kleisli

Three newtype wrappers around p a (f b) with different type parameter orderings, enabling different type class instances depending on which parameter is last.

Type	Parameter order	Primary instances
Kleisli p a f b	functor in b	Functor, Applicative, Monad, MonadTrans, Distributive, Representable
ProKleisli p f a b	profunctor in (a, b)	Profunctor, Strong, Choice, Closed, Category, Arrow, Sieve, Representable
ContraKleisli p b f a	contravariant in a	Contravariant, Divisible, Decidable

All three are representationally identical (p a (f b)) and connected by isomorphisms.

Usage

When p is specialised to (->), extensive instances are derived via Star, ReaderT, Arrow.Kleisli, and Op:

import Data.Kleisli

-- Kleisli: use as a functor/monad in the result
k :: Kleisli (->) Int Maybe Int
k = fmap (+1) (Kleisli Just)

-- ProKleisli: use as a profunctor/arrow
p :: ProKleisli (->) Maybe Int Int
p = arr (+1) >>> arr (*2)

-- ContraKleisli: use as a contravariant functor
c :: ContraKleisli (->) Int Maybe Int
c = contramap (+1) (ContraKleisli Just)

Type aliases

Convenient aliases eliminate common type parameters:

type Kleisli'  p a b = Kleisli p a Identity b       -- no functor layer
type KleisliA  a f b = Kleisli (->) a f b           -- specialise profunctor to (->)
type KleisliA' a b   = KleisliA a Identity b        -- both specialised

Analogous aliases exist for ProKleisli and ContraKleisli.

Isomorphisms

Lens Isos convert between the three orderings:

_Kleisli_ProKleisli    :: Iso (Kleisli p a f b) ... (ProKleisli p f a b) ...
_Kleisli_ContraKleisli :: Iso (Kleisli p a f b) ... (ContraKleisli p b f a) ...

Identity-eliminating isos map through the Identity wrapper:

kleisli' :: Iso (Kleisli' p a b) ... (p a b) ...

Building

cabal build
cabal test doctest
cabal bench