Bifunctor

definition example

A bifunctor is a Functor $F:\mathcal{C}\times \mathcal{D}\to \mathcal{E}$ out of a Product Category: it maps a pair of objects to an object and a pair of arrows to an arrow, functorially in each variable. In Haskell: class Bifunctor f where bimap :: (a -> a') -> (b -> b') -> (f a b -> f a' b').

Sources: DaoFP §8.3 (“Bifunctors”), Exercise 8.3.4; 7 Sketches §4.4.3 (the monoidal product $\otimes :\mathcal{C}\times \mathcal{C}\to \mathcal{C}$ is a functor).

Examples: the Product (,) with bimap g h (a, b) = (g a, h b); the Coproduct Either; MoreThanA a b = More a (Maybe b); the tensor product $\otimes$ of any Monoidal Category; the Hom Functor is a mixed-variance bifunctor ${\mathcal{C}}^{\mathrm{op}}\times \mathcal{C}\to \mathbf{Set}$, i.e. a Profunctor. Functoriality of sums and products is discussed in DaoFP §4.4, §5.1 (“Functoriality”).

LeanHaskell

-- a bifunctor is a functor out of a product category, or a curried functor C ⥤ D ⥤ E
#check CategoryTheory.Functor.prod'   -- hmm: see `CategoryTheory.uncurry`, `CategoryTheory.curry`
#check CategoryTheory.curry           -- (C × D ⥤ E) ⥤ (C ⥤ D ⥤ E)

class Bifunctor f where
  bimap :: (a -> a') -> (b -> b') -> (f a b -> f a' b')
 
instance Bifunctor (,) where
  bimap g h (a, b) = (g a, h b)
 
instance Bifunctor Either where
  bimap g _ (Left a)  = Left (g a)
  bimap _ h (Right b) = Right (h b)
 
data MoreThanA a b = More a (Maybe b)
instance Bifunctor MoreThanA where
  bimap g h (More a mb) = More (g a) (fmap h mb)