Data.Functor.Coyoneda

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

Methods

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

MonadTrans Coyoneda

Methods

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

Monad m => Monad (Coyoneda m)

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

fail :: String -> Coyoneda m a

Functor (Coyoneda f)

Methods

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

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

MonadFix f => MonadFix (Coyoneda f)

Methods

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

Applicative f => Applicative (Coyoneda f)

Methods

pure :: a -> Coyoneda f a

(<*>) :: 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

Foldable f => Foldable (Coyoneda f)

Methods

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

Methods

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

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

Methods

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

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

Associated Types

type Rep (Coyoneda f :: * -> *) :: *

Methods

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

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

Alternative f => Alternative (Coyoneda f)

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)

Methods

mzero :: Coyoneda f a

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

Comonad w => Comonad (Coyoneda w)

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

Traversable1 f => Traversable1 (Coyoneda f)

Methods

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

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

Plus f => Plus (Coyoneda f)

Methods

zero :: Coyoneda f a

Alt f => Alt (Coyoneda f)

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)

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

Bind m => Bind (Coyoneda m)

Methods

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

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

Extend w => Extend (Coyoneda w)

Methods

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

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

Foldable1 f => Foldable1 (Coyoneda f)

Methods

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

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

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

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

Methods

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

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

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

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)

Methods

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

readList :: ReadS [Coyoneda f a]

readPrec :: ReadPrec (Coyoneda f a)

readListPrec :: ReadPrec [Coyoneda f a]

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

Methods

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

show :: Coyoneda f a -> String

showList :: [Coyoneda f a] -> ShowS

type Rep (Coyoneda f)

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