Canonical Gradings of Monads

We define a notion of grading monoid T in monoidal category C, relative to class morphisms M (which provide M-subobject). show that, under reasonable conditions (including that forms factorization system), there is canonical T. Our application graded monads and models computational effects. demonstrate our results by characterizing the gradings number monads, for which C endofunctors with composition. also we can obtain grades algebraic operations.