Copyright	(C) 2011-2016 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	non-portable (rank-2 polymorphism)
Safe Haskell	Safe-Inferred
Language	Haskell2010

Control.Monad.Co

Contents

Monads from Comonads
Monad Transformers from Comonads
Klesili from CoKleisli

Description

Monads from Comonads

http://comonad.com/reader/2011/monads-from-comonads/

Co can be viewed as a right Kan lift along a Comonad .

In general you can "sandwich" a monad in between two halves of an adjunction. That is to say, if you have an adjunction F -| G : C -> D then not only does GF form a monad, but GMF forms a monad for M a monad in D . Therefore if we have an adjunction F -| G : Hask -> Hask^op then we can lift a Comonad in Hask which is a Monad in Hask^op to a Monad in Hask .

For any r , the Contravariant functor / presheaf (-> r) :: Hask^op -> Hask is adjoint to the "same" Contravariant functor (-> r) :: Hask -> Hask^op . So we can sandwich a Monad in Hask^op in the middle to obtain w (a -> r-) -> r+ , and then take a coend over r to obtain forall r. w (a -> r) -> r . This gives rise to Co . If we observe that we didn't care what the choices we made for r were to finish this construction, we can upgrade to forall r. w (a -> m r) -> m r in a manner similar to how ContT is constructed yielding CoT .

We could consider unifying the definition of Co and Rift , but there are many other arguments for which Rift can form a Monad , and this wouldn't give rise to CoT .

Synopsis

type Co w = CoT w Identity
co :: Functor w => ( forall r. w (a -> r) -> r) -> Co w a
runCo :: Functor w => Co w a -> w (a -> r) -> r
newtype CoT w m a = CoT {
- runCoT :: forall r. w (a -> m r) -> m r
}
liftCoT0 :: Comonad w => ( forall a. w a -> s) -> CoT w m s
liftCoT0M :: ( Comonad w, Monad m) => ( forall a. w a -> m s) -> CoT w m s
lowerCoT0 :: ( Functor w, Monad m) => CoT w m s -> w a -> m s
lowerCo0 :: Functor w => Co w s -> w a -> s
liftCoT1 :: ( forall a. w a -> a) -> CoT w m ()
liftCoT1M :: Monad m => ( forall a. w a -> m a) -> CoT w m ()
lowerCoT1 :: ( Functor w, Monad m) => CoT w m () -> w a -> m a
lowerCo1 :: Functor w => Co w () -> w a -> a
diter :: Functor f => a -> (a -> f a) -> Density ( Cofree f) a
dctrlM :: Monad m => ( forall a. w a -> m (w a)) -> CoT ( Density w) m ()
posW :: ComonadStore s w => CoT w m s
peekW :: ComonadStore s w => s -> CoT w m ()
peeksW :: ComonadStore s w => (s -> s) -> CoT w m ()
askW :: ComonadEnv e w => CoT w m e
asksW :: ComonadEnv e w => (e -> a) -> CoT w m a
traceW :: ComonadTraced e w => e -> CoT w m ()

Monads from Comonads

type Co w = CoT w Identity Source #

co :: Functor w => ( forall r. w (a -> r) -> r) -> Co w a Source #

runCo :: Functor w => Co w a -> w (a -> r) -> r Source #

Monad Transformers from Comonads

newtype CoT w m a Source #

Co w a ~ Rift w Identity a

Constructors

CoT
Fields runCoT :: forall r. w (a -> m r) -> m r

Instances

Instances details

( Comonad w, MonadReader e m) => MonadReader e ( CoT w m) Source #
Instance details Defined in Control.Monad.Co Methods ask :: CoT w m e Source # local :: (e -> e) -> CoT w m a -> CoT w m a Source # reader :: (e -> a) -> CoT w m a Source #
( Comonad w, MonadState s m) => MonadState s ( CoT w m) Source #
Instance details Defined in Control.Monad.Co Methods get :: CoT w m s Source # put :: s -> CoT w m () Source # state :: (s -> (a, s)) -> CoT w m a Source #
( Comonad w, MonadWriter e m) => MonadWriter e ( CoT w m) Source #
Instance details Defined in Control.Monad.Co Methods writer :: (a, e) -> CoT w m a Source # tell :: e -> CoT w m () Source # listen :: CoT w m a -> CoT w m (a, e) Source # pass :: CoT w m (a, e -> e) -> CoT w m a Source #
( Comonad w, MonadError e m) => MonadError e ( CoT w m) Source #
Instance details Defined in Control.Monad.Co Methods throwError :: e -> CoT w m a Source # catchError :: CoT w m a -> (e -> CoT w m a) -> CoT w m a Source #
Comonad w => MonadTrans ( CoT w :: ( Type -> Type ) -> Type -> Type ) Source #
Instance details Defined in Control.Monad.Co Methods lift :: Monad m => m a -> CoT w m a Source #
Comonad w => Monad ( CoT w m) Source #
Instance details Defined in Control.Monad.Co Methods (>>=) :: CoT w m a -> (a -> CoT w m b) -> CoT w m b Source # (>>) :: CoT w m a -> CoT w m b -> CoT w m b Source # return :: a -> CoT w m a Source #
Functor w => Functor ( CoT w m) Source #
Instance details Defined in Control.Monad.Co Methods fmap :: (a -> b) -> CoT w m a -> CoT w m b Source # (<$) :: a -> CoT w m b -> CoT w m a Source #
( Comonad w, MonadFail m) => MonadFail ( CoT w m) Source #
Instance details Defined in Control.Monad.Co Methods fail :: String -> CoT w m a Source #
Comonad w => Applicative ( CoT w m) Source #
Instance details Defined in Control.Monad.Co Methods pure :: a -> CoT w m a Source # (<>) :: CoT w m (a -> b) -> CoT w m a -> CoT w m b Source # liftA2 :: (a -> b -> c) -> CoT w m a -> CoT w m b -> CoT w m c Source # (>) :: CoT w m a -> CoT w m b -> CoT w m b Source # (<*) :: CoT w m a -> CoT w m b -> CoT w m a Source #
( Comonad w, MonadIO m) => MonadIO ( CoT w m) Source #
Instance details Defined in Control.Monad.Co Methods liftIO :: IO a -> CoT w m a Source #
Extend w => Apply ( CoT w m) Source #
Instance details Defined in Control.Monad.Co Methods (<.>) :: CoT w m (a -> b) -> CoT w m a -> CoT w m b Source # (.>) :: CoT w m a -> CoT w m b -> CoT w m b Source # (<.) :: CoT w m a -> CoT w m b -> CoT w m a Source # liftF2 :: (a -> b -> c) -> CoT w m a -> CoT w m b -> CoT w m c Source #
Extend w => Bind ( CoT w m) Source #
Instance details Defined in Control.Monad.Co Methods (>>-) :: CoT w m a -> (a -> CoT w m b) -> CoT w m b Source # join :: CoT w m ( CoT w m a) -> CoT w m a Source #

Klesili from CoKleisli

liftCoT0 :: Comonad w => ( forall a. w a -> s) -> CoT w m s Source #

liftCoT0M :: ( Comonad w, Monad m) => ( forall a. w a -> m s) -> CoT w m s Source #

lowerCoT0 :: ( Functor w, Monad m) => CoT w m s -> w a -> m s Source #

lowerCo0 :: Functor w => Co w s -> w a -> s Source #

liftCoT1 :: ( forall a. w a -> a) -> CoT w m () Source #

liftCoT1M :: Monad m => ( forall a. w a -> m a) -> CoT w m () Source #

lowerCoT1 :: ( Functor w, Monad m) => CoT w m () -> w a -> m a Source #

lowerCo1 :: Functor w => Co w () -> w a -> a Source #

diter :: Functor f => a -> (a -> f a) -> Density ( Cofree f) a Source #

dctrlM :: Monad m => ( forall a. w a -> m (w a)) -> CoT ( Density w) m () Source #

posW :: ComonadStore s w => CoT w m s Source #

peekW :: ComonadStore s w => s -> CoT w m () Source #

peeksW :: ComonadStore s w => (s -> s) -> CoT w m () Source #

askW :: ComonadEnv e w => CoT w m e Source #

asksW :: ComonadEnv e w => (e -> a) -> CoT w m a Source #

traceW :: ComonadTraced e w => e -> CoT w m () Source #