Functors, Applicatives, And Monads In Pictures
updated: May 20, 2013
Here's a simple value:
And we know how to apply a function to this value:
Simple enough. Lets extend this by saying that any value can be in a context. For now you can think of a context as a box that you can put a value in:
Now when you apply a function to this value, you'll get different results depending on the context. This is the idea that Functors, Applicatives, Monads, Arrows etc are all based on. The
Maybe data type defines two related contexts:
data Maybe a = Nothing | Just a
In a second we'll see how function application is different when something is a
Just a versus a
Nothing. First let's talk about Functors!
When a value is wrapped in a context, you can't apply a normal function to it:
This is where
fmap comes in.
fmap is from the street,
fmap is hip to contexts.
fmap knows how to apply functions to values that are wrapped in a context. For example, suppose you want to apply
Just 2. Use
> fmap (+3) (Just 2) Just 5
fmap shows us how it's done! But how does
fmap know how to apply the function?
Just what is a Functor, really?
Functor is a typeclass. Here's the definition:
Functor is any data type that defines how
fmap applies to it. Here's how
So we can do this:
> fmap (+3) (Just 2) Just 5
fmap magically applies this function, because
Maybe is a Functor. It specifies how
fmap applies to
instance Functor Maybe where fmap func (Just val) = Just (func val) fmap func Nothing = Nothing
Here's what is happening behind the scenes when we write
fmap (+3) (Just 2):
So then you're like, alright
fmap, please apply
(+3) to a
> fmap (+3) Nothing Nothing
Like Morpheus in the Matrix,
fmap knows just what to do; you start with
Nothing, and you end up with
fmap is zen. Now it makes sense why the
Maybe data type exists. For example, here's how you work with a database record in a language without
post = Post.find_by_id(1) if post return post.title else return nil end
But in Haskell:
fmap (getPostTitle) (findPost 1)
findPost returns a post, we will get the title with
getPostTitle. If it returns
Nothing, we will return
Nothing! Pretty neat, huh?
<$> is the infix version of
fmap, so you will often see this instead:
getPostTitle <$> (findPost 1)
Here's another example: what happens when you apply a function to a list?
Lists are functors too! Here's the definition:
instance Functor  where fmap = map
Okay, okay, one last example: what happens when you apply a function to another function?
fmap (+3) (+1)
Here's a function:
Here's a function applied to another function:
The result is just another function!
> import Control.Applicative > let foo = fmap (+3) (+2) > foo 10 15
So functions are Functors too!
instance Functor ((->) r) where fmap f g = f . g
When you use fmap on a function, you're just doing function composition!
Applicatives take it to the next level. With an applicative, our values are wrapped in a context, just like Functors:
But our functions are wrapped in a context too!
Yeah. Let that sink in. Applicatives don't kid around.
<*>, which knows how to apply a function wrapped in a context to a value wrapped in a context:
Just (+3) <*> Just 2 == Just 5
<*> can lead to some interesting situations. For example:
> [(*2), (+3)] <*> [1, 2, 3] [2, 4, 6, 4, 5, 6]
Here's something you can do with Applicatives that you can't do with Functors. How do you apply a function that takes two arguments to two wrapped values?
> (+) <$> (Just 5) Just (+5) > Just (+5) <$> (Just 4) ERROR ??? WHAT DOES THIS EVEN MEAN WHY IS THE FUNCTION WRAPPED IN A JUST
> (+) <$> (Just 5) Just (+5) > Just (+5) <*> (Just 3) Just 8
Functor aside. "Big boys can use functions with any number of arguments," it says. "Armed
<*>, I can take any function that expects any number of unwrapped values. Then I pass it all wrapped values, and I get a wrapped value out! AHAHAHAHAH!"
> (*) <$> Just 5 <*> Just 3 Just 15
And hey! There's a function called
liftA2 that does the same thing:
> liftA2 (*) (Just 5) (Just 3) Just 15
How to learn about Monads:
- Get a PhD in computer science.
- Throw it away because you don't need it for this section!
Monads add a new twist.
Functors apply a function to a wrapped value:
Applicatives apply a wrapped function to a wrapped value:
Monads apply a function that returns a wrapped value to a wrapped value. Monads have a function
>>= (pronounced "bind") to do this.
Let's see an example. Good ol'
Maybe is a monad:
half is a function that only works on even numbers:
half x = if even x then Just (x `div` 2) else Nothing
What if we feed it a wrapped value?
We need to use
>>= to shove our wrapped value into the function. Here's a photo of
Here's how it works:
> Just 3 >>= half Nothing > Just 4 >>= half Just 2 > Nothing >>= half Nothing
What's happening inside?
Monad is another typeclass. Here's a partial definition:
class Monad m where (>>=) :: m a -> (a -> m b) -> m b
Maybe is a Monad:
instance Monad Maybe where Nothing >>= func = Nothing Just val >>= func = func val
Here it is in action with a
And if you pass in a
Nothing it's even simpler:
You can also chain these calls:
> Just 20 >>= half >>= half >>= half Nothing
Cool stuff! So now we know that
Maybe is a
Applicative, and a
Now let's mosey on over to another example: the
Specifically three functions.
getLine takes no arguments and gets user input:
getLine :: IO String
readFile takes a string (a filename) and returns that file's contents:
readFile :: FilePath -> IO String
putStrLn takes a string and prints it:
putStrLn :: String -> IO ()
All three functions take a regular value (or no value) and return a wrapped value. We can chain all of these using
getLine >>= readFile >>= putStrLn
Aw yeah! Front row seats to the monad show!
Haskell also provides us with some syntactical sugar for monads, called
foo = do filename <- getLine contents <- readFile filename putStrLn contents
- A functor is a data type that implements the
- An applicative is a data type that implements the
- A monad is a data type that implements the
Maybeimplements all three, so it is a functor, an applicative, and a monad.
What is the difference between the three?
- functors: you apply a function to a wrapped value using
- applicatives: you apply a wrapped function to a wrapped value using
- monads: you apply a function that returns a wrapped value, to a wrapped value using
So, dear friend (I think we are friends by this point), I think we both agree that monads are easy and a SMART IDEA(tm). Now that you've wet your whistle on this guide, why not pull a Mel Gibson and grab the whole bottle. Check out LYAH's section on Monads. There's a lot of things I've glossed over because Miran does a great job going in-depth with this stuff.
This post has been translated into:
- Another Chinese translation
- Chinese, Kotlin
- Kotlin. This author also translated this Kotlin version into Chinese.
- Kotlin (translated from the Swift translation)
If you translate this post, send me an email and I'll add it to this list!
For more monads and pictures, check out three useful monads.