Data.Functor.Coyoneda

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.

Documentation

data Coyoneda f a where

A covariant Functor suitable for Yoneda reduction

Constructors

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

Instances

ComonadTrans Coyoneda

Defined in Data.Functor.Coyoneda

Methods

lower :: Comonad w => Coyoneda w a -> w a

MonadTrans Coyoneda

Defined in Data.Functor.Coyoneda

Methods

lift :: Monad m => m a -> Coyoneda m a

Monad m => Monad (Coyoneda m)

Defined in Data.Functor.Coyoneda

Methods

(>>=) :: Coyoneda m a -> (a -> Coyoneda m b) -> Coyoneda m b

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

return :: a -> Coyoneda m a

Functor (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

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

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

MonadFix f => MonadFix (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

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

Applicative f => Applicative (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

pure :: a -> Coyoneda f a

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

liftA2 :: (a -> b -> c) -> Coyoneda f a -> Coyoneda f b -> Coyoneda f c

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

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

Foldable f => Foldable (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

fold :: Monoid m => Coyoneda f m -> m

foldMap :: Monoid m => (a -> m) -> Coyoneda f a -> m

foldMap' :: Monoid m => (a -> m) -> Coyoneda f a -> m

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

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

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

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

foldr1 :: (a -> a -> a) -> Coyoneda f a -> a

foldl1 :: (a -> a -> a) -> Coyoneda f a -> a

toList :: Coyoneda f a -> [a]

null :: Coyoneda f a -> Bool

length :: Coyoneda f a -> Int

elem :: Eq a => a -> Coyoneda f a -> Bool

maximum :: Ord a => Coyoneda f a -> a

minimum :: Ord a => Coyoneda f a -> a

sum :: Num a => Coyoneda f a -> a

product :: Num a => Coyoneda f a -> a

Traversable f => Traversable (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

traverse :: Applicative f0 => (a -> f0 b) -> Coyoneda f a -> f0 (Coyoneda f b)

sequenceA :: Applicative f0 => Coyoneda f (f0 a) -> f0 (Coyoneda f a)

mapM :: Monad m => (a -> m b) -> Coyoneda f a -> m (Coyoneda f b)

sequence :: Monad m => Coyoneda f (m a) -> m (Coyoneda f a)

Distributive f => Distributive (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

distribute :: Functor f0 => f0 (Coyoneda f a) -> Coyoneda f (f0 a)

collect :: Functor f0 => (a -> Coyoneda f b) -> f0 a -> Coyoneda f (f0 b)

distributeM :: Monad m => m (Coyoneda f a) -> Coyoneda f (m a)

collectM :: Monad m => (a -> Coyoneda f b) -> m a -> Coyoneda f (m b)

Representable f => Representable (Coyoneda f)

Defined in Data.Functor.Coyoneda

Associated Types

type Rep (Coyoneda f)

Methods

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

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

Eq1 f => Eq1 (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

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

Ord1 f => Ord1 (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

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

Read1 f => Read1 (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Coyoneda f a)

liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Coyoneda f a]

liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (Coyoneda f a)

liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [Coyoneda f a]

(Functor f, Show1 f) => Show1 (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Coyoneda f a -> ShowS

liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Coyoneda f a] -> ShowS

Alternative f => Alternative (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

empty :: Coyoneda f a

(<|>) :: Coyoneda f a -> Coyoneda f a -> Coyoneda f a

some :: Coyoneda f a -> Coyoneda f [a]

many :: Coyoneda f a -> Coyoneda f [a]

MonadPlus f => MonadPlus (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

mzero :: Coyoneda f a

mplus :: Coyoneda f a -> Coyoneda f a -> Coyoneda f a

Comonad w => Comonad (Coyoneda w)

Defined in Data.Functor.Coyoneda

Methods

extract :: Coyoneda w a -> a

duplicate :: Coyoneda w a -> Coyoneda w (Coyoneda w a)

extend :: (Coyoneda w a -> b) -> Coyoneda w a -> Coyoneda w b

Plus f => Plus (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

zero :: Coyoneda f a

Traversable1 f => Traversable1 (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

traverse1 :: Apply f0 => (a -> f0 b) -> Coyoneda f a -> f0 (Coyoneda f b)

sequence1 :: Apply f0 => Coyoneda f (f0 b) -> f0 (Coyoneda f b)

Alt f => Alt (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

(<!>) :: Coyoneda f a -> Coyoneda f a -> Coyoneda f a

some :: Applicative (Coyoneda f) => Coyoneda f a -> Coyoneda f [a]

many :: Applicative (Coyoneda f) => Coyoneda f a -> Coyoneda f [a]

Apply f => Apply (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

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

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

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

liftF2 :: (a -> b -> c) -> Coyoneda f a -> Coyoneda f b -> Coyoneda f c

Bind m => Bind (Coyoneda m)

Defined in Data.Functor.Coyoneda

Methods

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

join :: Coyoneda m (Coyoneda m a) -> Coyoneda m a

Foldable1 f => Foldable1 (Coyoneda f)

Defined in Data.Functor.Coyoneda

Methods

fold1 :: Semigroup m => Coyoneda f m -> m

foldMap1 :: Semigroup m => (a -> m) -> Coyoneda f a -> m

toNonEmpty :: Coyoneda f a -> NonEmpty a

Extend w => Extend (Coyoneda w)

Defined in Data.Functor.Coyoneda

Methods

duplicated :: Coyoneda w a -> Coyoneda w (Coyoneda w a)

extended :: (Coyoneda w a -> b) -> Coyoneda w a -> Coyoneda w b

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

Defined in Data.Functor.Coyoneda

Methods

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

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

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

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

(Functor f, Eq1 f, Eq a) => Eq (Coyoneda f a)

Defined in Data.Functor.Coyoneda

Methods

(==) :: Coyoneda f a -> Coyoneda f a -> Bool

(/=) :: Coyoneda f a -> Coyoneda f a -> Bool

(Functor f, Ord1 f, Ord a) => Ord (Coyoneda f a)

Defined in Data.Functor.Coyoneda

Methods

compare :: Coyoneda f a -> Coyoneda f a -> Ordering

(<) :: Coyoneda f a -> Coyoneda f a -> Bool

(<=) :: Coyoneda f a -> Coyoneda f a -> Bool

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

(>=) :: Coyoneda f a -> Coyoneda f a -> Bool

max :: Coyoneda f a -> Coyoneda f a -> Coyoneda f a

min :: Coyoneda f a -> Coyoneda f a -> Coyoneda f a

Read (f a) => Read (Coyoneda f a)

Defined in Data.Functor.Coyoneda

Methods

readsPrec :: Int -> ReadS (Coyoneda f a)

readList :: ReadS [Coyoneda f a]

readPrec :: ReadPrec (Coyoneda f a)

readListPrec :: ReadPrec [Coyoneda f a]

(Functor f, Show1 f, Show a) => Show (Coyoneda f a)

Defined in Data.Functor.Coyoneda

Methods

showsPrec :: Int -> Coyoneda f a -> ShowS

show :: Coyoneda f a -> String

showList :: [Coyoneda f a] -> ShowS

type Rep (Coyoneda f)

Defined in Data.Functor.Coyoneda

type Rep (Coyoneda f) = Rep f

liftCoyoneda :: f a -> Coyoneda f a

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

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

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

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

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