Monad

A monad is a triple $(T,\eta ,\mu )$ where:

• $T:\mathcal{C}\to \mathcal{C}$ is an endofunctor
• $\eta :{\text{Id}}_{\mathcal{C}}\Rightarrow T$ (unit)
• $\mu :T^2\Rightarrow T$ (multiplication)

Satisfying:

1. Associativity: $\mu \circ T\mu =\mu \circ {\mu }_T$
2. Unit: $\mu \circ T\eta =\mu \circ {\eta }_T={\text{id}}_T$

See also

• Functor
• Category
• Natural Transformation
• Adjunction