kan-extensions-5.2: Kan extensions, Kan lifts, the Yoneda lemma, and (co)density (co)monads

Copyright(C) 2011-2016 Edward Kmett
LicenseBSD-style (see the file LICENSE)
MaintainerEdward Kmett <ekmett@gmail.com>
Stabilityprovisional
Portabilitynon-portable (rank-2 polymorphism)
Safe HaskellSafe
LanguageHaskell98

Control.Monad.Co

Contents

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

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
(Comonad w, MonadReader e m) => MonadReader e (CoT w m) Source # 
Instance details

Defined in Control.Monad.Co

Methods

ask :: CoT w m e #

local :: (e -> e) -> CoT w m a -> CoT w m a #

reader :: (e -> a) -> CoT w m a #

(Comonad w, MonadState s m) => MonadState s (CoT w m) Source # 
Instance details

Defined in Control.Monad.Co

Methods

get :: CoT w m s #

put :: s -> CoT w m () #

state :: (s -> (a, s)) -> CoT w m a #

(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 #

tell :: e -> CoT w m () #

listen :: CoT w m a -> CoT w m (a, e) #

pass :: CoT w m (a, e -> e) -> CoT w m a #

(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 #

catchError :: CoT w m a -> (e -> CoT w m a) -> CoT w m a #

Comonad w => MonadTrans (CoT w :: (* -> *) -> * -> *) Source # 
Instance details

Defined in Control.Monad.Co

Methods

lift :: Monad m => m a -> CoT w m a #

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 #

(>>) :: CoT w m a -> CoT w m b -> CoT w m b #

return :: a -> CoT w m a #

fail :: String -> CoT w m a #

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 #

(<$) :: a -> CoT w m b -> CoT w m a #

(Comonad w, MonadFail m) => MonadFail (CoT w m) Source # 
Instance details

Defined in Control.Monad.Co

Methods

fail :: String -> CoT w m a #

Comonad w => Applicative (CoT w m) Source # 
Instance details

Defined in Control.Monad.Co

Methods

pure :: a -> CoT w m a #

(<*>) :: CoT w m (a -> b) -> CoT w m a -> CoT w m b #

liftA2 :: (a -> b -> c) -> CoT w m a -> CoT w m b -> CoT w m c #

(*>) :: CoT w m a -> CoT w m b -> CoT w m b #

(<*) :: CoT w m a -> CoT w m b -> CoT w m a #

(Comonad w, MonadIO m) => MonadIO (CoT w m) Source # 
Instance details

Defined in Control.Monad.Co

Methods

liftIO :: IO a -> CoT w m a #

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 #

(.>) :: CoT w m a -> CoT w m b -> CoT w m b #

(<.) :: CoT w m a -> CoT w m b -> CoT w m a #

liftF2 :: (a -> b -> c) -> CoT w m a -> CoT w m b -> CoT w m c #

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 #

join :: CoT w m (CoT w m a) -> CoT w m a #

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 #