Algebra over a monad

Summary

An algebra over a monad $T$ on a category $\mathcal{C}$ is an object $A$ with a structure map $a \colon T(A) \to A$. The structure map must be compatible with the unit and the multiplication of $T$. These algebras, with the maps that respect the structure maps, form the Eilenberg–Moore category of $T$. Many algebraic structures are algebras over a monad on sets. Examples are monoids for the list monad, complete join-semilattices for the power set monad, and modules for the free module monad.