I Product Systems Of Weak Crossover Study

The article defines the general notion of crossed coproducts in weak contexts; Also,the crossed coproducts introduced can apply to weak A-cocleft coextensions assoociated to weak entwined structures in braided monoidal category and weak Hopf algebras living in symmetric monoidal category,the dissertation consists of five sections:The first section is the introduction;The second section, for an coalgebra C and an object V, living in a braided monoidal category with egualizers and coequlizers, we introduce the notion of crossed coproduct system and prove that the coproduct induced by it is coassociativ if satisfies the twisted and the cocycle conditions;The third section,we introduce the notion of weak A-cocleft coextension, provides an example of crossed coproduct system satisfying the twisted and the cocycle conditions,and obtain that the notion of the crossed coproduct is induced by the crossed coproduct systems;The four section,we define the notion of crossed coproduct system with counity, prove the main result of this paper which shows the relation between the coproduct and the crossed coproduct systems;The five section,we apply our theory to the context of weak Hopf algebras, obtain that our construction is valid to develop a theory of crossed coproducts for weak Hopf algebras. The final example of this section, we prove that for all morphism of weak Hopf algebras with coalgebra splitting, it is possible to obtain a crossed coproduct system that is also an example of the cocleft theory developed in Section 3.