Weak Hom-Hopf algebras and their (co)representations

In this paper we will introduce the notion of a weak Hom-Hopf algebra, generalizing both weak Hopf algebras and Hom-Hopf algebras. Then we study the category Rep(H) of Hom-modules with bijective Hom-structure maps over a weak Hom-Hopf algebra H and show that the tensor functor of a (weak) Hom-bialgebra is actually a (weak) bimonad on a Hom-type category. At last, we prove that if H is a quasitriangular weak Hom-bialgebra (resp. ribbon weak Hom-Hopf algebra), then Rep(H) is a braided monoidal category (resp. ribbon category).