Crossed Products in Weak Contexts

We define the general notion of crossed products in a weak context, which generalizes the ones defined by Blattner, Cohen and Montgomery, Doi and Takeuchi in the context of Hopf algebras and the one given by Brzezinski. Also, the crossed products obtained by the authors, for weak Hopf algebras living in a symmetric monoidal category and weak C-cleft extensions associated to weak entwined structures, are particular instances of this theory.