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

Abstract 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.

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