory all reflect basic"modal"properties of singular-cohesive homotopy theory. Classical literature has mostly been concerned with compact Lie structure groups. Where these are truncated, our classification recovers and generalizes results of Lashof, May, Segal and Rezk. A key non-classical example is the infinite projective unitary structure group, in which case we are classifying degree-3 twists of equivariant KU-theory, recovering a result of Lueck&Uribe. Our theorem enhances this to conjugation-equivariance, where we are classifying the geometric twists of equivariant KR-theory, restricting on"O-planes"to the geometric twists of KO-theory. This is the generality in which equivariant K-theory twists are conjectured to model the B-field in string theory on orbi-orientifolds.