During development it is common occurrence to modify deeply nested structures. One of the best known libraries for this purpose is lens, but it's quite overkill for some purposes.

This library describes simple and composable combinators that are built on top of very basic concept:

f . h . g

Where f and g are fixed. It is possible to reduce it to just:

(f .) . (. g)

Which is the core pattern used by all functions defined in this module.

Trying to generalize this pattern further ends as (f <$>) . (<$> g), where <$> = fmap. Other combinations of substituting . for fmap will end up less or equally generic. Type of such expression is:

\f g -> (f <$>) . (<$> g)
    :: Functor f => (b -> c) -> f a -> (a -> b) -> f c

Which doesn't give us much more power. Instead of going for such generalization we kept the original ((f .) . (. g)) which we named between or ~@~ in its infix form.

This module reexports Data.Function.Between.Lazy that uses standard definition of (.) function as a basis of all combinators. There is also module Data.Function.Between.Strict, that uses strict definition of function composition.

(f . h) ~@~ (i . g) === (f ~@~ g) . (h ~@~ i)

This shows us that it is possible to define (f ~@~ g) and (h ~@~ i) separately, for reusability, and then compose them.

The fun doesn't end on functions that take just one parameter, because ~@~ lets you build up things like:

(f ~@~ funOnY) ~@~ funOnX
    === g x y -> f (g (funOnX x) (funOnY y))

As you can se above g is a function that takes two parameters. Now we can define (f ~@~ funOnY) separately, then when ever we need we can extend it to higher arity function by appending (~@~ funOnX). Special case when funOnY = funOnX is very interesting, in example function on can be defined using between as:

on :: (b -> b -> c) -> (a -> b) -> a -> a -> c
on f g = (id ~@~ g ~@~ g) f
    -- or: ((. g) ~@~ g) f

We can also define function on3 that takes function with arity three as its first argument:

on3 :: (b -> b -> b -> d) -> (a -> b) -> a -> a -> a -> d
on3 f g = (id ~@~ g ~@~ g ~@~ g) f
    -- or: ((. g) ~@~ g ~@~ g) f

If we once again consider generalizing above examples by using three different functions g1 =/= g2 =/= g3 instead of just one g then we get:

on' :: (b -> b1 -> c)
    -> (a2 -> b2)
    -> (a1 -> b1)
    -> a1 -> a2 -> c
on' f g1 g2 = (id ~@~ g2 ~@~ g1) f

on3'
    :: (b1 -> b2 -> b3 -> c)
    -> (a3 -> b3)
    -> (a2 -> b2)
    -> (a1 -> b1)
    -> a1 -> a2 -> a3 -> c
on3' f g1 g2 g3 = (id ~@~ g3 ~@~ g2 ~@~ g1) f

Which allows us to interpret ~@~ in terms like "apply this function to the n-th argument before passing it to the function f". We just have to count the arguments backwards. In example if want to apply function g to third argument, but no other, then we can use:

\g f -> (id ~@~ g ~@~ id ~@~ id) f
    --   ^      ^     ^      ^- Applied to the first argument.
    --   |      |     '- Applied to the second argument.
    --   |      '- Applied to the third argument.
    --   '- Applied to the result.
    :: (a3 -> b3) -> (a1 -> a2 -> b3 -> c) -> a1 -> a2 -> a3 -> c

Or we can use ~@@~, which is just flipped version of ~@~ and then it would be:

\g f -> (id ~@@~ id ~@@~ g ~@@~ id) f
    --   ^       ^       ^      ^- Applied to the result.
    --   |       |       '- Applied to the third argument.
    --   |       '- Applied to the second argument.
    --   '- Applied to the first argument.
    :: (a3 -> b3) -> (a1 -> a2 -> b3 -> c) -> a1 -> a2 -> a3 -> c

Another interesting situation is when f and g in (f ~@~ g) form an isomorphism. Then we can construct a mapping function that takes function operating on one type and transform it in to a function that operates on a different type. As we shown before it is also possible to map functions with higher arity then one.

Simplicity of how between combinator can be used to define set of functions by reusing previous definitions makes it also very suitable for usage in TemplateHaskell and generic programming.

When we use (f ~@~ g) where f and g form an isomorphism of two types, and if f is a constructor and g a selector of newtype, then (f ~@~ g) is a mapping function that allows us to manipulate value wrapped inside a newtype.

newtype T t a = T {fromT :: a}

mapT
    :: (a -> b)
    -> T t a -> T t' b
mapT = T ~@~ fromT

Note that mapT above is generalized version of fmap of obvious Functor instance for newtype T.

Interestingly, we can use between to define higher order mapping functions by simple chaining:

mapT2
    :: (a -> b -> c)
    -> T t1 a -> T t2 b -> T t3 c
mapT2 = mapT ~@~ fromT
    -- or: T ~@~ fromT ~@~ fromT
    -- or: mapT between2l fromT

mapT3
    :: (a -> b -> c -> d)
    -> T t1 a -> T t2 b -> T t3 c -> T t4 d
mapT3 = mapT2 ~@~ fromT
    -- or: T ~@~ fromT ~@~ fromT ~@~ fromT
    -- or: mapT between3l fromT

Dually to definition of mapT and mapT2 we can also define:

comapT :: (T a -> T b) -> a -> b
comapT = fromT ~@~ T
    -- or: T ~@@~ fromT

comapT2 :: (T a -> T b -> T c) -> a -> b -> c
comapT2 = fromT ~@~ T ~@~ T
    -- or: comapT ~@~ T
    -- or: T ~@@~ T ~@@~ fromT
    -- or: T ~@@~ comapT
    -- or: fromT between2l T

In code above we can read code like:

fromT ~@~ T ~@~ T

or

T ~@@~ T ~@@~ fromT

as "Apply T to first and second argument before passing it to a function and apply fromT to its result."

Here is another example with a little more complex type wrapped inside a newtype:

newtype T e a = T {fromT :: Either e a}

mapT
    :: (Either e a -> Either e' b)
    -> T e a -> T e' b
mapT = T ~@~ fromT

mapT2
    :: (Either e1 a -> Either e2 b -> Either e3 c)
    -> T e1 a -> T e2 b -> T e3 c
mapT2 = mapT ~@~ fromT

This last example is typical for monad transformers:

newtype ErrorT e m a = ErrorT {runErrorT :: m (Either e a)}

mapErrorT
    :: (m (Either e a) -> m' (Either e' b))
    -> ErrorT e m a -> ErrorT e' m' b
mapErrorT = ErrorT ~@~ runErrorT

mapErrorT2
    :: (m1 (Either e1 a) -> m2 (Either e2 b) -> m3 (Either e3 c))
    -> ErrorT e1 m1 a -> ErrorT e2 m2 b -> ErrorT e3 m3 c
mapErrorT2 = mapErrorT ~@~ runErrorT

Library lens is notorious for its huge list of (mostly transitive) dependencies. However it is easy to define a lot of things without the need to depend on lens directly. This module defines few functions that will make it even easier.

Lens for a simple newtype:

newtype T a = T {fromT :: a}

t :: Functor f => (a -> f b) -> T a -> f (T b)
t = fmap T ~@~ fromT

To simplify things we can use function <~@~:

t :: Functor f => (a -> f b) -> T a -> f (T b)
t = T <~@~ fromT

Now, lets define lenses for generic data type, e.g. something like:

data D a b = D {_x :: a, _y :: b}

Their types in lens terms would be:

x :: Lens (D a c) (D b c) a b
y :: Lens (D c a) (D c b) a b

Here is how implementation can look like:

x :: Functor f => (a -> f b) -> D a c -> f (D b c)
x = _x ~@@^> s b -> s{_x = b}

Alternative definitions:

x = (\s b -> s{_x = b}) <^@~ _x
x f s = (_x ~@@~> b -> s{_x = b}) f s
x f s = ((\b -> s{_x = b}) <~@~ _x) f s
x f s = (const _x ^@@^> \s' b -> s'{_x = b}) f s s
x f s = ((\s' b -> s'{_x = b}) <^@^ const _x) f s s

And now for y we do mostly the same:

y :: Functor f => (a -> f b) -> D c a -> f (D c b)
y = _y ~@@^> s b -> s{_y = b}

Above example shows us that we are able to define function equivalent to lens from lens package as follows:

lens
    :: (s -> a)
    -- ^ Selector function.
    -> (s -> b -> t)
    -- ^ Setter function.
    -> (forall f. Functor f => (a -> f b) -> s -> f t)
    -- ^ In /lens/ terms this is Lens s t a b
lens = (~@@^>)

Alternative definitions:

lens get set f s = (const get ^@@^> set) f s s
lens get set f s = (set <^@^ const get) f s s
lens get set f s = (get ~@~> set s) f s
lens get set f s = (set s <~@~ get) f s

Some other functions from lens package can be defined using ~@~:

set :: ((a -> Identity b) -> s -> Identity t) -> b -> s -> t
set = (runIdentity .) ~@~ (const . Identity)

over :: ((a -> Identity b) -> s -> Identity t) -> (a -> b) -> s -> t
over = (runIdentity .) ~@~ (Identity .)

Data type Identity is defined in transformers package or in base >= 4.8.

Leses are basically just functions with a nice trick to them. If you look at the core pattern used in lens library is:

type Optical p q f s t a b = p a (f b) -> q s (f t)

Which is just a function c -> d where c = p a (f b) and d = q s (f t). In most common situations p and q are instantiated to be -> making the Optical type colapse in to something more specific:

type LensLike f s t a b = (a -> f b) -> s -> f t

Where f is some instance of Functor and that is how we get Lens, which is just:

type Lens s t a b = forall f. Functor f => (a -> f b) -> s -> f t

These lenses are called Laarhoven Lenses, after Twan van Laarhoven who introduced them in CPS based functional references article.

We can choose even stronger constraints then Functor, in example Applicative, then we get a Traversal, and, of course, it doesn't end with it, there is a lot more to choose from.

What is important, in the above lens pattern, is that it's a function that can be composed using function composition (.) operator (remember that it's just a function c -> d). As a consequence between can be used as well. Small example:

>>> (1, ((2, 3), (4, 5))) ^. (_2 ~@~ _2) _1
3
>>> (1, ((2, 3), (4, 5))) ^. (_2 ~@~ _2) _2
5

This shows us that ~@~ can be used to compose two lenses, or other abstractions from that library, but with a hole in between, where another one can be injected.

Lets imagine following example:

data MyData f a b = MyData
    { _foo :: f a
    , _bar :: f b
    }

Lets have lenses for MyData:

foo :: Lens (MyData h a b) (MyData h a' b) (h a) (h a')
bar :: Lens (MyData h a b) (MyData h a b') (h b) (h b')

Following instance of data type MyData is what our example will be based upon:

-- We use type proxy to instantiate 'h' in to concrete functor.
myData
    :: Applicative h
    => proxy h
    -> MyData h (Int, Int) (String, String)
myData _ = MyData
    { _foo = pure (1, 2)
    , _bar = pure ("hello", "world")
    }

We don't know exactly what h will be instantiated to, but we can already provide following lenses:

foo1in
    :: (Field1 s t a1 b1, Functor f)
    => LensLike f (h a) (h a') s t
    -> LensLike f (MyData h a b) (MyData h a' b) a1 b1
foo1in = foo ~@~ _1

foo2in
    :: (Field2 s t a1 b1, Functor f)
    => LensLike f (h a) (h a') s t
    -> LensLike f (MyData h a b) (MyData h a' b) a1 b1
foo2in = foo ~@~ _2

bar1in
    :: (Field1 s t a1 b1, Functor f)
    => LensLike f (h b) (h b') s t
    -> LensLike f (MyData h a b) (MyData h a b') a1 b1
bar1in = bar ~@~ _1

bar2in
    :: (Field2 s t a1 b1, Functor f)
    => LensLike f (h b) (h b') s t
    -> LensLike f (MyData h a b) (MyData h a b') a1 b1
bar2in = bar ~@~ _2

Don't get scared by the type signatures, just focus on the pattern here.

>>> myData (Proxy :: Proxy ((,) ())) ^. foo1in _2
1
>>> myData (Proxy :: Proxy ((,) ())) ^. foo2in _2
2
>>> myData (Proxy :: Proxy Maybe) ^. bar1in _Just
"hello"
>>> myData (Proxy :: Proxy Maybe) ^. bar2in _Just
"world"

When it comes to standard data types, then, at the hart of every Iso, Lens and Prism, lies a simple trick. A hole is inserted between getter (i.e. destructor) function and setter (i.e. constructor) function. Difference between various constructs in e.g. lens library is the specialization of that hole, which in turn constraints type signature a little bit.

Example:

data Coords2D = Coords2D {_x :: Int, _y :: Int}

x :: Lens' Coords2D Int
x f s = setter s <$> f (getter s)
  where
    getter = _x
    setter s b = s{_x = b}

As we can see, in the above example, there is a function function inserted in between getter and setter functions. That function contains an unknown function f.

If we gather all the code in between getter and setter functions and put in to one place, then we would get:

x :: Lens' Coords2D Int
x = setter `f` getter
  where
    getter = _x
    setter s b = s{_x = b}
    f set get h s = set s <$> h (get h)

Now we can see that the original hole (function f) has moved little bit further down and is now called h. Function f now is a Lens smart constructor that takes getter and setter and creates a Lens. This leads us to a question. What would happen if we won't specialize f, at all, and leave it to a user to decide what it should be? This is what we would get:

preX :: ((Coords2D -> Int) -> (Coords2D -> Int -> Coords2D) -> r) -> r
preX f = _x `f` \s b -> s{_x = b}

Now we can move things arount a bit:

preX :: ((Int -> Coords2D -> Coords2D) -> (Coords2D -> Int) -> r) -> r
preX f = (\b s -> s{_x = b}) `f` _x

This can also be rewritten to use ~$~ combinator:

preX :: ((Int -> Coords2D -> Coords2D) -> (Coords2D -> Int) -> r) -> r
preX = (\b s -> s{_x = b}) ~$~ _x

Or even using its flipped variant ~$$~:

preX :: ((Int -> Coords2D -> Coords2D) -> (Coords2D -> Int) -> r) -> r
preX = _x ~$$~ \b s -> s{_x = b}

We call such function a PreLens, since it is actually a precursors to a Lens.

preX :: PreLens' r Coords2D Int
preX = _x ~$$~ \b s -> s{_x = b}

It is also function with the most generic type signature of a function that is capable of creating a lens from getter and setter, if f is specialized appropriately:

x :: Lens' Coords2D Int
x = preX ((<^@~) . flip)

Notice that preX, in the above code snipped, got specialized in to:

preX :: PreLens' (Lens' Coords2D Int) Coords2D Int

Function preX is takes a lens smart constructor regardles of what lens kind. It can be Laarhoven Lens, Store Comonad-coalgebra or any other representation. It can also take a function that gets either getter or setter, or even a function that combines those functions with others.

This trick of putting a hole between constructor (anamorphism) and destructor (catamorphism) is also the reason why Laarhoven's Lenses can be introduced as a generalization of zipper idiom. For more information see also:

• From Zipper To Lens
• CPS based functional references, introduction of Laarhoven's Lenses.

There are other packages out there that provide similar combinators.

You may have noticed similarity between:

dimap :: Profunctor p => (a -> b) -> (c -> d) -> p b c -> p a d

and

between :: (c -> d) -> (a -> b) -> (b -> c) -> a -> d

If you also consider that there is also instance Profunctor (->), then between becomes specialized flipped dimap for Profunctor (->).

Profunctors are a powerful abstraction and Edward Kmett's implementation also includes low level optimizations that use the coercible feature of GHC. For more details see its package documentation.

Package pointless-fun provides few similar combinators, to between, in both strict and lazy variants:

(~>) :: (a -> b) -> (c -> d) -> (b -> c) -> a -> d
(!~>) :: (a -> b) -> (c -> d) -> (b -> c) -> a -> d

Comare it with:

between :: (c -> d) -> (a -> b) -> (b -> c) -> a -> d

And you see that (~>) is flipped between and (!~>) is similar to (strict) between, but our (strict) between is even less lazy in its implementation then (!~>).