Why Functor class has no return function?

问题:

From categorical point of view, functor is pair of two maps (one between objects and another between arrows of categories), following some axioms.

I have assumed, what every Functor instance is similar to mathematical definition i.e. can map both objects and functions, but Haskell's Functor class has only function fmap which maps functions.

Why so?

UPD In other words:

Every Monad type M has an function return :: a -> M a.

And Functor type F has no function return :: a -> F a, but only F x constructor.

回答1:

First of all, there are two levels: types and values. As objects of Hask are types, you can map them only with type constructors, which have the * -> * kind:

• α -> F α (for Functor F),
• β -> M β (for Monad M).

Then for a functor you need a map on morphisms (i.e. functions, which are values): it's just fmap :: (α -> β) -> (F α -> F β).

So far, I guess, I'm not saying anything new. But important thing is that return :: α -> M α of Monad is not a mapper of a type α to the M α as you may think. Regarding to the math definition of a monad, return corresponds to a natural transformation from Id functor to the M functor. Just that this Id functor is kind of implicit. The standard definition of monad requires also another natural transformation M ◦ M -> M. So translating it to Haskell would be like

class Functor m => Monad m where
    return :: Id α -> m α
    join :: m (m α) -> m α

(As a side-note: these two natural transformations are actually the unit and multiplication, which make monad a monoid in the category of endofunctors)

The actual definition differs but is equivalent. See Haskell/wiki on that.

If you take the composition-like operator derived form the standard bind >>= :: m α -> (α -> m β) -> m β:

(>=>) :: Monad m => (α -> m β) -> (β -> m γ) -> (α -> m γ)
f >=> g = \a => f a >>= g

you can see, that it's all actually about the Kleisli category. See also the article on nLab about monads in computer science.

回答2:

Objects of a category are not the same as objects in a OO programming language (we prefer to call those values in Haskell; what they mean in category theory was discussed here). Rather, the objects of Hask are types. Haskell Functors are endofunctors in Hask, i.e. associate types to types, by the following means:

Prelude> :k Maybe
Maybe :: * -> *
Prelude> :k Int
Int :: *
Prelude> :k Maybe Int
Maybe Int :: *

OTOH, the arrows of Hask are in fact values, of some function type a -> b. These are associated in the following way:

fmap :: ( Functor (f ::   t     ->     f t       {- type-level  -} ) )
             =>         (a->b)  ->  fmap(a->b)   {- value-level -}
                     ≡  (a->b)  ->  (f a->f b)

回答3:

If you have

instance Functor F where
    fmap = ...

Then the type constructor F is the action on objects (which are types) taking a type T to the type F T, and fmap is the action on morphisms (which are functions) taking a function f :: T -> U to fmap f :: F T -> F U.

回答4:

Though you were using those fancy categorical terms in your question and should be completely satisfied with the existing answers, here is an attempt for a rather trivial explanation:

Suppose there would be a function return (or pure or unit or ...) in the Functor type class.

Now try to define some common instances of Functor: [] (Lists), Maybe, ((,) a) (Tuples with a left component)

Easy enough, eh?

Here are the ordinary Functor instances:

instance Functor [] where
   fmap f (x : xs) = f x : fmap xs
   fmap _ []       = []

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

instance Functor ((,) a) where
   fmap f (x, y) = (x, f y)

What about return for Functor now?

Lists:

instance Functor [] where
   return x = [x]

Alright. What about Maybe?

instance Functor Maybe where
   return x = Just x

Okay. Now Tuples:

instance Functor ((,) a) where
   return x = (??? , x)

You see, it is unknown which value should be filled into the left component of that tuple. The instance declaration says it has a type a but we do not know a value from that type. Maybe the type a is the Unit type with only one value. But if its Bool, should we take True or False? If it is Either Int Bool should we take Left 0 or Right False or Left 1?

So you see, if you had a return on Functors, you could not define a lot of valid functor instances in general (You would need to impose a constraint of something like a FunctorEmpty type class).

If you look at the documentation for Functor and Monad you will see that there are indeed instances for Functor ((,) a) but not for Monad ((,) a). This is because you just can't define return for that thing.