alternative, filterable, distributive and partial functors and profunctors

distributive

Given distributive categories
  * -C>, >C<, 1C, +C+, 0C
  * -D>, >D<, 1D, +D+, 0D

An alternative functor consists of
  * A functor
    - F : C -> D
  * morphisms
    - eps : 1D -D> F(1C)
    - mu : F(a) >D< F(b) -D> F(a >C< b)
    - eta : 1D -D> F(0C)
    - nu : F(a) >D< F(b) -D> F(a +C+ b)
  * laws
    - (F, eps, mu) is a lax monoidal functor
    - (F, eta, nu) is a lax monoidal functor

A lax distributive functor consists of
  * A functor
    - F : C -> D
  * morphisms
    - eps : 1D -D> F(1C)
    - mu : F(a) >D< F(b) -D> F(a >C< b)
    - iota' : F(0C) -D> 0D
    - kappa' : F(a +C+ b) -D> F(a) +D+ F(b)
  * laws
    - (F, eps, mu) is a lax monoidal functor
    - (F, iota', kappa') is an oplax monoidal functor

A filterable functor consists of
  * A functor
    - F : C -> D
  * morphisms
    - iota : 0D -D> F(0C)
    - kappa : F(a) +D+ F(b) -D> F(a +C+ b)
    - eta' : F(0C) -D> 1D
    - nu' : F(a +C+ b) -D> F(a) >D< F(b)
  * laws
    - (F, iota, kappa) is a lax monoidal functor
    - (F, eta', nu') is an oplax monoidal functor

A partial distributive functor consists of
  * A functor
    - F : C -> D
  * morphisms
    - eps : 1D -D> F(1C)
    - mu : F(a) >D< F(b) -D> F(a >C< b)
    - iota : 0D -D> F(0C)
    - kappa : F(a) +D+ F(b) -D> F(a +C+ b)
    - eta : F(0C) -D> 0D
    - nu : F(a +C+ b) -D> F(a) +D+ F(b)
    - eta' : F(0C) -D> 1D
    - nu' : F(a +C+ b) -D> F(a) >D< F(b)
  * laws
    - (F, eps, mu, eta, nu) is an alternative functor
    - (F, iota, kappa, eta', nu') is a filterable functor
    - (F, eta, eta', nu, nu') is a strong monoidal functor

Construct distributive categories with
  * C^op x D
    - (a >C< c, b >D< d)
    - (1C, 1D)
    - (a +C+ c, b +D+ d)
    - (0C, 0D)
  * Set
    - Functions ->
    - Cartesian product ><
    - One point set 1
    - Disjoint union |
    - Empty set 0

A lax distributive profunctor consists of
  * A profunctor
    - P : C^op x D -> Set
  * functions
    - eps : 1 -> P(1C, 1D)
    - mu : P(a,b) >< P(c,d) -> P(a >C< c, b >D< d)
    - eta : 1 -> P(0C, 0D)
    - nu : P(a,b) >< P(c,d) -> P(a +C+ c, b +D+ d)
  * law
    - (P, eps, mu, eta, nu) is a lax alternative functor

A partial profunctor consists of
  * A profunctor
    - P : C^op x D -> Set
  * functions
    - iota : 0 -> P(0C, 0D)
    - kappa : P(a,b) | P(c,d) -> P(a +C+ c, b +D+ d)
    - eta' : P(0C, 0D) -> 1
    - nu' : P(a +C+ c, b +D+ d) -> P(a,b) >< P(c,d)
  * law
    - (P, iota, kappa, eta', nu') is a filterable functor

A partial distributive profunctor consists of
  * A profunctor
    - P : C^op x D -> Set
  * functions
    - eps : 1 -> P(1C, 1D)
    - mu : P(a,b) >< P(c,d) -> P(a >C< c, b >D< d)
    - iota : 0 -> P(0C, 0D)
    - kappa : P(a,b) | P(c,d) -> P(a +C+ c, b +D+ d)
    - eta : 1 -> P(0C, 0D)
    - nu : P(a,b) >< P(c,d) -> P(a +C+ c, b +D+ d)
    - eta' : P(0C, 0D) -> 1
    - nu' : P(a +C+ c, b +D+ d) -> P(a,b) >< P(c,d)
  * law
    - (P, eps, mu, iota, kappa, eta, nu, eta', nu') is a syntax functor

Proposition:
Given a lax distributive functor F,
both F^* and F_* are lax distributive profunctors.

Example:
In Haskell, the `Alternative` typeclass is equivalent to
an alternative functor from ((->), (,), (), Either, Void) to ((->), (,), ()).

```
eps :: Applicative f => () -> f ()
eps _ = pure ()

mu :: Applicative f => (f a, f b) -> f (a,b)
mu (fa, fb) = (,) <$> fa <*> fb

eta :: Alternative f => () -> f Void
eta _ = empty

nu :: Alternative f => (f a, f b) -> f (Either a b)
nu (fa, fb) = Left <$> fa <|> Right <$> fb
```

Examples of lax distributive functors can be found too,
using the `Adjunction` typeclass in Haskell.

```
eta :: Adjunction f u => f Void -> Void
eta = unabsurdL

nu :: Adjunction f u => f (Either a b) -> Either (f a) (f b)
nu = cozipL
```