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

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




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 sandwhich 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.


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




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


(Comonad w, MonadState s m) => MonadState s (CoT w m) 
(Comonad w, MonadReader e m) => MonadReader e (CoT w m) 
(Comonad w, MonadError e m) => MonadError e (CoT w m) 
(Comonad w, MonadWriter e m) => MonadWriter e (CoT w m) 
Comonad w => MonadTrans (CoT w) 
Comonad w => Monad (CoT w m) 
Functor w => Functor (CoT w m) 
Comonad w => Applicative (CoT w m) 
(Comonad w, MonadIO m) => MonadIO (CoT w m) 
Extend w => Apply (CoT w m) 
Extend w => Bind (CoT w m) 

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

posW :: (ComonadStore s w, Monad m) => CoT w m s Source

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

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

askW :: (ComonadEnv e w, Monad m) => CoT w m e Source

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

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