Control.Invertible.MonadArrow

A symmetric version of the Kleisli monad transformer arrow. This admits three isomorphic MonadBijection types:

• MonadArrow (<->) m a b

• Bijection (MonadFunction m) a b

• m a <-> m b

The Alimarine paper just calls it "MoT" for Monad Transformer.

Documentation

newtype MonadArrow a m b c

Bidirectional Kleisli-like monad arrow transformer.

Constructors

MonadArrow

Fields

• runMonadArrow :: a (m b) (m c)

Instances

Category * a => Category * (MonadArrow a m)

Methods

id :: cat a a

(.) :: cat b c -> cat a b -> cat a c

Groupoid * a => Groupoid * (MonadArrow a m)

Methods

inv :: k1 a b -> k1 b a

Semigroupoid * a => Semigroupoid * (MonadArrow a m)

Methods

o :: c j k1 -> c i j -> c i k1

Monad m => Arrow (MonadArrow ((->) LiftedRep LiftedRep) m)

Methods

arr :: (b -> c) -> MonadArrow (LiftedRep -> LiftedRep) m b c

first :: MonadArrow (LiftedRep -> LiftedRep) m b c -> MonadArrow (LiftedRep -> LiftedRep) m (b, d) (c, d)

second :: MonadArrow (LiftedRep -> LiftedRep) m b c -> MonadArrow (LiftedRep -> LiftedRep) m (d, b) (d, c)

(***) :: MonadArrow (LiftedRep -> LiftedRep) m b c -> MonadArrow (LiftedRep -> LiftedRep) m b' c' -> MonadArrow (LiftedRep -> LiftedRep) m (b, b') (c, c')

(&&&) :: MonadArrow (LiftedRep -> LiftedRep) m b c -> MonadArrow (LiftedRep -> LiftedRep) m b c' -> MonadArrow (LiftedRep -> LiftedRep) m b (c, c')

Monad m => Arrow (MonadArrow (<->) m)

Methods

arr :: (b -> c) -> MonadArrow (<->) m b c

first :: MonadArrow (<->) m b c -> MonadArrow (<->) m (b, d) (c, d)

second :: MonadArrow (<->) m b c -> MonadArrow (<->) m (d, b) (d, c)

(***) :: MonadArrow (<->) m b c -> MonadArrow (<->) m b' c' -> MonadArrow (<->) m (b, b') (c, c')

(&&&) :: MonadArrow (<->) m b c -> MonadArrow (<->) m b c' -> MonadArrow (<->) m b (c, c')

MonadPlus m => ArrowZero (MonadArrow ((->) LiftedRep LiftedRep) m)

Methods

zeroArrow :: MonadArrow (LiftedRep -> LiftedRep) m b c

MonadPlus m => ArrowZero (MonadArrow (<->) m)

Methods

zeroArrow :: MonadArrow (<->) m b c

MonadPlus m => ArrowPlus (MonadArrow ((->) LiftedRep LiftedRep) m)

Methods

(<+>) :: MonadArrow (LiftedRep -> LiftedRep) m b c -> MonadArrow (LiftedRep -> LiftedRep) m b c -> MonadArrow (LiftedRep -> LiftedRep) m b c

MonadPlus m => ArrowPlus (MonadArrow (<->) m)

Methods

(<+>) :: MonadArrow (<->) m b c -> MonadArrow (<->) m b c -> MonadArrow (<->) m b c

Monad m => ArrowChoice (MonadArrow ((->) LiftedRep LiftedRep) m)

Methods

left :: MonadArrow (LiftedRep -> LiftedRep) m b c -> MonadArrow (LiftedRep -> LiftedRep) m (Either b d) (Either c d)

right :: MonadArrow (LiftedRep -> LiftedRep) m b c -> MonadArrow (LiftedRep -> LiftedRep) m (Either d b) (Either d c)

(+++) :: MonadArrow (LiftedRep -> LiftedRep) m b c -> MonadArrow (LiftedRep -> LiftedRep) m b' c' -> MonadArrow (LiftedRep -> LiftedRep) m (Either b b') (Either c c')

(|||) :: MonadArrow (LiftedRep -> LiftedRep) m b d -> MonadArrow (LiftedRep -> LiftedRep) m c d -> MonadArrow (LiftedRep -> LiftedRep) m (Either b c) d

Monad m => ArrowChoice (MonadArrow (<->) m)

Methods

left :: MonadArrow (<->) m b c -> MonadArrow (<->) m (Either b d) (Either c d)

right :: MonadArrow (<->) m b c -> MonadArrow (<->) m (Either d b) (Either d c)

(+++) :: MonadArrow (<->) m b c -> MonadArrow (<->) m b' c' -> MonadArrow (<->) m (Either b b') (Either c c')

(|||) :: MonadArrow (<->) m b d -> MonadArrow (<->) m c d -> MonadArrow (<->) m (Either b c) d

Monad m => BiArrow' (MonadArrow (<->) m)

(BiArrow a, Monad m) => BiArrow (MonadArrow a m)

Methods

(<->) :: (b -> c) -> (c -> b) -> MonadArrow a m b c

invert :: MonadArrow a m b c -> MonadArrow a m c b

type MonadFunction = MonadArrow (->)

Specialization of MonadArrow to function arrows.

type MonadBijection m = MonadArrow (<->) m

type MonadBijection' m = Bijection (MonadFunction m)

type MonadBijection'' m a b = m a <-> m b

monadBijection :: MonadBijection' m a b <-> MonadBijection m a b

Convert between isomorphic representations of MonadBijections.

monadBijection' :: MonadBijection'' m a b <-> MonadBijection' m a b

Convert between isomorphic representations of MonadBijections.