nLab
monads in Haskell

Definition

A monad in Haskell is defined to be a type class with two methods:

$\mathrm{class}\mathrm{Monad}m\mathrm{where}$

$>>=::\mathrm{ma}\to (a\to \mathrm{mb})\to \mathrm{mb}$

$\mathrm{return}::a\to \mathrm{ma}$

such that

$x>>=f\equiv \mathrm{fx}$

$a>>=\mathrm{return}\equiv a$

$(a>>=f)>>=g\equiv a>>=(\lambda x\to \mathrm{fx}>>=g)$

Remark

$m$ being a monad $(m:a\to a,\mu :m^2\to m,ϵ:{\mathrm{id}}_a\to m)$ on some object $a$ in a 2-category can be expressed in Haskell by

$\mathrm{class}\mathrm{Monad}m\mathrm{where}$

$\mathrm{map}::(a\to b)\to \mathrm{ma}\to \mathrm{mb}$

$\mathrm{return}::a\to \mathrm{ma}$

$\mathrm{join}::m(\mathrm{ma})\to \mathrm{ma}$

We have the following translation of Haskell and category theoretical language?:

$\mathrm{join}:=\mu$

$\mathrm{return}:=ϵ$

Then from the category theoretical properties of $(m,\mu ,ϵ)$ we obtain (mixing the two languages)

$\mathrm{map}g\circ ϵ\equiv ϵ\circ g$

$\mathrm{map}\circ \mu \equiv \mu \circ \mathrm{map}(\mathrm{map}g)$

$\mu \circ \mathrm{map}\mu \equiv \mu \circ \mu$

$\mu \circ ϵ\equiv \mu \circ \mathrm{map}ϵ=\mathrm{id}$

And with the following definitions

$\mathrm{map}f:=(\lambda \to a>>=(\mathrm{return}\circ f))$

$\mathrm{join}a=a>>\mathrm{id}$

or alternatively

$a>>=f:=\mathrm{join}(\mathrm{map}\mathrm{fa})$

one can verify that the two given definitions of a monad are equivalent.

References

category theory/monads, haskellwiki, wiki