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Fix a monoidal category C. The 2-category of monads in the 2-category ofC-actegories, colax C-equivarant functors, and C-equivariant naturaltransformations of colax functors, may be recast in terms of pairs consistingof a usual monad and a distributive law between the monad and the action of C,morphisms of monads respecting the distributive law, and transformations ofmonads satisfying some compatibility with the actions and distributive lawsinvolved. The monads in this picture may be generalized to actions of monoidalcategories, and actions of PRO-s in particular. If C is a PRO as well, then inspecial cases one gets various distributive laws of a given classical type, forexample between a comonad and an endofunctor or between a monad and a comonad.The usual pentagons are in general replaced by multigons, and there are also``mixed'' multigons involving two distinct distributive laws. Beck's bijectionbetween the distributive laws and lifts of one monad to the Eilenberg-Moorecategory of another monad is here extended to an isomorphism of 2-categories.The lifts of maps of above mentioned pairs are colax C-equivariant. We finishwith a short treatment of relative distributive laws between two pseudoalgebrastructures which are relative with respect to the distributivity of twopseudomonads involved, what gives a hint toward the generalizations.

Equivariant monads and equivariant lifts versus a 2-category of distributive laws