Monadic functor

Summary

A functor $U \colon \mathcal{D} \to \mathcal{C}$ is monadic when it has a left adjoint and $\mathcal{D}$ is, up to equivalence, the category of algebras of the resulting monad. So the objects of $\mathcal{D}$ are exactly objects of $\mathcal{C}$ with algebraic structure described by the monad. The forgetful functors from groups, rings, modules and compact Hausdorff spaces to sets are monadic. Beck's monadicity theorem characterizes monadic functors.