Hom.hs

{-# LANGUAGE NoImplicitPrelude #-}
module Homs where 

(.) :: (b -> c) -> (a -> b) -> (a -> c)
g . f = \a -> g (f a)

-- the class of covariant functors
class Functor f where 
  fmap :: (a -> b) -> f a -> f b 
  -- laws: fmap f . fmap g ~ fmap (f . g) 
  
-- the class of contravariant functors 
class Contravariant f where 
  contramap :: (a -> b) -> f b -> f a
  
-- Covariant hom functors
newtype Cov n x = Cov (n -> x) 

-- Contravariant hom functor
newtype Con n x = Con (x -> n) 

instance Functor (Cov n) where 
  fmap xy (Cov nx) = Cov (xy . nx) 
   -- C(n,x) -> C(n,y) via post composition with f : x -> y in C
   
instance Contravariant (Con n) where 
  contramap xy (Con ny) = Con (ny . xy) 
  -- C(n,y) -> C(n,x) via precomposition with f : x -> y in C^op