# 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!

## 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`<*>`

or`liftA`

**monads:**you apply a function that returns a wrapped value, to a wrapped value using`>>=`

or`liftM`

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.

## Translations

This post has been translated into:

Human languages:

- Chinese
- Another Chinese translation
- Chinese, Kotlin
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Programming languages:

- Javascript
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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.