Why can applicative functors have side effects, but functors can't?

Question

I\'m feeling rather silly asking this question, but it\'s been on my mind for a while and I can\'t find any answers.

So the question is: why can applicative functors

Answer 1

Let's first rename side effects to effects. All kinds of values can have effects. A functor is a type that allows you to map a function over whatever is produced by that effect.

When a functor is not applicative it doesn't allow you to use a certain composition style for the effects. Let's pick a (contrived) example:

data Contrived :: * -> * where
    AnInt :: Int -> Contrived Int
    ABool :: Bool -> Contrived Bool
    None  :: Contrived a

This is easily a functor:

instance Functor Contrived where
    fmap f (AnInt x) = AnInt (f x)
    fmap f (ABool x) = ABool (f x)
    fmap _ None      = None

However, there is no sensible implementation for pure, so this type is not an applicative functor. It is similar to Maybe in that it has the effect that there may not be a result value. But you can't compose it using applicative combinators.