kan-extensions-4.2.2: Kan extensions, Kan lifts, various forms of the Yoneda lemma, and (co)density (co)monads

Copyright(C) 2008-2013 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 Voightländer for more information about this type.

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

Constructors

Codensity 

Fields

runCodensity :: forall b. (a -> m b) -> m b
 

lowerCodensity :: Monad m => Codensity m a -> m 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