A little Haskell cheat sheet - Functor vs Applicative vs Monad

When learning about Haskell (from the really good book “Learn You a Haskell for Great Good” by Miran Lipovača) I came across the four typeclasses from the title - Functor, Applicative and Monad.
They all seem to be very cool and powerful, but after reading through all of them it was a bit hard to wrap my mind around it - what is exactly the difference between them, when should I prefer one over another?

This is why I wrote this small cheat sheet - mostly for myself to better remember and understand it, but if it helps somebody else as well even better!

So let’s dig in:

Functor

Motivation

If we have map for lists and its obvious how useful it is, why wouldn’t we have the same thing for any other type of data structure, like e.g. tree? So, what map is for lists, fmap is for any data structure.

More formally, from Data.Functor Hackage doc page:

Functors: uniform action over a parametrized type, generalizing the map function on lists.

One-line explanation

Functor is for mapping over parametrized types, such as [], Maybe, trees, …

Typeclass definition

class Functor f where
    fmap :: (a -> b) -> f a -> f b
    (<$>) :: (a -> b) -> f a -> f b -- inline version of fmap

Functor typeclass consists of one function only, fmap. fmap works like this:

• input 1: (a -> b) - function that does the mapping (transformation)
• input 2: f a - value with a context (“wrapped” value)
• output: f b - transformed value with the same type of context

<$> is just the infix version of fmap.

When to use it

When we have a value (e.g. Just 10) or data structure (e.g. tree) that provides context along with the data, and we want to transform that data while preserving the type of context.

When not to use it

With functor, we can not change the type of the given context - e.g. after applying a transformation the list will still be a list, Maybe, will still be a Maybe etc.

Examples

Mapping over a function

This is an interesting example since mapping over a function is a bit less intuitive than mapping over “container” types such as Maybe or Either. But, function is nothing else than a (parametrized) type with two parameters, (->) a b, just like Either a b. So when we map over a function, the result will again be a function. And it works in the exactly the same way, with b being affected. In the context of a function it can mean only one thing: the mapping function is applied to the result of the function that is being mapped over, which is a function composition. We can also see in the Prelude source code that is exactly how the function type implements Functor typeclass:

instance Functor ((->) r) where
    fmap = (.)

Let’s take a look at a few specific examples:

• (+1) <$> (*3) - just as we mentioned, this will simply compose these two functions. So for arg given, it will return (arg * 3) + 1.
• (+) <$> (*3) - very similar to the previous example, but with a twist that the mapping function takes two arguments, which means it returns a new function when given a single argument. For arg1 and arg2 given, it will return (arg1 * 3) + arg2.

More example ideas

• Show mapping over a function (e.g. parametrized newtype, like Parser in Real World Haskell).

Sources and additional reading

• LYAH Functor explanation
• Functor design pattern - seems really cool and in-depth but have not read it yet

Applicative

• Map function in a context to the value in a context
• Can be chained together
• We can apply “normal” function to the values with context - “lifting” a function

Motivation

What if we wanted to apply “normal” function to the values with the context? E.g. what if we had two Maybe values and wanted to use (+) on them? Given Just 3 and Just 5 we’d expect to receive Just 8 as a result, while given Just 3 and Nothing we’d expect Nothing.

This is exactly what Applicative type class helps us achieve.

There is probably some more motivation but this is what I have for now.

One-line explanation

All Applicative instances must also be Functor instances. Given that, with Applicative we can apply function in a context to the value in a context.

Typeclass definition

class (Functor f) => Applicative f where
    pure :: a -> f a
    (<*>) :: f (a -> b) -> f a -> f b

We can see that <*> is very similar to <$>, only that given function also has a context. pure is a method that enables us to bring some value in a default context (e.g. a “normal” function we want to use on values with the context).

Examples

    (+) <$> Just 3 <*> Just 5 = Just 8

Here we can see how we “lifted” a function (+) to work with Maybe values. We have to start with <$> first since we are starting with a “normal” function without context, so (+) <$> Just 3 produces Just (3+) (or we could have started with pure (+) <*> Just 3). After that we can just add all the other arguments by chaining them with <*> function.

Monad

Motivation

I am not sure how well did I get the motivation, but here is what I have for now: when dealing with values with a context, monad is the most general tool for it. When we have values with context and they are flowing from one method to another, somebody always has to “handle” the context.

In order to avoid doing it manually (which would increase the complexity a lot, and its a repetitive process), the monads step in - with them, we can “pretend” we are dealing with “normal” functions while they appropriately handle the context for us.

One-line explanation

When dealing with values with context, Monad typeclass helps us by automatically handling the context for us.

Typeclass definition

class Monad m where
    return :: a -> m a
    (>>=) :: m a -> (a -> m b) -> m b
    (>>) :: m a -> m b -> m b
    fail :: String -> m a

• return is just like pure in Applicative, putting value in a default, minimal context.
• (>>=), pronounced “bind”, is the main Monad function. It allows us to “feed” a value with a context into a function that takes a pure value but returns also a value with a context.
• (>>) is used when we want to “execute” one monad before another, without actually caring for its result (because it probably changes some state, has some side-effect).
• fail - apparently never used explicitly in code, but by Haskell for some special cases. Did not see it in action so far.

Examples

• Maybe example -> Show how would handling context look like without monad
• Parsec example
• IO example

WIP, ideas

I could write about the situations that drove the inception of the above concepts. E.g.:

1. First we just had functions that returned a single, simple value.
2. But soon, we also needed functions that returned multiple different values (e.g. because they can fail)
3. The values with context were created to address that need.
4. But then we needed to handle such types of values throughout the code -> we needed new tools to help us with that.