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

Copyright (C) 2011-2016 Edward Kmett BSD-style (see the file LICENSE) Edward Kmett provisional non-portable (rank-2 polymorphism) Safe Haskell98

Description

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 #

Co w a ~ Rift w Identity a


Constructors

 CoT FieldsrunCoT :: forall r. w (a -> m r) -> m r
Instances

# 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 () #