The natural promotion of the classical concept of (principal) fiber bundles to "higher structures", namely to equivariant principal infinity-bundles internal to a singular-cohesive infinity-topos, turns out to be a natural foundation for generalized cohomology theory in the full beauty of "twisted equivariant differential non abelian cohomology of orbifolds", and as such for much of the higher homotopical mathematics needed at the interface of algebraic topology, geometry and mathematical quantum physics.

This talk gives some introduction and overview, based on joint work with H. Sati
(arXiv:2008.01101, arXiv:2112.13654).