kan-extensions-5.1: Kan extensions, Kan lifts, the Yoneda lemma, and (co)density (co)monads

Copyright(C) 2011-2016 Edward Kmett
LicenseBSD-style (see the file LICENSE)
MaintainerEdward Kmett <ekmett@gmail.com>
Stabilityprovisional
PortabilityGADTs, MPTCs, fundeps
Safe HaskellTrustworthy
LanguageHaskell98

Data.Functor.Coyoneda

Contents

Description

The co-Yoneda lemma for a covariant Functor f states that Coyoneda f is naturally isomorphic to f.

Synopsis

Documentation

data Coyoneda f a where Source #

A covariant Functor suitable for Yoneda reduction

Constructors

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

Instances

ComonadTrans Coyoneda Source # 

Methods

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

MonadTrans Coyoneda Source # 

Methods

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

Monad m => Monad (Coyoneda m) Source # 

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) Source # 

Methods

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

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

MonadFix f => MonadFix (Coyoneda f) Source # 

Methods

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

Applicative f => Applicative (Coyoneda f) Source # 

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) Source # 

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) Source # 

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) Source # 

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) Source # 

Associated Types

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

Methods

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

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

(Functor f, Eq1 f) => Eq1 (Coyoneda f) Source # 

Methods

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

(Functor f, Ord1 f) => Ord1 (Coyoneda f) Source # 

Methods

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

Read1 f => Read1 (Coyoneda f) Source # 

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) Source # 

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) Source # 

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) Source # 

Methods

mzero :: Coyoneda f a #

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

Comonad w => Comonad (Coyoneda w) Source # 

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) Source # 

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) #

Foldable1 f => Foldable1 (Coyoneda f) Source # 

Methods

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

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

toNonEmpty :: Coyoneda f a -> NonEmpty a #

Plus f => Plus (Coyoneda f) Source # 

Methods

zero :: Coyoneda f a #

Alt f => Alt (Coyoneda f) Source # 

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) Source # 

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) Source # 

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) Source # 

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) Source # 

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) Source # 

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) Source # 

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

Methods

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

show :: Coyoneda f a -> String #

showList :: [Coyoneda f a] -> ShowS #

type Rep (Coyoneda f) Source # 
type Rep (Coyoneda f) = Rep f

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

Yoneda "expansion"

liftCoyoneda . lowerCoyonedaid
lowerCoyoneda . liftCoyonedaid
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 . lanToCoyonedaid
lanToCoyoneda . coyonedaToLanid