Definition. Monad [dm-000U]

A monad on the category \textbf {C} consists of an endofunctor \mathcal {M} : \textbf {C} \rightarrow \textbf {C} and two operations (natural transformations) for all \mathbf {C}-objects X: \mu _X : \mathcal {M}(\mathcal {M}(X)) \rightarrow \mathcal {M}(X) and \eta _X : X \rightarrow \mathcal {M}(X), such that the following diagrams commute:

In equational form, these laws are

1. \mu _X \circ \mathcal {M}(\mu _X) = \mu _X \circ \mu _{\mathcal {M}(X)}
2. \mu _X \circ \mathcal {M}(\eta _X) = \mu _X \circ \eta _{\mathcal {M}(X)} = \mathsf {id}_{\mathcal {M}(X)}

In computer science, \eta _X is sometimes called unit, pure, or return, and \mu _X is usually called join.