distributive-0.5.1: 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 | Safe
Language | Haskell98

Data.Distributive

Description

Synopsis

- class Functor g => Distributive g where
- 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
- fmapCollect :: Distributive f => (a -> b) -> f a -> f 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 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

Distributive U1 # |
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 Par1 # |
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 Identity # |
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 Min # |
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 Max # |
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 First # |
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 # |
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 Complex # |
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 Dual # |
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 Sum # |
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 Product # |
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 ((->) e) # |
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 (Rec1 f) # |
Methods distribute :: Functor f => f (Rec1 f a) -> Rec1 f (f a) # collect :: Functor f => (a -> Rec1 f b) -> f a -> Rec1 f (f 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 (Proxy *) # |
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 a, Distributive b) => Distributive ((:*:) a b) # |
Methods distribute :: Functor f => f ((a :*: b) a) -> (a :*: b) (f a) # collect :: Functor f => (a -> (a :*: b) b) -> f a -> (a :*: b) (f b) # distributeM :: Monad m => m ((a :*: b) a) -> (a :*: b) (m a) # collectM :: Monad m => (a -> (a :*: b) b) -> m a -> (a :*: b) (m b) #
(Distributive a, Distributive b) => Distributive ((:.:) a b) # |
Methods distribute :: Functor f => f ((a :.: b) a) -> (a :.: b) (f a) # collect :: Functor f => (a -> (a :.: b) b) -> f a -> (a :.: b) (f b) # distributeM :: Monad m => m ((a :.: b) a) -> (a :.: b) (m a) # collectM :: Monad m => (a -> (a :.: b) b) -> m a -> (a :.: b) (m b) #
Distributive (Tagged * t) # |
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 (Reverse * f) # |
Methods distribute :: Functor f => f (Reverse * f a) -> Reverse * f (f a) # collect :: Functor f => (a -> Reverse * f b) -> f a -> Reverse * f (f 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 (Backwards * f) # |
Methods distribute :: Functor f => f (Backwards * f a) -> Backwards * f (f a) # collect :: Functor f => (a -> Backwards * f b) -> f a -> Backwards * f (f 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) # |
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 f => Distributive (M1 i c f) # |
Methods distribute :: Functor f => f (M1 i c f a) -> M1 i c f (f a) # collect :: Functor f => (a -> M1 i c f b) -> f a -> M1 i c f (f 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) #
(Distributive f, Distributive g) => Distributive (Product * f g) # |
Methods distribute :: Functor f => f (Product * f g a) -> Product * f g (f a) # collect :: Functor f => (a -> Product * f g b) -> f a -> Product * f g (f 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 g => Distributive (ReaderT * e g) # |
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 g) => Distributive (Compose * * f g) # |
Methods distribute :: Functor f => f (Compose * * f g a) -> Compose * * f g (f a) # collect :: Functor f => (a -> Compose * * f g b) -> f a -> Compose * * f g (f 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) #

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

fmapCollect :: Distributive f => (a -> b) -> f a -> f b #

`fmapCollect` is a viable default definition for `fmap` given a `Distributive` instance defined in terms of `collect`.