Portability | RankNTypes |
---|---|

Stability | provisional |

Maintainer | Edward Kmett <ekmett@gmail.com> |

Safe Haskell | Safe-Infered |

Corepresentable endofunctors represented by their polymorphic lenses

The polymorphic lenses of the form `(forall x. Lens (f x) x)`

each
represent a distinct path into a functor `f`

. If the functor is entirely
characterized by assigning values to these paths, then the functor is
representable.

Consider the following example.

import Control.Lens import Control.Lens.Representable import Control.Lens.TH import Data.Distributive

data Pair a = Pair { _x :: a, _y :: a }

makeLenses ''Pair

instance Representable Pair where rep f = Pair (f x) (f y)

From there, you can get definitions for a number of instances for free.

instance Applicative Pair where pure = pureRep (<*>) = apRep

instance Monad Pair where return = pureRep (>>=) = bindRep

instance Distributive Pair where distribute = distributeRep

- class Functor f => Representable f where
- type Rep f = forall a. Simple Lens (f a) a
- fmapRep :: Representable f => (a -> b) -> f a -> f b
- pureRep :: Representable f => a -> f a
- apRep :: Representable f => f (a -> b) -> f a -> f b
- bindRep :: Representable f => f a -> (a -> f b) -> f b
- distributeRep :: (Representable f, Functor w) => w (f a) -> f (w a)
- newtype Key f = Key {}
- keys :: Representable f => f (Key f)
- tabulated :: Representable f => (Key f -> a) :~> f a
- mapWithRep :: Representable f => (Rep f -> a -> b) -> f a -> f b
- foldMapWithRep :: (Representable f, Foldable f, Monoid m) => (Rep f -> a -> m) -> f a -> m
- foldrWithRep :: (Representable f, Foldable f) => (Rep f -> a -> b -> b) -> b -> f a -> b
- traverseWithRep :: (Representable f, Traversable f, Applicative g) => (Rep f -> a -> g b) -> f a -> g (f b)
- traverseWithRep_ :: (Representable f, Foldable f, Applicative g) => (Rep f -> a -> g b) -> f a -> g ()
- forWithRep :: (Representable f, Traversable f, Applicative g) => f a -> (Rep f -> a -> g b) -> g (f b)
- mapMWithRep :: (Representable f, Traversable f, Monad m) => (Rep f -> a -> m b) -> f a -> m (f b)
- mapMWithRep_ :: (Representable f, Foldable f, Monad m) => (Rep f -> a -> m b) -> f a -> m ()
- forMWithRep :: (Representable f, Traversable f, Monad m) => f a -> (Rep f -> a -> m b) -> m (f b)

# Representable Functors

class Functor f => Representable f whereSource

Representable Functors.

A `Functor`

`f`

is `Representable`

if it is isomorphic to `(x -> a)`

for some x. All such functors can be represented by choosing `x`

to be
the set of lenses that are polymorphic in the contents of the `Functor`

,
that is to say `x = Rep f`

is a valid choice of `x`

for every
`Representable`

`Functor`

.

Note: Some sources refer to covariant representable functors as
corepresentable functors, and leave the "representable" name to
contravariant functors (those are isomorphic to `(a -> x)`

for some `x`

).

As the covariant case is vastly more common, and both are often referred to
as representable functors, we choose to call these functors `Representable`

here.

Representable Identity | |

Eq e => Representable ((->) e) | NB: The Eq requirement on this instance is a consequence of a lens
rather than |

# Using Lenses as Representations

type Rep f = forall a. Simple Lens (f a) aSource

The representation of a `Representable`

`Functor`

as Lenses

# Default definitions

fmapRep :: Representable f => (a -> b) -> f a -> f bSource

pureRep :: Representable f => a -> f aSource

apRep :: Representable f => f (a -> b) -> f a -> f bSource

bindRep :: Representable f => f a -> (a -> f b) -> f bSource

`bindRep`

is a valid default default definition for '(>>=)' for a
representable functor.

bindRep m f = rep $ \i -> f(m^.i)^.i

Usage for a representable functor `Foo`

:

instance Monad ... where return = pureRep (>>=) = bindRep

distributeRep :: (Representable f, Functor w) => w (f a) -> f (w a)Source

A default definition for `distribute`

for a `Representable`

`Functor`

distributeRep wf = rep $ \i -> fmap (^.i) wf

Typical Usage:

instance Distributive ... where distribute = distributeRep

# Wrapped Representations

Sometimes you need to store a path lens into a container, but at least at this time, impredicative polymorphism in GHC is somewhat lacking.

This type provides a way to, say, store a list of polymorphic lenses.

keys :: Representable f => f (Key f)Source

A `Representable`

`Functor`

has a fixed shape. This fills each position
in it with a `Key`

tabulated :: Representable f => (Key f -> a) :~> f aSource

# Traversal with representation

mapWithRep :: Representable f => (Rep f -> a -> b) -> f a -> f bSource

Map over a `Representable`

`Functor`

with access to the lens for the
current position

mapWithKey f m = rep $ \i -> f i (m^.i)

foldMapWithRep :: (Representable f, Foldable f, Monoid m) => (Rep f -> a -> m) -> f a -> mSource

Fold over a `Representable`

`Functor`

with access to the current path
as a lens, yielding a `Monoid`

foldrWithRep :: (Representable f, Foldable f) => (Rep f -> a -> b -> b) -> b -> f a -> bSource

Fold over a `Representable`

`Functor`

with access to the current path
as a lens.

traverseWithRep :: (Representable f, Traversable f, Applicative g) => (Rep f -> a -> g b) -> f a -> g (f b)Source

Traverse a `Representable`

`Functor`

with access to the current path

traverseWithRep_ :: (Representable f, Foldable f, Applicative g) => (Rep f -> a -> g b) -> f a -> g ()Source

Traverse a `Representable`

`Functor`

with access to the current path
as a lens, discarding the result

forWithRep :: (Representable f, Traversable f, Applicative g) => f a -> (Rep f -> a -> g b) -> g (f b)Source

Traverse a `Representable`

`Functor`

with access to the current path
and a lens (and the arguments flipped)

mapMWithRep :: (Representable f, Traversable f, Monad m) => (Rep f -> a -> m b) -> f a -> m (f b)Source

`mapM`

over a `Representable`

`Functor`

with access to the current path
as a lens

mapMWithRep_ :: (Representable f, Foldable f, Monad m) => (Rep f -> a -> m b) -> f a -> m ()Source

`mapM`

over a `Representable`

`Functor`

with access to the current path
as a lens, discarding the result

forMWithRep :: (Representable f, Traversable f, Monad m) => f a -> (Rep f -> a -> m b) -> m (f b)Source

`mapM`

over a `Representable`

`Functor`

with access to the current path
as a lens (with the arguments flipped)