Functors for Newbies

This is a tutorial on functors. It is inspired by the already-excellently written article here.

What is a Functor?

In Haskell, there is a Functor type class. If a type is an instance of the Functor class, it is essentially a container type that adds context to any value.

The Functor class is defined as follows:

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

. That looks a bit weird, so I will rewrite it:

class Functor box where
    fmap :: (apples -> oranges) -> box apples -> box oranges

. So, for a container type to be a real Functor, it has to be able to take in a function, and be able to modify the value contained inside itself without changing the context. This is extremely useful. Let’s look at some functors.

The List Functor

Imagine you have a list with some numbers in it: xs = [1, 2, 3]. Also imagine you already wrote this function below:

makeStatement :: Int -> String
makeStatement a
    | odd a = "Yuck."
    | otherwise = "Yay!"

. Now, you’d like to use makeStatement on the list xs, which has a type [Int], so that every Int inside the list gets transformed according to makeStatement. Thankfully, List already has an instance of the Functor class, so we can do it like this:

myFunc :: [Int] -> [String]
myFunc xs = fmap makeStatement xs

. Though, since List defines fmap with a shorter-named map, we can just use map instead of fmap to save ourselves a keystroke.¹ While we’re at it, we might as well drop the xs because it’s redundant.

myFunc :: [Int] -> [String]
myFunc = map makeStatement

This is immensely useful. Without the Functor class, you’d have to use pattern matching to manually extract the values contained in the list first before applying functions on them. Imagine a sophisticated, recursive binary tree with millions of elements — as long as the container type has an instance for Functor, you can just use a one-liner fmap to universally apply the function at hand to every single element inside the tree (and rest assured that the context for all the individual values and their interrelationships remain untouched — a win win!).

For empty lists, You can still pass along a function with fmap into them, but nothing will happen because they are empty (that is, there are no values inside to modify with the function).

map makeStatement [] -- same as []

The Maybe Functor

Let’s look at the Maybe type just to prove that they are also functors.

Maybe is a container, except that it can only contain just 1 element (unlike List which can hold multiple elements). Maybe has two constructors: Just and Nothing; you can use Just to put a value into the Maybe type, like how you can use the (:[]) function to put a single item into a list. Alternatively, you can use Nothing to denote the empty Maybe container (like [] for lists).

x :: Maybe Int
x = Just 10

bigX :: Maybe Int
bigX = fmap (*2) x -- returns Just 20

Again, using fmap on an empty container will return the empty container as-is:

fmap (*2) Nothing -- same as Nothing

The Tuple Functor?

You might be wondering — what about tuples? Well, because tuples hold different types in a single container, it is impossible to implement fmap for it. Recall the type signature required for fmap:

class Functor box where
    fmap :: (apples -> oranges) -> box apples -> box oranges

. Notice that the container must contain a single type apples — tuples by their nature have arbitrary numbers of different types (they have apples and oranges in them already). This is the reason why we don’t (and can’t) have an fmap function for tuples.

Conclusion

Functors are awesome. If you ever define your own custom container type, be sure to make an instance for Functor to make life easy for everyone.

Copyright (C) 2013-2023 Linus Arver. All rights reserved.