Monad

category-theory

Definition

Monad

A monad on a category $\mathcal{C}$ is an endofunctor $T:\mathcal{C}\to \mathcal{C}$ equipped with two natural transformations:

• Unit (return): $\eta :{\mathrm{Id}}_{\mathcal{C}}\Rightarrow T$
• Multiplication (join): $\mu :T^2\Rightarrow T$

satisfying associativity and unitality:

$$\mu \circ T\mu =\mu \circ \mu T\text{(associativity)}$$ $$\mu \circ T\eta =\mu \circ \eta T={\mathrm{id}}_T\text{(unitality)}$$

Programming

In functional programming, a monad structures computations with effects. The operations are usually written:

Category Theory	Programming
$T$	type constructor M a
$\eta$ (unit)	return :: a -> M a
$\mu$ (join)	join :: M (M a) -> M a
$Tf\circ \eta$	fmap then return
Kleisli composition	>=> or bind (>>=)

Monads as programmable semicolons

A monad captures a pattern of sequencing. The bind operation (>>=) threads a value through a chain of effectful computations, where the exact meaning of “effect” (state, I/O, exceptions, non-determinism) is determined by the particular monad instance.