Understand Functor, Applicative and Monad

Functor

• A Functor is any data type that defines how fmap applies to it.
• Lists are functors too!
• Functions are Functors too. When you use fmap on a function, you're just doing function composition!
• Functors apply a function to a wrapped value

Applicatives

• Applicatives take it to the next level. With an applicative, our values are wrapped in a context.
• Applicative pushes Functor aside.
• Control.Applicative defines <*>
• Applicatives apply a wrapped function to a wrapped value

Monads

• Monads apply a function that returns a wrapped value to a wrapped value.
• Monads have a function >>= (pronounced "bind") to do this.
• Monad is another typeclass.
• Maybe is a Functor, an Applicative, and a Monad
• Haskell also provides us with some syntactical sugar for monads, called do notation

Conclusion

• Functor, Applicative, Monad are typeclass, the value implements them is one Functor or Applicative or Monad
• A functor is a data type that implements the Functor typeclass.
• An applicative is a data type that implements the Applicative typeclass.
• A monad is a data type that implements the Monad typeclass.
• A Maybe implements all three, so it is a functor, an applicative, and a monad.
• syntax sugar
  1. functors: you apply a function to a wrapped value using fmap or <$>
  2. applicatives: you apply a wrapped function to a wrapped value using <*> or liftA
  3. monads: you apply a function that returns a wrapped value, to a wrapped value using >>= or liftM

# normal
f a -> b
# functor: 
f <a> -> <b>
# applicative: 
<f> <a> -> <b>
# monad: f a -> 
<a> (f a -> <b>) -> <b>

所以

• functor 解包参数
• applicative 解包函数
• monad 把一个既不是normal也不是functor情形的函数和一个包裹参数颠倒顺序

另外一种视角

1. Functor: apply a function to a container.
2. Applicative: apply a multi-argument function to multiple containers.
3. Monad: like Applicative but I can decide what to do next after each step.

所以可以把这个 container 看作“更多数量的元素”，把monad看做可中断的流水线。

Links

• Functors, Applicatives, And Monads In Pictures - adit.io
• Creating monads in haskell - Stack Overflow
• All About Monads - HaskellWiki
• Briefly on the purpose of Functors, Applicatives and Monads