Hopf (bi-)modules and crossed modules in braided monoidal categories

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. A generalization of the Majid-Radford criterion for a braided Hopf algebra to be a cross product is obtained as an application of these results. (C) 1998 Elsevier Science B.V.