## Definition

A monad in a category ${\displaystyle {\mathcal {C}}}$ is a triple ${\displaystyle (F:{\mathcal {C}}\to {\mathcal {C}},\eta :Id\to F,\mu :F\circ F\to F)}$.

### Axioms

1. ${\displaystyle \mu \circ F(\mu )=\mu \circ \mu _{F}}$
2. ${\displaystyle \mu \circ \eta _{F}=id=\mu \circ F(\eta )}$

### Examples

• In any category ${\displaystyle {\mathcal {C}}}$ with arbitrary products, for any object ${\displaystyle R}$ of ${\displaystyle {\mathcal {C}}}$ there is a monad with the object mapping taking the object ${\displaystyle A}$ of ${\displaystyle {\mathcal {C}}}$ to ${\displaystyle R^{{\mathcal {C}}(A,R)}}$ corresponding to the CPS monad in Haskell.

Translating the definition of a monad into Haskell using this terminology would give us

class Functor m => Monad m where
return :: alpha -> m alpha
join :: m (m alpha) -> m alpha

(join, by the way, is one of the most under-appreciated of Haskell library functions; learning it is necessary both for true mastery of Haskell monads. See Monad/join for further explication). The complete collection of class laws (including the natural transformation laws) in Haskell would be

fmap g . return = return . g
fmap g . join = join . fmap (fmap g)
join . fmap join = join . join
join . return = id = join . fmap return

Haskell, of course, actually gives us

return :: alpha -> m alpha
(>>=) :: m alpha -> (alpha -> m beta) -> m beta

The relationship between these two signatures is given by the set of equations

fmap f a = a >>= return . f
join a = a >>= id
a >>= f = join (fmap f a)

return x >>= f = f x
a >>= return = a
(a >>= f) >>= g = a >>= \ x -> f x >>= g

We can take the relationship given above as definitional, in either direction, and derive the appropriate set of laws. Taking fmap and join as primitive, we get

return x >>= f
= join (fmap f (return x))
= join (return (f x))
= f x
a >>= return
= join (fmap return a)
= a
(a >>= f) >>= g
= join (fmap g (join (fmap f a)))
= join (join (fmap (fmap g) (fmap f a)))
= join (fmap join (fmap (fmap g) (fmap f a)))
= join (fmap (join . fmap g . f) a)
= a >>= join . fmap g . f
= a >>= \ x -> join (fmap g (f x))
= a >>= \ x -> f x >>= g

Taking (>>=) as primitive, we get

fmap f (return x)
= return x >>= return . f
= return (f x)
fmap f (join a)
= (a >>= id) >>= return . f
= a >>= \ x -> id x >>= return . f
= a >>= \ x -> x >>= return . f
= a >>= fmap f
= a >>= \ x -> id (fmap f x)
= a >>= \ x -> return (fmap f x) >>= id
= (a >>= return . fmap f) >>= id
= join (fmap (fmap f) a)
join (join a)
= (a >>= id) >>= id
= a >>= \ x -> x >>= id
= a >>= \ x -> join x
= a >>= \ x -> return (join x) >>= id
= (a >>= return . join) >>= id
= join (fmap join a)
join (return a)
= return a >>= id
= id a
= a
join (fmap return a)
= (a >>= return . return) >>= id
= a >>= \ x -> return (return x) >>= id
= a >>= \ x -> return x
= a >>= return
= a