Data.Functor.Contravariant.Coyoneda

The co-Yoneda lemma for presheafs states that f is naturally isomorphic to Coyoneda f.

Documentation

data Coyoneda f a where

A Contravariant functor (aka presheaf) suitable for Yoneda reduction.

http://ncatlab.org/nlab/show/Yoneda+reduction

Constructors

Coyoneda :: (a -> b) -> f b -> Coyoneda f a

Instances

Contravariant (Coyoneda f)

Defined in Data.Functor.Contravariant.Coyoneda

Methods

contramap :: (a -> b) -> Coyoneda f b -> Coyoneda f a

(>$) :: b -> Coyoneda f b -> Coyoneda f a

Representable f => Representable (Coyoneda f)

Defined in Data.Functor.Contravariant.Coyoneda

Associated Types

type Rep (Coyoneda f)

Methods

tabulate :: (a -> Rep (Coyoneda f)) -> Coyoneda f a

index :: Coyoneda f a -> a -> Rep (Coyoneda f)

contramapWithRep :: (b -> Either a (Rep (Coyoneda f))) -> Coyoneda f a -> Coyoneda f b

Adjunction f g => Adjunction (Coyoneda f) (Coyoneda g)

Defined in Data.Functor.Contravariant.Coyoneda

Methods

unit :: a -> Coyoneda g (Coyoneda f a)

counit :: a -> Coyoneda f (Coyoneda g a)

leftAdjunct :: (b -> Coyoneda f a) -> a -> Coyoneda g b

rightAdjunct :: (a -> Coyoneda g b) -> b -> Coyoneda f a

type Rep (Coyoneda f)

Defined in Data.Functor.Contravariant.Coyoneda

type Rep (Coyoneda f) = Rep f

liftCoyoneda :: f a -> Coyoneda f a

Coyoneda "expansion" of a presheaf

liftCoyoneda . lowerCoyoneda ≡ id
lowerCoyoneda . liftCoyoneda ≡ id

lowerCoyoneda :: Contravariant f => Coyoneda f a -> f a

Coyoneda reduction on a presheaf

hoistCoyoneda :: (forall a. f a -> g a) -> Coyoneda f b -> Coyoneda g b

Lift a natural transformation from f to g to a natural transformation from Coyoneda f to Coyoneda g.