Control.Monad.Co

kan-extensions-5.0.1: Kan extensions, Kan lifts, various forms of the Yoneda lemma, and (co)density (co)monads

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
Language | Haskell98

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 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`.

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

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

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

# Monad Transformers from Comonads

newtype CoT w m a

    Co w a ~ Rift w Identity a

Constructors

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

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

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 ()