contravariant-1.2: Contravariant functors

Data.Functor.Contravariant.Divisible

Synopsis

Contravariant Applicative

class Contravariant f => Divisible f whereSource

A Divisible contravariant functor is the contravariant analogue of Applicative.

In denser jargon, a Divisible contravariant functor is a monoid object in the category of presheaves from Hask to Hask, equipped with Day convolution mapping the Cartesian product of the source to the Cartesian product of the target.

By way of contrast, an Applicative functor can be viewed as a monoid object in the category of copresheaves from Hask to Hask, equipped with Day convolution mapping the Cartesian product of the source to the Cartesian product of the target.

Given the canonical diagonal morphism:

delta a = (a,a)

divide delta should be associative with conquer as a unit

divide delta m conquer = m
divide delta conquer m = m
divide delta (divide delta m n) o = divide delta m (divide delta n o)

With more general arguments you'll need to reassociate and project using the monoidal structure of the source category. (Here fst and snd are used in lieu of the more restricted lambda and rho, but this construction works with just a monoidal category.)

divide f m conquer = contramap (fst . f) m
divide f conquer m = contramap (snd . f) m
divide f (divide g m n) o = divide f' m (divide id n o) where
f' a = case f a of (bc,d) -> case g bc of (b,c) -> (a,(b,c))

Methods

divide :: (a -> (b, c)) -> f b -> f c -> f aSource

conquer :: f aSource

The underlying theory would suggest that this should be:

conquer :: (a -> ()) -> f a

However, as we are working over a Cartesian category (Hask) and the Cartesian product, such an input morphism is uniquely determined to be const mempty, so we elide it.

Instances

 Divisible Equivalence Divisible Comparison Divisible Predicate Monoid r => Divisible (Op r) (Divisible f, Applicative g) => Divisible (ComposeCF f g) (Applicative f, Divisible g) => Divisible (ComposeFC f g)

divided :: Divisible f => f a -> f b -> f (a, b)Source

conquered :: Divisible f => f ()Source

Redundant, but provided for symmetry.

conquered = 'conquer

liftD :: Divisible f => (a -> b) -> f b -> f aSource

This is the divisible analogue of liftA. It gives a viable default definition for contramap in terms of the members of Divisible.

liftD f = divide ((,) () . f) conquer

Contravariant Alternative

class Divisible f => Decidable f whereSource

A Divisible contravariant functor is a monoid object in the category of presheaves from Hask to Hask, equipped with Day convolution mapping the cartesian product of the source to the Cartesian product of the target.

choose Left m (lose f)  = m
choose Right (lose f) m = m
choose f (choose g m n) o = divide f' m (divide id n o) where
f' bcd = either (either id (Right . Left) . g) (Right . Right) . f

In addition, we expect the same kind of distributive law as is satisfied by the usual covariant Alternative, w.r.t Applicative, which should be fully formulated and added here at some point!

Methods

lose :: (a -> Void) -> f aSource

The only way to win is not to play.

choose :: (a -> Either b c) -> f b -> f c -> f aSource

Instances

 Decidable Equivalence Decidable Comparison Decidable Predicate Monoid r => Decidable (Op r)

chosen :: Decidable f => f b -> f c -> f (Either b c)Source