Weak bialgebra is like bialgebra, except that the requirement that the counit map is an algebra map and the unit is a coalgebra map are appropriately relaxed. Like bialgebras, this notion can also be properly generalized to a monoidal/bicategorical context leading to weak bimonads.

Distributive laws among monads are monads in appropriate bicategory/2-category of monads. Similarly, one can understand weak distributive laws.