Transformation

A natural transformation is a concept from category theory for a mapping between two functors and their objects that preserves a naturality condition. In Haskell the naturality condition boils down to parametricity, so a natural transformation between two functors f and g is represented as

type NaturalTransformation f g = ∀a. f a → g a

This type appears in several Haskell libraries, most obviously in natural-transformations. There are times, however, when we crave more control. Sometimes what we want to do depends on which type a is hiding in that f a we're given. Sometimes, in other words, we need an unnatural transformation.

This means we have to abandon parametricity for ad-hoc polymorphism, and that means type classes. There are two steps to defining a transformation:

• an instance of the base class Transformation declares the two functors being mapped, much like a function type signature,
• while the actual mapping of values is performed by an arbitrary number of instances of the method $, a bit like multiple equation clauses that make up a single function definition.

The module is meant to be imported qualified.

Documentation

class Transformation t

A Transformation, natural or not, maps one functor to another.

Associated Types

type Domain t :: Type -> Type

type Codomain t :: Type -> Type

Instances

(Transformation t1, Transformation t2, Domain t1 ~ Domain t2) => Transformation (Either t1 t2)

Defined in Transformation

Associated Types

type Domain (Either t1 t2) :: Type -> Type

type Codomain (Either t1 t2) :: Type -> Type

(Transformation t, Transformation u, Domain t ~ Codomain u) => Transformation (Compose t u)

Defined in Transformation

Associated Types

type Domain (Compose t u) :: Type -> Type

type Codomain (Compose t u) :: Type -> Type

Transformation t => Transformation (Folded f t)

Defined in Transformation

Associated Types

type Domain (Folded f t) :: Type -> Type

type Codomain (Folded f t) :: Type -> Type

Transformation t => Transformation (Mapped f t)

Defined in Transformation

Associated Types

type Domain (Mapped f t) :: Type -> Type

type Codomain (Mapped f t) :: Type -> Type

(Transformation t, Codomain t ~ Compose m n) => Transformation (Traversed f t)

Defined in Transformation

Associated Types

type Domain (Traversed f t) :: Type -> Type

type Codomain (Traversed f t) :: Type -> Type

Transformation (Fold p m)

Defined in Transformation.Rank2

Associated Types

type Domain (Fold p m) :: Type -> Type

type Codomain (Fold p m) :: Type -> Type

Transformation (Map p q)

Defined in Transformation.Rank2

Associated Types

type Domain (Map p q) :: Type -> Type

type Codomain (Map p q) :: Type -> Type

(Transformation t1, Transformation t2, Domain t1 ~ Domain t2) => Transformation (t1, t2)

Defined in Transformation

Associated Types

type Domain (t1, t2) :: Type -> Type

type Codomain (t1, t2) :: Type -> Type

Transformation (Feeder a b f)

Defined in Transformation.AG.Dimorphic

Associated Types

type Domain (Feeder a b f) :: Type -> Type

type Codomain (Feeder a b f) :: Type -> Type

Transformation (Traversal p q m)

Defined in Transformation.Rank2

Associated Types

type Domain (Traversal p q m) :: Type -> Type

type Codomain (Traversal p q m) :: Type -> Type

Transformation (Arrow p q x)

Defined in Transformation

Associated Types

type Domain (Arrow p q x) :: Type -> Type

type Codomain (Arrow p q x) :: Type -> Type

class Transformation t => At t x where

An unnatural Transformation can behave differently at different points.

Methods

($) :: t -> Domain t x -> Codomain t x infixr 0

Apply the transformation t at type x to map Domain to the Codomain functor.

Instances

(Transformation (Auto t), p ~ Domain (Auto t), q ~ Codomain (Auto t), q ~ Semantics a b, Foldable (g q), Monoid a, Monoid b, Foldable p, Attribution (Auto t) a b g q p) => At (Auto t) (g (Semantics a b) (Semantics a b))

Defined in Transformation.AG.Dimorphic

Methods

($) :: Auto t -> Domain (Auto t) (g (Semantics a b) (Semantics a b)) -> Codomain (Auto t) (g (Semantics a b) (Semantics a b))

(Transformation (Keep t), p ~ Domain (Keep t), q ~ Codomain (Keep t), q ~ PreservingSemantics p a b, Foldable (g q), Monoid a, Monoid b, Foldable p, Functor p, Attribution (Keep t) a b g q p) => At (Keep t) (g (PreservingSemantics p a b) (PreservingSemantics p a b))

Defined in Transformation.AG.Dimorphic

Methods

($) :: Keep t -> Domain (Keep t) (g (PreservingSemantics p a b) (PreservingSemantics p a b)) -> Codomain (Keep t) (g (PreservingSemantics p a b) (PreservingSemantics p a b))

(Revelation (Auto t), Domain (Auto t) ~ f, Codomain (Auto t) ~ Semantics (Auto t), Apply (g (Semantics (Auto t))), Attribution (Auto t) g (Semantics (Auto t)) f) => At (Auto t) (g (Semantics (Auto t)) (Semantics (Auto t)))

Defined in Transformation.AG.Generics

Methods

($) :: Auto t -> Domain (Auto t) (g (Semantics (Auto t)) (Semantics (Auto t))) -> Codomain (Auto t) (g (Semantics (Auto t)) (Semantics (Auto t)))

(Revelation (Keep t), p ~ Domain (Keep t), Apply (g q), q ~ Codomain (Keep t), q ~ PreservingSemantics (Keep t) p, s ~ Semantics (Keep t), Atts (Inherited (Keep t)) (g q q) ~ Atts (Inherited (Keep t)) (g s s), Atts (Synthesized (Keep t)) (g q q) ~ Atts (Synthesized (Keep t)) (g s s), g q (Synthesized (Keep t)) ~ g s (Synthesized (Keep t)), g q (Inherited (Keep t)) ~ g s (Inherited (Keep t)), Attribution (Keep t) g q p) => At (Keep t) (g (PreservingSemantics (Keep t) p) (PreservingSemantics (Keep t) p))

Defined in Transformation.AG.Generics

Methods

($) :: Keep t -> Domain (Keep t) (g (PreservingSemantics (Keep t) p) (PreservingSemantics (Keep t) p)) -> Codomain (Keep t) (g (PreservingSemantics (Keep t) p) (PreservingSemantics (Keep t) p))

(Transformation (Auto t), p ~ Domain (Auto t), q ~ Codomain (Auto t), q ~ Semantics a, Foldable (g q), Monoid a, Foldable p, Attribution (Auto t) a g q p) => At (Auto t) (g (Semantics a) (Semantics a))

Defined in Transformation.AG.Monomorphic

Methods

($) :: Auto t -> Domain (Auto t) (g (Semantics a) (Semantics a)) -> Codomain (Auto t) (g (Semantics a) (Semantics a))

(Transformation (Keep t), p ~ Domain (Keep t), q ~ Codomain (Keep t), q ~ PreservingSemantics p a, Foldable (g q), Monoid a, Foldable p, Functor p, Attribution (Keep t) a g q p) => At (Keep t) (g (PreservingSemantics p a) (PreservingSemantics p a))

Defined in Transformation.AG.Monomorphic

Methods

($) :: Keep t -> Domain (Keep t) (g (PreservingSemantics p a) (PreservingSemantics p a)) -> Codomain (Keep t) (g (PreservingSemantics p a) (PreservingSemantics p a))

(At t x, At u x, Domain t ~ Domain u) => At (Either t u) x

Defined in Transformation

Methods

($) :: Either t u -> Domain (Either t u) x -> Codomain (Either t u) x

(At t x, At u x, Domain t ~ Codomain u) => At (Compose t u) x

Defined in Transformation

Methods

($) :: Compose t u -> Domain (Compose t u) x -> Codomain (Compose t u) x

(At t x, Foldable f, Codomain t ~ (Const m :: Type -> Type), Monoid m) => At (Folded f t) x

Defined in Transformation

Methods

($) :: Folded f t -> Domain (Folded f t) x -> Codomain (Folded f t) x

(At t x, Functor f) => At (Mapped f t) x

Defined in Transformation

Methods

($) :: Mapped f t -> Domain (Mapped f t) x -> Codomain (Mapped f t) x

(At t x, Traversable f, Codomain t ~ Compose m n, Applicative m) => At (Traversed f t) x

Defined in Transformation

Methods

($) :: Traversed f t -> Domain (Traversed f t) x -> Codomain (Traversed f t) x

At (Fold p m) x

Defined in Transformation.Rank2

Methods

($) :: Fold p m -> Domain (Fold p m) x -> Codomain (Fold p m) x

At (Map p q) x

Defined in Transformation.Rank2

Methods

($) :: Map p q -> Domain (Map p q) x -> Codomain (Map p q) x

(At t x, At u x, Domain t ~ Domain u) => At (t, u) x

Defined in Transformation

Methods

($) :: (t, u) -> Domain (t, u) x -> Codomain (t, u) x

At (Feeder a b f) g

Defined in Transformation.AG.Dimorphic

Methods

($) :: Feeder a b f -> Domain (Feeder a b f) g -> Codomain (Feeder a b f) g

At (Traversal p q m) x

Defined in Transformation.Rank2

Methods

($) :: Traversal p q m -> Domain (Traversal p q m) x -> Codomain (Traversal p q m) x

At (Arrow p q x) x

Defined in Transformation

Methods

($) :: Arrow p q x -> Domain (Arrow p q x) x -> Codomain (Arrow p q x) x

apply :: t `At` x => t -> Domain t x -> Codomain t x

Alphabetical synonym for $

data Compose t u

Composition of two transformations

Constructors

Compose t u

Instances

(Transformation t, Transformation u, Domain t ~ Codomain u) => Transformation (Compose t u)

Defined in Transformation

Associated Types

type Domain (Compose t u) :: Type -> Type

type Codomain (Compose t u) :: Type -> Type

(At t x, At u x, Domain t ~ Codomain u) => At (Compose t u) x

Defined in Transformation

Methods

($) :: Compose t u -> Domain (Compose t u) x -> Codomain (Compose t u) x

type Codomain (Compose t u)

Defined in Transformation

type Codomain (Compose t u) = Codomain t

type Domain (Compose t u)

Defined in Transformation

type Domain (Compose t u) = Domain u

newtype Mapped (f :: Type -> Type) t

Transformation under a Functor

Constructors

Mapped t

Instances

Transformation t => Transformation (Mapped f t)

Defined in Transformation

Associated Types

type Domain (Mapped f t) :: Type -> Type

type Codomain (Mapped f t) :: Type -> Type

(At t x, Functor f) => At (Mapped f t) x

Defined in Transformation

Methods

($) :: Mapped f t -> Domain (Mapped f t) x -> Codomain (Mapped f t) x

type Codomain (Mapped f t)

Defined in Transformation

type Codomain (Mapped f t) = Compose f (Codomain t)

type Domain (Mapped f t)

Defined in Transformation

type Domain (Mapped f t) = Compose f (Domain t)

newtype Folded (f :: Type -> Type) t

Transformation under a Foldable

Constructors

Folded t

Instances

Transformation t => Transformation (Folded f t)

Defined in Transformation

Associated Types

type Domain (Folded f t) :: Type -> Type

type Codomain (Folded f t) :: Type -> Type

(At t x, Foldable f, Codomain t ~ (Const m :: Type -> Type), Monoid m) => At (Folded f t) x

Defined in Transformation

Methods

($) :: Folded f t -> Domain (Folded f t) x -> Codomain (Folded f t) x

type Codomain (Folded f t)

Defined in Transformation

type Codomain (Folded f t) = Codomain t

type Domain (Folded f t)

Defined in Transformation

type Domain (Folded f t) = Compose f (Domain t)

newtype Traversed (f :: Type -> Type) t

Transformation under a Traversable

Constructors

Traversed t

Instances

(Transformation t, Codomain t ~ Compose m n) => Transformation (Traversed f t)

Defined in Transformation

Associated Types

type Domain (Traversed f t) :: Type -> Type

type Codomain (Traversed f t) :: Type -> Type

(At t x, Traversable f, Codomain t ~ Compose m n, Applicative m) => At (Traversed f t) x

Defined in Transformation

Methods

($) :: Traversed f t -> Domain (Traversed f t) x -> Codomain (Traversed f t) x

type Codomain (Traversed f t)

Defined in Transformation

type Codomain (Traversed f t) = Compose (ComposeOuter (Codomain t)) (Compose f (ComposeInner (Codomain t)))

type Domain (Traversed f t)

Defined in Transformation

type Domain (Traversed f t) = Compose f (Domain t)

type family ComposeOuter (c :: Type -> Type) :: Type -> Type where ...

Equations

ComposeOuter (Compose p q) = p

type family ComposeInner (c :: Type -> Type) :: Type -> Type where ...

Equations

ComposeInner (Compose p q) = q