Prearrow

definition example

A pre-arrow is a Monad in the Bicategory of Profunctors $\mathbf{Prof}$: a Profunctor p with a composition (>>>) :: p a x -> p x b -> p a b (the multiplication $\mu :P\diamond P\to P$) and an embedding of ordinary functions arr :: (a -> b) -> p a b (the unit $\eta :\mathcal{C}(-,=)\to P$), satisfying associativity and unit laws. Haskell’s Arrow class is a pre-arrow that is also a Tambara Module (first :: p a b -> p (a, c) (b, c)).

Sources: DaoFP §17.8 (“Prearrows as monads in Prof”), §18 (Tambara modules and arrows).

• Examples: functions (->); Kleisli arrows Kleisli m a b = a -> m b of a monad; Star f and Costar f.

Haskell

newtype Kleisli m a b = Kleisli (a -> m b)
instance Monad m => Profunctor (Kleisli m) where
  dimap f g (Kleisli h) = Kleisli (fmap g . h . f)
instance Monad m => PreArrow (Kleisli m) where
  Kleisli f >>> Kleisli g = Kleisli (\a -> f a >>= g)
  arr f = Kleisli (return . f)