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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.

Bicovariant differential calculi and cross products on braided Hopf algebras