Given an arbitrary locally finitely presentable category $K$ and finitary monads $T$ and $S$ on $K$, we characterize monad morphisms $\alpha: S\to T$ with the property that the induced functor $\alpha_*: K^T \to K^ S$ 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: $\alpha$ 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 $\alpha$ and clarify the connection between various notions of epimorphisms between monads.
Keywords: Definable operation, monad morphism, locally finitely presentable category
2000 MSC: 18C20, 18C35
Theory and Applications of Categories,
Vol. 18, 2007,
No. 14, pp 372-399.