Playing around Hom(X,_) in Hask

hom_covariant.hs

module Main where

import Control.Monad

-- Directly composing morphisms as if we were not in End_{Hask}.
regular :: (->) Int Int
regular = (+) 1 . (+) 1 . (+) 1 . (+) 1

-- Actually the same, but explicitly using action of Hom-functor on arrows.
functorial :: (->) Int Int
functorial = (+) 1 <$> (+) 1 <$> (+) 1 <$> (+) 1

-- A bit of a sudden, but where is a Monad instance for Hom-functor. It operates X as a context of reader passing through.
-- Since it's just an arrow, nothing prevents from accumulating this value.
-- But we cannot derive essentially new way of composing functions like `f >=> f >=> f` or so.
monadic :: (->) Int Int
monadic = (+) 1 >>= ((+) >=> (+) >=> (+))

-- And if instances of Monad and Functor are defined it's possible to define Applicative.
applicative :: (->) Int Int
applicative = (+) <*> (+) 1 <$> (+) 1

main :: IO ()
main = do
  print $ regular 1
  print $ functorial 1
  print $ monadic 1
  print $ applicative 1