Data.Functor.Yoneda

The covariant form of the Yoneda lemma states that f is naturally isomorphic to Yoneda f.

This is described in a rather intuitive fashion by Dan Piponi in

http://blog.sigfpe.com/2006/11/yoneda-lemma.html

Documentation

newtype Yoneda f a

Yoneda f a can be viewed as the partial application of fmap to its second argument.

Constructors

Yoneda

Fields

• runYoneda :: forall b. (a -> b) -> f b

Instances

ComonadTrans Yoneda

Methods

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

MonadTrans Yoneda

Methods

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

(Functor f, MonadFree f m) => MonadFree f (Yoneda m)

Methods

wrap :: f (Yoneda m a) -> Yoneda m a

Monad m => Monad (Yoneda m)

Methods

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

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

return :: a -> Yoneda m a

fail :: String -> Yoneda m a

Functor (Yoneda f)

Methods

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

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

MonadFix m => MonadFix (Yoneda m)

Methods

mfix :: (a -> Yoneda m a) -> Yoneda m a

Applicative f => Applicative (Yoneda f)

Methods

pure :: a -> Yoneda f a

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

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

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

Foldable f => Foldable (Yoneda f)

Methods

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

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

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

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

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

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

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

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

toList :: Yoneda f a -> [a]

null :: Yoneda f a -> Bool

length :: Yoneda f a -> Int

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

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

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

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

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

Traversable f => Traversable (Yoneda f)

Methods

traverse :: Applicative b => (a -> b c) -> Yoneda f a -> b (Yoneda f c)

sequenceA :: Applicative a => Yoneda f (a b) -> a (Yoneda f b)

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

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

Distributive f => Distributive (Yoneda f)

Methods

distribute :: Functor a => a (Yoneda f b) -> Yoneda f (a b)

collect :: Functor c => (a -> Yoneda f b) -> c a -> Yoneda f (c b)

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

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

Representable g => Representable (Yoneda g)

Associated Types

type Rep (Yoneda g :: * -> *) :: *

Methods

tabulate :: (Rep (Yoneda g) -> a) -> Yoneda g a

index :: Yoneda g a -> Rep (Yoneda g) -> a

Alternative f => Alternative (Yoneda f)

Methods

empty :: Yoneda f a

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

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

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

MonadPlus m => MonadPlus (Yoneda m)

Methods

mzero :: Yoneda m a

mplus :: Yoneda m a -> Yoneda m a -> Yoneda m a

Comonad w => Comonad (Yoneda w)

Methods

extract :: Yoneda w a -> a

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

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

Plus f => Plus (Yoneda f)

Methods

zero :: Yoneda f a

Traversable1 f => Traversable1 (Yoneda f)

Methods

traverse1 :: Apply b => (a -> b c) -> Yoneda f a -> b (Yoneda f c)

sequence1 :: Apply a => Yoneda f (a b) -> a (Yoneda f b)

Alt f => Alt (Yoneda f)

Methods

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

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

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

Apply f => Apply (Yoneda f)

Methods

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

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

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

Bind m => Bind (Yoneda m)

Methods

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

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

Extend w => Extend (Yoneda w)

Methods

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

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

Foldable1 f => Foldable1 (Yoneda f)

Methods

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

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

Adjunction f g => Adjunction (Yoneda f) (Yoneda g)

Methods

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

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

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

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

Eq (f a) => Eq (Yoneda f a)

Methods

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

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

Ord (f a) => Ord (Yoneda f a)

Methods

compare :: Yoneda f a -> Yoneda f a -> Ordering

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

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

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

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

max :: Yoneda f a -> Yoneda f a -> Yoneda f a

min :: Yoneda f a -> Yoneda f a -> Yoneda f a

(Functor f, Read (f a)) => Read (Yoneda f a)

Methods

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

readList :: ReadS [Yoneda f a]

readPrec :: ReadPrec (Yoneda f a)

readListPrec :: ReadPrec [Yoneda f a]

Show (f a) => Show (Yoneda f a)

Methods

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

show :: Yoneda f a -> String

showList :: [Yoneda f a] -> ShowS

type Rep (Yoneda g) = Rep g

liftYoneda :: Functor f => f a -> Yoneda f a

The natural isomorphism between f and Yoneda f given by the Yoneda lemma is witnessed by liftYoneda and lowerYoneda

liftYoneda . lowerYoneda ≡ id
lowerYoneda . liftYoneda ≡ id

lowerYoneda (liftYoneda fa) =         -- definition
lowerYoneda (Yoneda (f -> fmap f a)) -- definition
(f -> fmap f fa) id                  -- beta reduction
fmap id fa                            -- functor law
fa

lift = liftYoneda

lowerYoneda :: Yoneda f a -> f a

maxF :: (Functor f, Ord (f a)) => Yoneda f a -> Yoneda f a -> Yoneda f a

minF :: (Functor f, Ord (f a)) => Yoneda f a -> Yoneda f a -> Yoneda f a

maxM :: (Monad m, Ord (m a)) => Yoneda m a -> Yoneda m a -> Yoneda m a

minM :: (Monad m, Ord (m a)) => Yoneda m a -> Yoneda m a -> Yoneda m a

as a right Kan extension

yonedaToRan :: Yoneda f a -> Ran Identity f a

Yoneda f can be viewed as the right Kan extension of f along the Identity functor.

yonedaToRan . ranToYoneda ≡ id
ranToYoneda . yonedaToRan ≡ id

ranToYoneda :: Ran Identity f a -> Yoneda f a

as a right Kan lift

yonedaToRift :: Yoneda f a -> Rift Identity f a

Yoneda f can be viewed as the right Kan lift of f along the Identity functor.

yonedaToRift . riftToYoneda ≡ id
riftToYoneda . yonedaToRift ≡ id

riftToYoneda :: Rift Identity f a -> Yoneda f a