Bosonization for dual quasi-bialgebras and preantipode

To every dual quasi-bialgebra H and every bialgebra R in the category of Yetter-Drinfeld modules over H, one can associate a dual quasi-bialgebra, called bosonization. In this paper, using the fundamental theorem, we characterize as bosonizations the dual quasi-bialgebras with a projection onto a dual quasi-bialgebra with a preantipode. As an application we investigate the structure of the graded coalgebra gr. A associated to a dual quasi-bialgebra A with the dual Chevalley property (e.g. A is pointed).