Copyright	(C) 2013-2016 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	GADTs, TFs, MPTCs
Safe Haskell	Trustworthy
Language	Haskell2010

Data.Functor.Contravariant.Coyoneda

Description

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

Synopsis

Documentation

data Coyoneda f a where Source #

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

Constructors

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

Instances details

Contravariant ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Contravariant.Coyoneda Methods contramap :: (a -> b) -> Coyoneda f b -> Coyoneda f a Source # (>$) :: b -> Coyoneda f b -> Coyoneda f a Source #
Representable f => Representable ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Contravariant.Coyoneda Associated Types type Rep ( Coyoneda f) Source # Methods tabulate :: (a -> Rep ( Coyoneda f)) -> Coyoneda f a Source # index :: Coyoneda f a -> a -> Rep ( Coyoneda f) Source # contramapWithRep :: (b -> Either a ( Rep ( Coyoneda f))) -> Coyoneda f a -> Coyoneda f b Source #
Adjunction f g => Adjunction ( Coyoneda f) ( Coyoneda g) Source #
Instance details Defined in Data.Functor.Contravariant.Coyoneda Methods unit :: a -> Coyoneda g ( Coyoneda f a) Source # counit :: a -> Coyoneda f ( Coyoneda g a) Source # leftAdjunct :: (b -> Coyoneda f a) -> a -> Coyoneda g b Source # rightAdjunct :: (a -> Coyoneda g b) -> b -> Coyoneda f a Source #
type Rep ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Contravariant.Coyoneda type Rep ( Coyoneda f) = Rep f

Coyoneda "expansion" of a presheaf

liftCoyoneda . lowerCoyoneda ≡ id
lowerCoyoneda . liftCoyoneda ≡ id

Coyoneda reduction on a presheaf

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

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