Data.Distributive

distributive-0.6.2.1: Distributive functors -- Dual to Traversable

Copyright | (C) 2011-2016 Edward Kmett
License | BSD-style (see the file LICENSE)
Maintainer | Edward Kmett <ekmett@gmail.com>
Stability | provisional
Portability | portable
Safe Haskell | Trustworthy
Language | Haskell2010

Data.Distributive

Description

Synopsis

- class Functor g => Distributive g where
  - distribute :: Functor f => f (g a) -> g (f a)
  - collect :: Functor f => (a -> g b) -> f a -> g (f b)
  - distributeM :: Monad m => m (g a) -> g (m a)
  - collectM :: Monad m => (a -> g b) -> m a -> g (m b)
- cotraverse :: (Distributive g, Functor f) => (f a -> b) -> f (g a) -> g b
- comapM :: (Distributive g, Monad m) => (m a -> b) -> m (g a) -> g b

# Documentation

class Functor g => Distributive g where #

This is the categorical dual of `Traversable`.

Due to the lack of non-trivial comonoids in Haskell, we can restrict ourselves to requiring a `Functor` rather than some Coapplicative class. Categorically every `Distributive` functor is actually a right adjoint, and so it must be `Representable` endofunctor and preserve all limits. This is a fancy way of saying it is isomorphic to `(->) x` for some x.

To be distributable a container will need to have a way to consistently zip a potentially infinite number of copies of itself. This effectively means that the holes in all values of that type, must have the same cardinality, fixed sized vectors, infinite streams, functions, etc. and no extra information to try to merge together.

Minimal complete definition

distribute | collect

Methods

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

The dual of `sequenceA`

    >>> distribute [(+1),(+2)] 1
    [2,3]

    distribute = collect id
    distribute . distribute = id

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

    collect f = distribute . fmap f
    fmap f = runIdentity . collect (Identity . f)
    fmap distribute . collect f = getCompose . collect (Compose . f)

distributeM :: Monad m => m (g a) -> g (m a) #

The dual of `sequence`

    distributeM = fmap unwrapMonad . distribute . WrapMonad

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

    collectM = distributeM . liftM f

#### Instances

Instances details

Distributive Complex # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (Complex a) -> Complex (f a) # collect :: Functor f => (a -> Complex b) -> f a -> Complex (f b) # distributeM :: Monad m => m (Complex a) -> Complex (m a) # collectM :: Monad m => (a -> Complex b) -> m a -> Complex (m b) #
Distributive Identity # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (Identity a) -> Identity (f a) # collect :: Functor f => (a -> Identity b) -> f a -> Identity (f b) # distributeM :: Monad m => m (Identity a) -> Identity (m a) # collectM :: Monad m => (a -> Identity b) -> m a -> Identity (m b) #
Distributive First # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (First a) -> First (f a) # collect :: Functor f => (a -> First b) -> f a -> First (f b) # distributeM :: Monad m => m (First a) -> First (m a) # collectM :: Monad m => (a -> First b) -> m a -> First (m b) #
Distributive Last # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (Last a) -> Last (f a) # collect :: Functor f => (a -> Last b) -> f a -> Last (f b) # distributeM :: Monad m => m (Last a) -> Last (m a) # collectM :: Monad m => (a -> Last b) -> m a -> Last (m b) #
Distributive Max # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (Max a) -> Max (f a) # collect :: Functor f => (a -> Max b) -> f a -> Max (f b) # distributeM :: Monad m => m (Max a) -> Max (m a) # collectM :: Monad m => (a -> Max b) -> m a -> Max (m b) #
Distributive Min # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (Min a) -> Min (f a) # collect :: Functor f => (a -> Min b) -> f a -> Min (f b) # distributeM :: Monad m => m (Min a) -> Min (m a) # collectM :: Monad m => (a -> Min b) -> m a -> Min (m b) #
Distributive Dual # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (Dual a) -> Dual (f a) # collect :: Functor f => (a -> Dual b) -> f a -> Dual (f b) # distributeM :: Monad m => m (Dual a) -> Dual (m a) # collectM :: Monad m => (a -> Dual b) -> m a -> Dual (m b) #
Distributive Product # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (Product a) -> Product (f a) # collect :: Functor f => (a -> Product b) -> f a -> Product (f b) # distributeM :: Monad m => m (Product a) -> Product (m a) # collectM :: Monad m => (a -> Product b) -> m a -> Product (m b) #
Distributive Sum # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (Sum a) -> Sum (f a) # collect :: Functor f => (a -> Sum b) -> f a -> Sum (f b) # distributeM :: Monad m => m (Sum a) -> Sum (m a) # collectM :: Monad m => (a -> Sum b) -> m a -> Sum (m b) #
Distributive Par1 # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (Par1 a) -> Par1 (f a) # collect :: Functor f => (a -> Par1 b) -> f a -> Par1 (f b) # distributeM :: Monad m => m (Par1 a) -> Par1 (m a) # collectM :: Monad m => (a -> Par1 b) -> m a -> Par1 (m b) #
(Distributive m, Monad m) => Distributive (WrappedMonad m) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (WrappedMonad m a) -> WrappedMonad m (f a) # collect :: Functor f => (a -> WrappedMonad m b) -> f a -> WrappedMonad m (f b) # distributeM :: Monad m0 => m0 (WrappedMonad m a) -> WrappedMonad m (m0 a) # collectM :: Monad m0 => (a -> WrappedMonad m b) -> m0 a -> WrappedMonad m (m0 b) #
Distributive (Proxy :: Type -> Type) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (Proxy a) -> Proxy (f a) # collect :: Functor f => (a -> Proxy b) -> f a -> Proxy (f b) # distributeM :: Monad m => m (Proxy a) -> Proxy (m a) # collectM :: Monad m => (a -> Proxy b) -> m a -> Proxy (m b) #
Distributive (U1 :: Type -> Type) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (U1 a) -> U1 (f a) # collect :: Functor f => (a -> U1 b) -> f a -> U1 (f b) # distributeM :: Monad m => m (U1 a) -> U1 (m a) # collectM :: Monad m => (a -> U1 b) -> m a -> U1 (m b) #
Distributive f => Distributive (Rec1 f) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f0 => f0 (Rec1 f a) -> Rec1 f (f0 a) # collect :: Functor f0 => (a -> Rec1 f b) -> f0 a -> Rec1 f (f0 b) # distributeM :: Monad m => m (Rec1 f a) -> Rec1 f (m a) # collectM :: Monad m => (a -> Rec1 f b) -> m a -> Rec1 f (m b) #
Distributive (Tagged t) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (Tagged t a) -> Tagged t (f a) # collect :: Functor f => (a -> Tagged t b) -> f a -> Tagged t (f b) # distributeM :: Monad m => m (Tagged t a) -> Tagged t (m a) # collectM :: Monad m => (a -> Tagged t b) -> m a -> Tagged t (m b) #
Distributive f => Distributive (Backwards f) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f0 => f0 (Backwards f a) -> Backwards f (f0 a) # collect :: Functor f0 => (a -> Backwards f b) -> f0 a -> Backwards f (f0 b) # distributeM :: Monad m => m (Backwards f a) -> Backwards f (m a) # collectM :: Monad m => (a -> Backwards f b) -> m a -> Backwards f (m b) #
Distributive g => Distributive (IdentityT g) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (IdentityT g a) -> IdentityT g (f a) # collect :: Functor f => (a -> IdentityT g b) -> f a -> IdentityT g (f b) # distributeM :: Monad m => m (IdentityT g a) -> IdentityT g (m a) # collectM :: Monad m => (a -> IdentityT g b) -> m a -> IdentityT g (m b) #
Distributive g => Distributive (ReaderT e g) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (ReaderT e g a) -> ReaderT e g (f a) # collect :: Functor f => (a -> ReaderT e g b) -> f a -> ReaderT e g (f b) # distributeM :: Monad m => m (ReaderT e g a) -> ReaderT e g (m a) # collectM :: Monad m => (a -> ReaderT e g b) -> m a -> ReaderT e g (m b) #
Distributive f => Distributive (Reverse f) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f0 => f0 (Reverse f a) -> Reverse f (f0 a) # collect :: Functor f0 => (a -> Reverse f b) -> f0 a -> Reverse f (f0 b) # distributeM :: Monad m => m (Reverse f a) -> Reverse f (m a) # collectM :: Monad m => (a -> Reverse f b) -> m a -> Reverse f (m b) #
(Distributive f, Distributive g) => Distributive (Product f g) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f0 => f0 (Product f g a) -> Product f g (f0 a) # collect :: Functor f0 => (a -> Product f g b) -> f0 a -> Product f g (f0 b) # distributeM :: Monad m => m (Product f g a) -> Product f g (m a) # collectM :: Monad m => (a -> Product f g b) -> m a -> Product f g (m b) #
(Distributive a, Distributive b) => Distributive (a :*: b) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f ((a :*: b) a0) -> (a :*: b) (f a0) # collect :: Functor f => (a0 -> (a :*: b) b0) -> f a0 -> (a :*: b) (f b0) # distributeM :: Monad m => m ((a :*: b) a0) -> (a :*: b) (m a0) # collectM :: Monad m => (a0 -> (a :*: b) b0) -> m a0 -> (a :*: b) (m b0) #
Distributive ((->) e) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f (e -> a) -> e -> f a # collect :: Functor f => (a -> e -> b) -> f a -> e -> f b # distributeM :: Monad m => m (e -> a) -> e -> m a # collectM :: Monad m => (a -> e -> b) -> m a -> e -> m b #
(Distributive f, Distributive g) => Distributive (Compose f g) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f0 => f0 (Compose f g a) -> Compose f g (f0 a) # collect :: Functor f0 => (a -> Compose f g b) -> f0 a -> Compose f g (f0 b) # distributeM :: Monad m => m (Compose f g a) -> Compose f g (m a) # collectM :: Monad m => (a -> Compose f g b) -> m a -> Compose f g (m b) #
(Distributive a, Distributive b) => Distributive (a :.: b) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f => f ((a :.: b) a0) -> (a :.: b) (f a0) # collect :: Functor f => (a0 -> (a :.: b) b0) -> f a0 -> (a :.: b) (f b0) # distributeM :: Monad m => m ((a :.: b) a0) -> (a :.: b) (m a0) # collectM :: Monad m => (a0 -> (a :.: b) b0) -> m a0 -> (a :.: b) (m b0) #
Distributive f => Distributive (M1 i c f) # |
Instance details Defined in Data.Distributive Methods distribute :: Functor f0 => f0 (M1 i c f a) -> M1 i c f (f0 a) # collect :: Functor f0 => (a -> M1 i c f b) -> f0 a -> M1 i c f (f0 b) # distributeM :: Monad m => m (M1 i c f a) -> M1 i c f (m a) # collectM :: Monad m => (a -> M1 i c f b) -> m a -> M1 i c f (m b) #

cotraverse :: (Distributive g, Functor f) => (f a -> b) -> f (g a) -> g b #

The dual of `traverse`

    cotraverse f = fmap f . distribute

comapM :: (Distributive g, Monad m) => (m a -> b) -> m (g a) -> g b #

The dual of `mapM`

    comapM f = fmap f . distributeM