N_P1Ch08b_BiFunctorComposition

Markdown of literate Haskell program. Program source: /src/CTNotes/P1Ch08b_BiFunctorComposition.lhs

Notes about CTFP Part 1 Chapter 8. Functoriality. BiFunctor composition as functor composition.

It is conceptually easier to think of bifunctor as simply a functor from a product category. This note uses this approach to explain bifunctor composition.

 {-# LANGUAGE TypeOperators #-}
 module CTNotes.P1Ch08b_BiFunctorComposition where
 import CTNotes.P1Ch08a_BiFunctorAsFunctor (Bifunctor, bimap)

Directly from the book

 newtype BiComp bf fu gu a b = BiComp (bf (fu a) (gu b))
 
 instance (Bifunctor bf, Functor fu, Functor gu) =>
   Bifunctor (BiComp bf fu gu) where
     bimap f1 f2 (BiComp x) = BiComp ((bimap (fmap f1) (fmap f2)) x)

Comp2 is a convenient type that subsumes binary operators on functors such as Data.Functor.Product and Data.Functor.Sum found in the base package.

 newtype Comp2 bf fu gu a = Comp2 { runComp2 :: bf (fu a) (gu a) }

 instance (Bifunctor bf, Functor fu, Functor gu) => Functor (Comp2 bf fu gu) where
   fmap f (Comp2 x) = Comp2 ((bimap (fmap f) (fmap f)) x)

 type Product f g a = Comp2 (,) f g a
 type Sum f g a = Comp2 Either f g a
 type (f :*: g) a  = Product f g a 
 type (f :+: g) a  = Sum f g a 

Note that Comp2 is a is equivalent to:

type Comp2 bf fu gu a = BiComp bf fu gu a a

this diagram illustrates what has just happened

          Δ                   fu, gu                  bf
Hask   ----->   Hask x Hask  -------->  Hask x Hask ----->  Hask 
 a                a x a               (fu a) x (gu a)       bf (fu a) (gu a)

I find diagrams like this more intuitive than the code. Being able to reason about code like this is, indeed, a nice incentive to study category theory.