scala212-category-theory-either-bifunctor

Reference: https://bartoszmilewski.com/2015/02/03/functoriality/

definition

Bifunctor is a two-argument functor (domain is product category).

The easiest way to think about bifunctors is that they are functors in both arguments. So instead of translating functorial laws - associativity and identity preservation - from functors to bifunctors, it’s enough to check them separately for each argument.

As far as we stay only in Cats category (functors as a morphisms) the above statement is true. But not in general: https://ncatlab.org/nlab/show/funny+tensor+product#separate_functoriality

class Bifunctor f where
    bimap :: (a -> c) -> (b -> d) -> f a b -> f c d
    bimap g h = first g . second h
    first :: (a -> c) -> f a b -> f c b
    first g = bimap g id
    second :: (b -> d) -> f a b -> f a d
    second = bimap id

• we can specify bimap in terms of first and second

  bimap g h = first g . second h
  

• we can specify first and second in terms of bimap

  first :: (a -> c) -> f a b -> f c b
  second :: (b -> d) -> f a b -> f a d
  

• we can specify three of them, but we have to assure that they are related in that way

project description

We provide implementation of bimap for Either:

// note that also we could use: either.map(second).swap.map(first).swap
def bimap[X1, Y1](first: X => X1, second: Y => Y1): Either[X1, Y1] = either match {
  case Left(x) => Left(first(x))
  case Right(y) => Right(second(y))
}

with derived implementations of first and second

def first[X1](f: Function[X, X1]): Either[X1, Y] = bimap(f, identity)

def second[Y1](f: Function[Y, Y1]): Either[X, Y1] = bimap(identity, f)

and tests

Right[Int, String]("a").bimap(_ * 2, _ + " mapped") should be(Right("a mapped"))
Left[Int, String](1).bimap(_ * 2, _ + " mapped") should be(Left(2))

def f(i: Int) = i * 2
Left[Int, String](1).first(f) should be(Left(2))

def f(s: String) = s + " mapped"
Right[Int, String]("a").second(f) should be(Right("a mapped"))