kan-extensions-5.0.2: Kan extensions, Kan lifts, various forms of the Yoneda lemma, and (co)density (co)monads

Copyright | (C) 2011-2016 Edward Kmett
License | BSD-style (see the file LICENSE)
Maintainer | Edward Kmett <ekmett@gmail.com>
Stability | provisional
Portability | GADTs, MPTCs, fundeps
Safe Haskell | Trustworthy
Language | Haskell98

Data.Functor.Coyoneda

Contents

- as a Left Kan extension

Description

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

Synopsis

- data Coyoneda f a where
  - Coyoneda :: (b -> a) -> f b -> Coyoneda f a
- liftCoyoneda :: f a -> Coyoneda f a
- lowerCoyoneda :: Functor f => Coyoneda f a -> f a
- lowerM :: Monad f => Coyoneda f a -> f a
- hoistCoyoneda :: (forall a. f a -> g a) -> Coyoneda f b -> Coyoneda g b
- coyonedaToLan :: Coyoneda f a -> Lan Identity f a
- lanToCoyoneda :: Lan Identity f a -> Coyoneda f a

# 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 # toNonEmpty :: Coyoneda f a -> NonEmpty a #
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 #