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 #

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

Constructors

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

Instances details

Contravariant (Coyoneda f) #
Instance details 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) #
Instance details 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) #
Instance details 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) #
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 #

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