| Copyright | (C) 2011 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 |
| Language | Haskell98 |
Control.Monad.Co
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 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.
- 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 :: (Comonad w, Monad m) => (forall a. w a -> m (w a)) -> CoT (Density w) m ()
- posW :: (ComonadStore s w, Monad m) => CoT w m s
- peekW :: (ComonadStore s w, Monad m) => s -> CoT w m ()
- peeksW :: (ComonadStore s w, Monad m) => (s -> s) -> CoT w m ()
- askW :: (ComonadEnv e w, Monad m) => CoT w m e
- asksW :: (ComonadEnv e w, Monad m) => (e -> a) -> CoT w m a
- traceW :: (ComonadTraced e w, Monad m) => e -> CoT w m ()
Monads from Comonads
Monad Transformers from Comonads
newtype CoT w m a
Instances
| (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
posW :: (ComonadStore s w, Monad m) => CoT w m s
peekW :: (ComonadStore s w, Monad m) => s -> CoT w m ()
peeksW :: (ComonadStore s w, Monad m) => (s -> s) -> CoT w m ()
askW :: (ComonadEnv e w, Monad m) => CoT w m e
asksW :: (ComonadEnv e w, Monad m) => (e -> a) -> CoT w m a
traceW :: (ComonadTraced e w, Monad m) => e -> CoT w m ()