A monad over a category $\mathcal{C}$ is a triple $(T,\eta,\mu)$ , where $T$ is an endofunctor of $\mathcal{C}$ , $\eta$ is a natural transformation from the identity functor on $\mathcal{C}$ , and $\mu$ is a natural transformations from $T\circ T$ to $T$ , such that the following two properties hold:

• $\mu\circ(\mu\circ T)\equiv\mu\circ(T\circ\mu)$
• $\mu\circ(T\circ\eta)\equiv\mathrm{id}_\mathcal{C}\equiv\mu\circ(\eta\circ T)$

These laws are illustrated in the following diagrams.

$\mu\circ(\mu\circ T)\equiv\mu\circ(T\circ\mu)$	$\mu\circ(T\circ\eta)\equiv\mathrm{id}_\mathcal{C}\equiv\mu\circ(\eta\circ T)$

As an application, monads have been successfully applied in the field of functional programming. A pure functional program can have no side effects, but some computations are frequently much simpler with such behavior. Thus a mathematical model of computation such as a monad is needed. In this case, monads serve to represent state transformations, mutable variables, and interactions between a program and its environment. For further information in this regard, see http://www.nomaware.com/monads/html/.