monad

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

• $\mu \circ (\mu \circ T)\equiv \mu \circ (T\circ \mu )$

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

These laws are illustrated in the following diagrams.