Dense Morphisms of Monads

Given an arbitrary locally finitely presentable category K and finitary monads T and S on K , we characterize monad morphisms α : S −→ T with the property that the induced functor α∗ : K T −→ K S between the categories of Eilenberg-Moore algebras is fully faithful. We call such monad morphisms dense and give a characterization of them in the spirit of Beth’s definability theorem: α is a dense monad morphism if and only if every T-operation is explicitly defined using S-operations. We also give a characterization in terms of epimorphic property of α and clarify the connection between various notions of epimorphisms between monads.