Bicovariant differential calculi and cross products on braided Hopf algebras

In a braided monoidal category C we consider Hopf bimodules and crossed modules over a braided Hopf algebra H. We show that both categories are equivalent. It is discussed that the category of Hopf bimodule bialgebras coincides up to isomorphism with the category of bialgebra projections over H. Using these results we generalize the Radford-Majid criterion and show that bialgebra cross products over the Hopf algebra H are precisely described by H-crossed module bialgebras. In specific braided monoidal abelian categories we define (bicovariant) braided differential calculi over H and apply the results on Hopf bimodules to construct a higher order bicovariant differential calculus over H out of any first order bicovariant differential calculus over H. This object is shown to be a bialgebra with universal properties