Preunits and weak crossed products

Using the notion of a preunit and the properties of idempotent morphisms, we give a general notion of a crossed product of an algebra AA and an object VV both living in a monoidal category CC. We endow A⊗VA⊗V with a multiplication and an idempotent morphism, whose image inherits the multiplication. Sufficient conditions for these multiplications to be associative are given. If the product on A⊗VA⊗V has a preunit, the related idempotent is given in terms of the preunit, and its image has an algebra structure. A characterization of crossed products with preunit is given, and it is used to recover classical examples of crossed products and to study crossed products in weak contexts. Finally crossed products of an algebra by a weak bialgebra are recovered using this theory.