Bounded cohomology of classifying spaces for families of subgroups

We introduce a bounded version of Bredon cohomology for groups relative to a family of subgroups. Our theory generalizes bounded cohomology and differs from Mineyev–Yaman’s relative bounded cohomology for pairs. We obtain cohomological characterizations of relative amenability and relative hyperbolicity, analogous to the results of Johnson and Mineyev for bounded cohomology.