  Hopf (bi-)modules and crossed modules over a bialgebra B in a braided
monoidal category C are considered. The (braided) monoidal equivalence of both
categories is proved provided B is a Hopf algebra (with invertible antipode).
Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra in C are
found to be isomorphic categories. As a consequence a generalization of the
Radford-Majid criterion for a braided Hopf algebra to be a cross product is
obtained. The results of this paper turn out to be fundamental for the
construction of (bicovariant) differential calculi on braided Hopf algebras.
