Data.Functor.Kan.Rift

Right and Left Kan lifts for functors over Hask, where they exist.

http://ncatlab.org/nlab/show/Kan+lift

Right Kan lifts

newtype Rift g h a

g . Rift g f => f

This could alternately be defined directly from the (co)universal propertly in which case, we'd get toRift = UniversalRift, but then the usage would suffer.

data UniversalRift g f a = forall z. Functor z =>
     UniversalRift (forall x. g (z x) -> f x) (z a)

We can witness the isomorphism between Rift and UniversalRift using:

riftIso1 :: Functor g => UniversalRift g f a -> Rift g f a
riftIso1 (UniversalRift h z) = Rift $ \g -> h $ fmap (\k -> k <$> z) g

riftIso2 :: Rift g f a -> UniversalRift g f a
riftIso2 (Rift e) = UniversalRift e id

riftIso1 (riftIso2 (Rift h)) =
riftIso1 (UniversalRift h id) =          -- by definition
Rift $ \g -> h $ fmap (\k -> k <$> id) g -- by definition
Rift $ \g -> h $ fmap id g               -- <$> = (.) and (.id)
Rift $ \g -> h g                         -- by functor law
Rift h                                   -- eta reduction

The other direction is left as an exercise for the reader.

There are several monads that we can form from Rift.

When g is corepresentable (e.g. is a right adjoint) then there exists x such that g ~ (->) x, then it follows that

Rift g g a ~
forall r. (x -> a -> r) -> x -> r ~
forall r. (a -> x -> r) -> x -> r ~
forall r. (a -> g r) -> g r ~
Codensity g r

When f is a left adjoint, so that f -| g then

Rift f f a ~
forall r. f (a -> r) -> f r ~
forall r. (a -> r) -> g (f r) ~
forall r. (a -> r) -> Adjoint f g r ~
Yoneda (Adjoint f g r)

An alternative way to view that is to note that whenever f is a left adjoint then f -| Rift f Identity, and since Rift f f is isomorphic to Rift f Identity (f a), this is the Monad formed by the adjunction.

Rift Identity m can be a Monad for any Monad m, as it is isomorphic to Yoneda m.

Constructors

Rift

Fields

runRift :: forall r. g (a -> r) -> h r

Instances

Functor g => Functor (Rift g h)

(Functor g, (~) (* -> *) g h) => Applicative (Rift g h)

toRift :: (Functor g, Functor k) => (forall x. g (k x) -> h x) -> k a -> Rift g h a

The universal property of Rift

fromRift :: Adjunction f u => (forall a. k a -> Rift f h a) -> f (k b) -> h b

When f -| u, then f -| Rift f Identity and

toRift . fromRift ≡ id
fromRift . toRift ≡ id

grift :: Adjunction f u => f (Rift f k a) -> k a

composeRift :: (Composition compose, Adjunction g u) => Rift f (Rift g h) a -> Rift (compose g f) h a

composeRift . decomposeRift ≡ id
decomposeRift . composeRift ≡ id

decomposeRift :: (Composition compose, Functor f, Functor g) => Rift (compose g f) h a -> Rift f (Rift g h) a

adjointToRift :: Adjunction f u => u a -> Rift f Identity a

Rift f Identity a is isomorphic to the right adjoint to f if one exists.

adjointToRift . riftToAdjoint ≡ id
riftToAdjoint . adjointToRift ≡ id

riftToAdjoint :: Adjunction f u => Rift f Identity a -> u a

Rift f Identity a is isomorphic to the right adjoint to f if one exists.

composedAdjointToRift :: (Functor h, Adjunction f u) => u (h a) -> Rift f h a

Rift f h a is isomorphic to the post-composition of the right adjoint of f onto h if such a right adjoint exists.

riftToComposedAdjoint :: Adjunction f u => Rift f h a -> u (h a)

Rift f h a is isomorphic to the post-composition of the right adjoint of f onto h if such a right adjoint exists.

riftToComposedAdjoint . composedAdjointToRift ≡ id
composedAdjointToRift . riftToComposedAdjoint ≡ id

liftRift :: Applicative f => f a -> Rift f f a

The natural isomorphism between f and Rift f f. lowerRift . liftRift ≡ id liftRift . lowerRift ≡ id

lowerRift (liftRift x)     -- definition
lowerRift (Rift (<*> x))   -- definition
(<*> x) (pure id)          -- beta reduction
pure id <*> x              -- Applicative identity law
x

lowerRift :: Applicative f => Rift f g a -> g a

Lower Rift by applying pure id to the continuation.

See liftRift.

rap :: Functor f => Rift f g (a -> b) -> Rift g h a -> Rift f h b

Indexed applicative composition of right Kan lifts.