Functors, Applicatives, And Monads In Pictures
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!
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 (+3)
to Just 2
. Use fmap
:
> fmap (+3) (Just 2)
Just 5
Bam!
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:
A Functor
is any data type that defines how fmap
applies to it. Here’s how fmap
works:
So we can do this:
> fmap (+3) (Just 2)
Just 5
And fmap
magically applies this function, because Maybe
is a Functor. It specifies how fmap
applies to Just
s and Nothing
s:
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 Nothing
?
> fmap (+3) Nothing
Nothing
Like Morpheus in the Matrix, fmap
knows just what to do; you start with Nothing
, and you end up with Nothing
! 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 Maybe
:
post = Post.find_by_id(1)
if post
return post.title
else
return nil
end
But in Haskell:
fmap (getPostTitle) (findPost 1)
If 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
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. Control.Applicative
defines <*>
, which knows how to apply a function wrapped in a context to a value wrapped in a context:
i.e:
Just (+3) <*> Just 2 == Just 5
Using <*>
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
Applicatives:
> (+) <$> (Just 5)
Just (+5)
> Just (+5) <*> (Just 3)
Just 8
Applicative
pushes Functor
aside. “Big boys can use functions with any number of arguments,” it says. “Armed <$>
and <*>
, 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
Monads
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:
Suppose 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
Where >>=
is:
So Maybe
is a Monad:
instance Monad Maybe where
Nothing >>= func = Nothing
Just val >>= func = func val
Here it is in action with a Just 3
!
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 Functor
, an Applicative
, and a Monad
.
Now let’s mosey on over to another example: the IO
monad:
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 do
notation:
foo = do
filename < getLine
contents < readFile filename
putStrLn contents
Conclusion
 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.
What is the difference between the three?

functors: you apply a function to a wrapped value using
fmap
or<$>

applicatives: you apply a wrapped function to a wrapped value using
<*>
orliftA

monads: you apply a function that returns a wrapped value, to a wrapped value using
>>=
orliftM
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 indepth with this stuff.
Translations
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For more monads and pictures, check out three useful monads.
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