In writing software, we often encounter scenarios where a value resides within a context, a container of sorts. Standard function application, so straightforward with simple values, presents a challenge in these situations. Consider the Maybe data type in Haskell, which encapsulates the possibility of a value’s absence. Applying functions to values wrapped in Maybe requires a different approach, as direct function application results in a type error.

For example, applying the function (+4) to the integer 2 is trivial:

However, attempting to apply the same function directly to a Maybe wrapped value results in a type error:

ghci> x = Just 2
ghci> (+4) x

<interactive>:25:1: error: [GHC-39999]
    • No instance for ‘Num (Maybe Integer)’ arising from a use of ‘it’
    • In the first argument of ‘print’, namely ‘it’
      In a stmt of an interactive GHCi command: print it

This is where fmap comes in. fmap provides a way to apply a function to a value within a wrapped context. In the case of Maybe, fmap allows us to apply (+4) to the integer inside a Just context:

ghci> fmap (+4) (Just 2)
Just 6

This leads us to the concept of Functors. The Functor typeclass ^{in Haskell represents any type that can be mapped over. Lists, for example, are instances of the Functor typeclass. The definition of the Functor typeclass is as follows:}

class Functor f where
    fmap :: (a -> b) -> f a -> f b

This typeclass defines only one function, fmap. This polymorphic function, works for any type constructor f that is an instance of the Functor typeclass. It takes :

1. A function of type a -> b, which transforms a value of type a into a value of type b.
2. A value of type f a, which represents a value of type a within a functorial context f. It then returns a value of type f b, which is a value of type b within the same functorial context f. Essentially, fmap applies the function to the value(s) inside the functorial context.

As demonstrated earlier, we can use fmap (+4) (Just 2) because Maybe is a functor. fmap applies the function to the values inside the Maybe context. The Maybe type itself is defined as:

data Maybe a = Nothing | Just a

instance Functor Maybe where
    fmap f (Just x) = Just (f x)
    fmap f Nothing = Nothing

This implementation reveals that if we have a value of Nothing, fmap will return Nothing. If we have a value wrapped in Just, fmap applies the function to the contents of the Just context.

In languages like Go, which lacks an explicit Maybe type, we often use error handling or other mechanisms to achieve similar results.

For instance, consider this Go code:

metadata, err := user.GetMetadata()
if err != nil {
    return err
}

if metadata == nil {
    return nil
}
return metadata.ProjectedValue

The Haskell equivalent:

getProjectedValue <$> (User.getMetadata "uuid")

Here, <$> is the infix version of fmap. If User.getMetadata returns Just Metadata, we can extract the projected value from it. If it returns Nothing, we’ll return Nothing.

Functor Laws

For a type to be considered a proper Functor, it must adhere to certain laws:

Identity Law

Applying fmap id should not change the functor’s value.

This law ensures that Functors don’t introduce any unexpected changes or side effects when traversing their structure with a function that shouldn’t modify anything.

As we can see from the Maybe implementation of fmap, this law holds.

instance Functor Maybe where
    fmap f (Just x) = Just (f x)
    fmap f Nothing = Nothing

Application of the identity law:

Maybe:

ghci> fmap id (Just "change")
Just "change"

ghci> id (Just "change")
Just "change"

ghci> fmap id Nothing
Nothing

ghci> id Nothing
Nothing

List:

ghci> fmap id [1, 2, 3, 4]
[1, 2, 3, 4]

ghci> id [1, 2, 3, 4]
[1, 2, 3, 4]

IO:

ghci> fmap id (return "test")
"test"
ghci> id (return "test")
"test"

Even in the IO Functor, fmap id leaves the action unchanged.

Composition Law

Applying fmap to a composed function should be the same as applying fmap separately in sequence.

This ensures that Functors behave consistently when dealing with function composition. This is crucial for maintaining predictability and allowing for easier reasoning about code that uses Functors.

fmap (f . g) == fmap f . fmap g

Application of the composition law:

Maybe:

ghci> let f = (*3)
ghci> let g = (+5)
ghci> let c = f . g

ghci> fmap c (Just 10)
Just 45

ghci> fmap f (fmap g (Just 10))
Just 45

List:

ghci> let f = (*3)
ghci> let g = (+5)
ghci> let c = f . g

ghci> fmap c [1, 2, 3]
[18,21,24]

ghci> fmap f (fmap g [1, 2, 3])
[18,21,24]

Conclusion

Functors in Haskell provide a powerful abstraction for applying functions to values wrapped in a context, such as Maybe, lists, or IO. By following the identity and composition laws, they ensure predictable behavior and maintain structure while enabling clean, concise, and expressive code. By leveraging fmap, we can seamlessly work with values inside different contexts, making functional programming more flexible and easier to reason about.