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

Copyright(C) 2008-2016 Edward Kmett
LicenseBSD-style (see the file LICENSE)
MaintainerEdward Kmett <ekmett@gmail.com>
Stabilityprovisional
Portabilitynon-portable (rank-2 polymorphism)
Safe HaskellTrustworthy
LanguageHaskell98

Control.Monad.Codensity

Description

 

Synopsis

Documentation

newtype Codensity m a Source #

Codensity f is the Monad generated by taking the right Kan extension of any Functor f along itself (Ran f f).

This can often be more "efficient" to construct than f itself using repeated applications of (>>=).

See "Asymptotic Improvement of Computations over Free Monads" by Janis Voigtländer for more information about this type.

http://www.iai.uni-bonn.de/~jv/mpc08.pdf

Constructors

Codensity 

Fields

Instances

MonadTrans Codensity Source # 

Methods

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

MonadReader r m => MonadState r (Codensity m) Source # 

Methods

get :: Codensity m r #

put :: r -> Codensity m () #

state :: (r -> (a, r)) -> Codensity m a #

MonadReader r m => MonadReader r (Codensity m) Source # 

Methods

ask :: Codensity m r #

local :: (r -> r) -> Codensity m a -> Codensity m a #

reader :: (r -> a) -> Codensity m a #

(Functor f, MonadFree f m) => MonadFree f (Codensity m) Source # 

Methods

wrap :: f (Codensity m a) -> Codensity m a #

Monad (Codensity f) Source # 

Methods

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

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

return :: a -> Codensity f a #

fail :: String -> Codensity f a #

Functor (Codensity k) Source # 

Methods

fmap :: (a -> b) -> Codensity k a -> Codensity k b #

(<$) :: a -> Codensity k b -> Codensity k a #

MonadFail f => MonadFail (Codensity f) Source # 

Methods

fail :: String -> Codensity f a #

Applicative (Codensity f) Source # 

Methods

pure :: a -> Codensity f a #

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

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

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

MonadIO m => MonadIO (Codensity m) Source # 

Methods

liftIO :: IO a -> Codensity m a #

Alternative v => Alternative (Codensity v) Source # 

Methods

empty :: Codensity v a #

(<|>) :: Codensity v a -> Codensity v a -> Codensity v a #

some :: Codensity v a -> Codensity v [a] #

many :: Codensity v a -> Codensity v [a] #

Alternative v => MonadPlus (Codensity v) Source # 

Methods

mzero :: Codensity v a #

mplus :: Codensity v a -> Codensity v a -> Codensity v a #

Plus v => Plus (Codensity v) Source # 

Methods

zero :: Codensity v a #

Alt v => Alt (Codensity v) Source # 

Methods

(<!>) :: Codensity v a -> Codensity v a -> Codensity v a #

some :: Applicative (Codensity v) => Codensity v a -> Codensity v [a] #

many :: Applicative (Codensity v) => Codensity v a -> Codensity v [a] #

Apply (Codensity f) Source # 

Methods

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

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

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

lowerCodensity :: Applicative f => Codensity f a -> f a Source #

This serves as the *left*-inverse (retraction) of lift.

lowerCodensity . liftid

In general this is not a full 2-sided inverse, merely a retraction, as Codensity m is often considerably "larger" than m.

e.g. Codensity ((->) s)) a ~ forall r. (a -> s -> r) -> s -> r could support a full complement of MonadState s actions, while (->) s is limited to MonadReader s actions.

codensityToAdjunction :: Adjunction f g => Codensity g a -> g (f a) Source #

The Codensity monad of a right adjoint is isomorphic to the monad obtained from the Adjunction.

codensityToAdjunction . adjunctionToCodensityid
adjunctionToCodensity . codensityToAdjunctionid

codensityToRan :: Codensity g a -> Ran g g a Source #

The Codensity Monad of a Functor g is the right Kan extension (Ran) of g along itself.

codensityToRan . ranToCodensityid
ranToCodensity . codensityToRanid

codensityToComposedRep :: Representable u => Codensity u a -> u (Rep u, a) Source #

The Codensity monad of a representable Functor is isomorphic to the monad obtained from the Adjunction for which that Functor is the right adjoint.

codensityToComposedRep . composedRepToCodensityid
composedRepToCodensity . codensityToComposedRepid
codensityToComposedRep = ranToComposedRep . codensityToRan

improve :: Functor f => (forall m. MonadFree f m => m a) -> Free f a Source #

Right associate all binds in a computation that generates a free monad

This can improve the asymptotic efficiency of the result, while preserving semantics.

See "Asymptotic Improvement of Computations over Free Monads" by Janis Voightländer for more information about this combinator.

http://www.iai.uni-bonn.de/~jv/mpc08.pdf