Copyright	(C) 2011-2013 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	MPTCs, fundeps
Safe Haskell	Trustworthy
Language	Haskell98

Data.Functor.Yoneda

Contents

as a right Kan extension
as a right Kan lift

Description

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

Synopsis

Documentation

newtype Yoneda f a Source

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 Source
Methods lower :: Comonad w => Yoneda w a -> w a
MonadTrans Yoneda Source
Methods lift :: Monad m => m a -> Yoneda m a
(Functor f, MonadFree f m) => MonadFree f (Yoneda m) Source
Methods wrap :: f (Yoneda m a) -> Yoneda m a
Monad m => Monad (Yoneda m) Source
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) Source
Methods fmap :: (a -> b) -> Yoneda f a -> Yoneda f b (<$) :: a -> Yoneda f b -> Yoneda f a
MonadFix m => MonadFix (Yoneda m) Source
Methods mfix :: (a -> Yoneda m a) -> Yoneda m a
Applicative f => Applicative (Yoneda f) Source
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) Source
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) Source
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) Source
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) Source
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) Source
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) Source
Methods mzero :: Yoneda m a mplus :: Yoneda m a -> Yoneda m a -> Yoneda m a
Comonad w => Comonad (Yoneda w) Source
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) Source
Methods zero :: Yoneda f a
Traversable1 f => Traversable1 (Yoneda f) Source
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) Source
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) Source
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) Source
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) Source
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) Source
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) Source
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) Source
Methods (==) :: Yoneda f a -> Yoneda f a -> Bool (/=) :: Yoneda f a -> Yoneda f a -> Bool
Ord (f a) => Ord (Yoneda f a) Source
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) Source
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) Source
Methods showsPrec :: Int -> Yoneda f a -> ShowS show :: Yoneda f a -> String showList :: [Yoneda f a] -> ShowS
type Rep (Yoneda g) = Rep g Source

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

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 Source

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

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

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

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

as a right Kan extension

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

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 Source

as a right Kan lift

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

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 Source