adjunctions-4.3: Adjunctions and representable functors

Copyright(c) Edward Kmett 2011-2014
LicenseBSD3
Maintainerekmett@gmail.com
Stabilityexperimental
Safe HaskellNone
LanguageHaskell98

Data.Functor.Rep

Contents

Description

Representable endofunctors over the category of Haskell types are isomorphic to the reader monad and so inherit a very large number of properties for free.

Synopsis

Representable Functors

class Distributive f => Representable f where #

A Functor f is Representable if tabulate and index witness an isomorphism to (->) x.

Every Distributive Functor is actually Representable.

Every Representable Functor from Hask to Hask is a right adjoint.

tabulate . index  ≡ id
index . tabulate  ≡ id
tabulate . return ≡ return

Minimal complete definition

tabulate, index

Associated Types

type Rep f :: * #

Methods

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

fmap f . tabulate ≡ tabulate . fmap f

index :: f a -> Rep f -> a #

Instances

Representable Identity # 

Associated Types

type Rep (Identity :: * -> *) :: * #

Methods

tabulate :: (Rep Identity -> a) -> Identity a #

index :: Identity a -> Rep Identity -> a #

Representable Complex # 

Associated Types

type Rep (Complex :: * -> *) :: * #

Methods

tabulate :: (Rep Complex -> a) -> Complex a #

index :: Complex a -> Rep Complex -> a #

Representable Dual # 

Associated Types

type Rep (Dual :: * -> *) :: * #

Methods

tabulate :: (Rep Dual -> a) -> Dual a #

index :: Dual a -> Rep Dual -> a #

Representable Sum # 

Associated Types

type Rep (Sum :: * -> *) :: * #

Methods

tabulate :: (Rep Sum -> a) -> Sum a #

index :: Sum a -> Rep Sum -> a #

Representable Product # 

Associated Types

type Rep (Product :: * -> *) :: * #

Methods

tabulate :: (Rep Product -> a) -> Product a #

index :: Product a -> Rep Product -> a #

Representable ((->) e) # 

Associated Types

type Rep ((->) e :: * -> *) :: * #

Methods

tabulate :: (Rep ((->) e) -> a) -> e -> a #

index :: (e -> a) -> Rep ((->) e) -> a #

Representable (Proxy *) # 

Associated Types

type Rep (Proxy * :: * -> *) :: * #

Methods

tabulate :: (Rep (Proxy *) -> a) -> Proxy * a #

index :: Proxy * a -> Rep (Proxy *) -> a #

Representable f => Representable (Cofree f) # 

Associated Types

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

Methods

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

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

Representable f => Representable (Co f) # 

Associated Types

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

Methods

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

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

Representable w => Representable (TracedT s w) # 

Associated Types

type Rep (TracedT s w :: * -> *) :: * #

Methods

tabulate :: (Rep (TracedT s w) -> a) -> TracedT s w a #

index :: TracedT s w a -> Rep (TracedT s w) -> a #

Representable m => Representable (IdentityT * m) # 

Associated Types

type Rep (IdentityT * m :: * -> *) :: * #

Methods

tabulate :: (Rep (IdentityT * m) -> a) -> IdentityT * m a #

index :: IdentityT * m a -> Rep (IdentityT * m) -> a #

Representable (Tagged * t) # 

Associated Types

type Rep (Tagged * t :: * -> *) :: * #

Methods

tabulate :: (Rep (Tagged * t) -> a) -> Tagged * t a #

index :: Tagged * t a -> Rep (Tagged * t) -> a #

(Representable f, Representable m) => Representable (ReaderT f m) # 

Associated Types

type Rep (ReaderT f m :: * -> *) :: * #

Methods

tabulate :: (Rep (ReaderT f m) -> a) -> ReaderT f m a #

index :: ReaderT f m a -> Rep (ReaderT f m) -> a #

(Representable f, Representable g) => Representable (Product * f g) # 

Associated Types

type Rep (Product * f g :: * -> *) :: * #

Methods

tabulate :: (Rep (Product * f g) -> a) -> Product * f g a #

index :: Product * f g a -> Rep (Product * f g) -> a #

Representable m => Representable (ReaderT * e m) # 

Associated Types

type Rep (ReaderT * e m :: * -> *) :: * #

Methods

tabulate :: (Rep (ReaderT * e m) -> a) -> ReaderT * e m a #

index :: ReaderT * e m a -> Rep (ReaderT * e m) -> a #

(Representable f, Representable g) => Representable (Compose * * f g) # 

Associated Types

type Rep (Compose * * f g :: * -> *) :: * #

Methods

tabulate :: (Rep (Compose * * f g) -> a) -> Compose * * f g a #

index :: Compose * * f g a -> Rep (Compose * * f g) -> a #

tabulated :: (Representable f, Representable g, Profunctor p, Functor h) => p (f a) (h (g b)) -> p (Rep f -> a) (h (Rep g -> b)) #

tabulate and index form two halves of an isomorphism.

This can be used with the combinators from the lens package.

tabulated :: Representable f => Iso' (Rep f -> a) (f a)

Wrapped representable functors

newtype Co f a #

Constructors

Co 

Fields

Instances

ComonadTrans Co # 

Methods

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

(Representable f, (~) * (Rep f) a) => MonadReader a (Co f) # 

Methods

ask :: Co f a #

local :: (a -> a) -> Co f a -> Co f a #

reader :: (a -> a) -> Co f a #

Representable f => Monad (Co f) # 

Methods

(>>=) :: Co f a -> (a -> Co f b) -> Co f b #

(>>) :: Co f a -> Co f b -> Co f b #

return :: a -> Co f a #

fail :: String -> Co f a #

Functor f => Functor (Co f) # 

Methods

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

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

Representable f => Applicative (Co f) # 

Methods

pure :: a -> Co f a #

(<*>) :: Co f (a -> b) -> Co f a -> Co f b #

(*>) :: Co f a -> Co f b -> Co f b #

(<*) :: Co f a -> Co f b -> Co f a #

(Representable f, Monoid (Rep f)) => Comonad (Co f) # 

Methods

extract :: Co f a -> a #

duplicate :: Co f a -> Co f (Co f a) #

extend :: (Co f a -> b) -> Co f a -> Co f b #

Representable f => Distributive (Co f) # 

Methods

distribute :: Functor f => f (Co f a) -> Co f (f a) #

collect :: Functor f => (a -> Co f b) -> f a -> Co f (f b) #

distributeM :: Monad m => m (Co f a) -> Co f (m a) #

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

Representable f => Apply (Co f) # 

Methods

(<.>) :: Co f (a -> b) -> Co f a -> Co f b #

(.>) :: Co f a -> Co f b -> Co f b #

(<.) :: Co f a -> Co f b -> Co f a #

Representable f => Bind (Co f) # 

Methods

(>>-) :: Co f a -> (a -> Co f b) -> Co f b #

join :: Co f (Co f a) -> Co f a #

(Representable f, Semigroup (Rep f)) => Extend (Co f) # 

Methods

duplicated :: Co f a -> Co f (Co f a) #

extended :: (Co f a -> b) -> Co f a -> Co f b #

Representable f => Representable (Co f) # 

Associated Types

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

Methods

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

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

type Rep (Co f) # 
type Rep (Co f) = Rep f

Default definitions

Functor

fmapRep :: Representable f => (a -> b) -> f a -> f b #

Distributive

distributeRep :: (Representable f, Functor w) => w (f a) -> f (w a) #

Apply/Applicative

apRep :: Representable f => f (a -> b) -> f a -> f b #

pureRep :: Representable f => a -> f a #

liftR2 :: Representable f => (a -> b -> c) -> f a -> f b -> f c #

liftR3 :: Representable f => (a -> b -> c -> d) -> f a -> f b -> f c -> f d #

Bind/Monad

bindRep :: Representable f => f a -> (a -> f b) -> f b #

MonadFix

mfixRep :: Representable f => (a -> f a) -> f a #

MonadZip

mzipRep :: Representable f => f a -> f b -> f (a, b) #

mzipWithRep :: Representable f => (a -> b -> c) -> f a -> f b -> f c #

MonadReader

askRep :: Representable f => f (Rep f) #

localRep :: Representable f => (Rep f -> Rep f) -> f a -> f a #

Extend

duplicatedRep :: (Representable f, Semigroup (Rep f)) => f a -> f (f a) #

extendedRep :: (Representable f, Semigroup (Rep f)) => (f a -> b) -> f a -> f b #

Comonad

duplicateRep :: (Representable f, Monoid (Rep f)) => f a -> f (f a) #

extendRep :: (Representable f, Monoid (Rep f)) => (f a -> b) -> f a -> f b #

extractRep :: (Representable f, Monoid (Rep f)) => f a -> a #

Comonad, with user-specified monoid

duplicateRepBy :: Representable f => (Rep f -> Rep f -> Rep f) -> f a -> f (f a) #

extendRepBy :: Representable f => (Rep f -> Rep f -> Rep f) -> (f a -> b) -> f a -> f b #

extractRepBy :: Representable f => Rep f -> f a -> a #