Copyright	(C) 2011-2016 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	GADTs, MPTCs, fundeps
Safe Haskell	Trustworthy
Language	Haskell2010

Data.Functor.Coyoneda

Contents

as a Left Kan extension

Description

Coyoneda f is the "free functor" over f . The co-Yoneda lemma for a covariant Functor f states that Coyoneda f is naturally isomorphic to f .

Synopsis

data Coyoneda f a where
- Coyoneda :: (b -> a) -> f b -> Coyoneda f a
liftCoyoneda :: f a -> Coyoneda f a
lowerCoyoneda :: Functor f => Coyoneda f a -> f a
lowerM :: Monad f => Coyoneda f a -> f a
hoistCoyoneda :: ( forall a. f a -> g a) -> Coyoneda f b -> Coyoneda g b
coyonedaToLan :: Coyoneda f a -> Lan Identity f a
lanToCoyoneda :: Lan Identity f a -> Coyoneda f a

Documentation

data Coyoneda f a where Source #

A covariant Functor suitable for Yoneda reduction

Constructors

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

Instances

Instances details

ComonadTrans Coyoneda Source #
Instance details Defined in Data.Functor.Coyoneda Methods lower :: Comonad w => Coyoneda w a -> w a Source #
MonadTrans Coyoneda Source #
Instance details Defined in Data.Functor.Coyoneda Methods lift :: Monad m => m a -> Coyoneda m a Source #
Monad m => Monad ( Coyoneda m) Source #
Instance details Defined in Data.Functor.Coyoneda Methods (>>=) :: Coyoneda m a -> (a -> Coyoneda m b) -> Coyoneda m b Source # (>>) :: Coyoneda m a -> Coyoneda m b -> Coyoneda m b Source # return :: a -> Coyoneda m a Source #
Functor ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods fmap :: (a -> b) -> Coyoneda f a -> Coyoneda f b Source # (<$) :: a -> Coyoneda f b -> Coyoneda f a Source #
MonadFix f => MonadFix ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods mfix :: (a -> Coyoneda f a) -> Coyoneda f a Source #
Applicative f => Applicative ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods pure :: a -> Coyoneda f a Source # (<>) :: Coyoneda f (a -> b) -> Coyoneda f a -> Coyoneda f b Source # liftA2 :: (a -> b -> c) -> Coyoneda f a -> Coyoneda f b -> Coyoneda f c Source # (>) :: Coyoneda f a -> Coyoneda f b -> Coyoneda f b Source # (<*) :: Coyoneda f a -> Coyoneda f b -> Coyoneda f a Source #
Foldable f => Foldable ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods fold :: Monoid m => Coyoneda f m -> m Source # foldMap :: Monoid m => (a -> m) -> Coyoneda f a -> m Source # foldMap' :: Monoid m => (a -> m) -> Coyoneda f a -> m Source # foldr :: (a -> b -> b) -> b -> Coyoneda f a -> b Source # foldr' :: (a -> b -> b) -> b -> Coyoneda f a -> b Source # foldl :: (b -> a -> b) -> b -> Coyoneda f a -> b Source # foldl' :: (b -> a -> b) -> b -> Coyoneda f a -> b Source # foldr1 :: (a -> a -> a) -> Coyoneda f a -> a Source # foldl1 :: (a -> a -> a) -> Coyoneda f a -> a Source # toList :: Coyoneda f a -> [a] Source # null :: Coyoneda f a -> Bool Source # length :: Coyoneda f a -> Int Source # elem :: Eq a => a -> Coyoneda f a -> Bool Source # maximum :: Ord a => Coyoneda f a -> a Source # minimum :: Ord a => Coyoneda f a -> a Source # sum :: Num a => Coyoneda f a -> a Source # product :: Num a => Coyoneda f a -> a Source #
Traversable f => Traversable ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods traverse :: Applicative f0 => (a -> f0 b) -> Coyoneda f a -> f0 ( Coyoneda f b) Source # sequenceA :: Applicative f0 => Coyoneda f (f0 a) -> f0 ( Coyoneda f a) Source # mapM :: Monad m => (a -> m b) -> Coyoneda f a -> m ( Coyoneda f b) Source # sequence :: Monad m => Coyoneda f (m a) -> m ( Coyoneda f a) Source #
Distributive f => Distributive ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods distribute :: Functor f0 => f0 ( Coyoneda f a) -> Coyoneda f (f0 a) Source # collect :: Functor f0 => (a -> Coyoneda f b) -> f0 a -> Coyoneda f (f0 b) Source # distributeM :: Monad m => m ( Coyoneda f a) -> Coyoneda f (m a) Source # collectM :: Monad m => (a -> Coyoneda f b) -> m a -> Coyoneda f (m b) Source #
Representable f => Representable ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Associated Types type Rep ( Coyoneda f) Source # Methods tabulate :: ( Rep ( Coyoneda f) -> a) -> Coyoneda f a Source # index :: Coyoneda f a -> Rep ( Coyoneda f) -> a Source #
Eq1 f => Eq1 ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods liftEq :: (a -> b -> Bool ) -> Coyoneda f a -> Coyoneda f b -> Bool Source #
Ord1 f => Ord1 ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods liftCompare :: (a -> b -> Ordering ) -> Coyoneda f a -> Coyoneda f b -> Ordering Source #
Read1 f => Read1 ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods liftReadsPrec :: ( Int -> ReadS a) -> ReadS [a] -> Int -> ReadS ( Coyoneda f a) Source # liftReadList :: ( Int -> ReadS a) -> ReadS [a] -> ReadS [ Coyoneda f a] Source # liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec ( Coyoneda f a) Source # liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [ Coyoneda f a] Source #
( Functor f, Show1 f) => Show1 ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods liftShowsPrec :: ( Int -> a -> ShowS ) -> ([a] -> ShowS ) -> Int -> Coyoneda f a -> ShowS Source # liftShowList :: ( Int -> a -> ShowS ) -> ([a] -> ShowS ) -> [ Coyoneda f a] -> ShowS Source #
Alternative f => Alternative ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods empty :: Coyoneda f a Source # (<\|>) :: Coyoneda f a -> Coyoneda f a -> Coyoneda f a Source # some :: Coyoneda f a -> Coyoneda f [a] Source # many :: Coyoneda f a -> Coyoneda f [a] Source #
MonadPlus f => MonadPlus ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods mzero :: Coyoneda f a Source # mplus :: Coyoneda f a -> Coyoneda f a -> Coyoneda f a Source #
Comonad w => Comonad ( Coyoneda w) Source #
Instance details Defined in Data.Functor.Coyoneda Methods extract :: Coyoneda w a -> a Source # duplicate :: Coyoneda w a -> Coyoneda w ( Coyoneda w a) Source # extend :: ( Coyoneda w a -> b) -> Coyoneda w a -> Coyoneda w b Source #
Plus f => Plus ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods zero :: Coyoneda f a Source #
Traversable1 f => Traversable1 ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods traverse1 :: Apply f0 => (a -> f0 b) -> Coyoneda f a -> f0 ( Coyoneda f b) Source # sequence1 :: Apply f0 => Coyoneda f (f0 b) -> f0 ( Coyoneda f b) Source #
Alt f => Alt ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods (<!>) :: Coyoneda f a -> Coyoneda f a -> Coyoneda f a Source # some :: Applicative ( Coyoneda f) => Coyoneda f a -> Coyoneda f [a] Source # many :: Applicative ( Coyoneda f) => Coyoneda f a -> Coyoneda f [a] Source #
Apply f => Apply ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods (<.>) :: Coyoneda f (a -> b) -> Coyoneda f a -> Coyoneda f b Source # (.>) :: Coyoneda f a -> Coyoneda f b -> Coyoneda f b Source # (<.) :: Coyoneda f a -> Coyoneda f b -> Coyoneda f a Source # liftF2 :: (a -> b -> c) -> Coyoneda f a -> Coyoneda f b -> Coyoneda f c Source #
Bind m => Bind ( Coyoneda m) Source #
Instance details Defined in Data.Functor.Coyoneda Methods (>>-) :: Coyoneda m a -> (a -> Coyoneda m b) -> Coyoneda m b Source # join :: Coyoneda m ( Coyoneda m a) -> Coyoneda m a Source #
Foldable1 f => Foldable1 ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda Methods fold1 :: Semigroup m => Coyoneda f m -> m Source # foldMap1 :: Semigroup m => (a -> m) -> Coyoneda f a -> m Source # toNonEmpty :: Coyoneda f a -> NonEmpty a Source #
Extend w => Extend ( Coyoneda w) Source #
Instance details Defined in Data.Functor.Coyoneda Methods duplicated :: Coyoneda w a -> Coyoneda w ( Coyoneda w a) Source # extended :: ( Coyoneda w a -> b) -> Coyoneda w a -> Coyoneda w b Source #
Adjunction f g => Adjunction ( Coyoneda f) ( Coyoneda g) Source #
Instance details Defined in Data.Functor.Coyoneda Methods unit :: a -> Coyoneda g ( Coyoneda f a) Source # counit :: Coyoneda f ( Coyoneda g a) -> a Source # leftAdjunct :: ( Coyoneda f a -> b) -> a -> Coyoneda g b Source # rightAdjunct :: (a -> Coyoneda g b) -> Coyoneda f a -> b Source #
( Eq1 f, Eq a) => Eq ( Coyoneda f a) Source #
Instance details Defined in Data.Functor.Coyoneda Methods (==) :: Coyoneda f a -> Coyoneda f a -> Bool Source # (/=) :: Coyoneda f a -> Coyoneda f a -> Bool Source #
( Ord1 f, Ord a) => Ord ( Coyoneda f a) Source #
Instance details Defined in Data.Functor.Coyoneda Methods compare :: Coyoneda f a -> Coyoneda f a -> Ordering Source # (<) :: Coyoneda f a -> Coyoneda f a -> Bool Source # (<=) :: Coyoneda f a -> Coyoneda f a -> Bool Source # (>) :: Coyoneda f a -> Coyoneda f a -> Bool Source # (>=) :: Coyoneda f a -> Coyoneda f a -> Bool Source # max :: Coyoneda f a -> Coyoneda f a -> Coyoneda f a Source # min :: Coyoneda f a -> Coyoneda f a -> Coyoneda f a Source #
Read (f a) => Read ( Coyoneda f a) Source #
Instance details Defined in Data.Functor.Coyoneda Methods readsPrec :: Int -> ReadS ( Coyoneda f a) Source # readList :: ReadS [ Coyoneda f a] Source # readPrec :: ReadPrec ( Coyoneda f a) Source # readListPrec :: ReadPrec [ Coyoneda f a] Source #
( Functor f, Show1 f, Show a) => Show ( Coyoneda f a) Source #
Instance details Defined in Data.Functor.Coyoneda Methods showsPrec :: Int -> Coyoneda f a -> ShowS Source # show :: Coyoneda f a -> String Source # showList :: [ Coyoneda f a] -> ShowS Source #
type Rep ( Coyoneda f) Source #
Instance details Defined in Data.Functor.Coyoneda type Rep ( Coyoneda f) = Rep f

liftCoyoneda :: f a -> Coyoneda f a Source #

Yoneda "expansion"

liftCoyoneda . lowerCoyoneda ≡ id
lowerCoyoneda . liftCoyoneda ≡ id

lowerCoyoneda (liftCoyoneda fa) = -- by definition
lowerCoyoneda (Coyoneda id fa)  = -- by definition
fmap id fa                      = -- functor law
fa

lift = liftCoyoneda

lowerCoyoneda :: Functor f => Coyoneda f a -> f a Source #

Yoneda reduction lets us walk under the existential and apply fmap .

Mnemonically, "Yoneda reduction" sounds like and works a bit like β-reduction.

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

You can view Coyoneda as just the arguments to fmap tupled up.

lower = lowerM = lowerCoyoneda

lowerM :: Monad f => Coyoneda f a -> f a Source #

Yoneda reduction given a Monad lets us walk under the existential and apply liftM .

You can view Coyoneda as just the arguments to liftM tupled up.

lower = lowerM = lowerCoyoneda

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 .

as a Left Kan extension

coyonedaToLan :: Coyoneda f a -> Lan Identity f a Source #

Coyoneda f is the left Kan extension of f along the Identity functor.

Coyoneda f is always a functor, even if f is not. In this case, it is called the free functor over f . Note the following categorical fine print: If f is not a functor, Coyoneda f is actually not the left Kan extension of f along the Identity functor, but along the inclusion functor from the discrete subcategory of Hask which contains only identity functions as morphisms to the full category Hask . (This is because f , not being a proper functor, can only be interpreted as a categorical functor by restricting the source category to only contain identities.)

coyonedaToLan . lanToCoyoneda ≡ id
lanToCoyoneda . coyonedaToLan ≡ id

lanToCoyoneda :: Lan Identity f a -> Coyoneda f a Source #