Data.Invertible.Bijection

The base representation for bidirectional arrows (bijections).

Documentation

data Bijection a b c

A representation of a bidirectional arrow (embedding-projection pair of arrows transformer): an arrow and its inverse. Most uses will prefer the specialized <-> type for function arrows.

To constitute a valid bijection, biTo and biFrom should be inverses:

• biTo . biFrom = id

• biFrom . biTo = id

It may be argued that the arguments should be in the opposite order due to the arrow syntax, but it makes more sense to me to have the forward function come first.

Constructors

(:<->:) infix 2

Fields

biTo :: a b c

biFrom :: a c b

Instances

Category * a => Category * (Bijection a)

Semigroupoid * a => Groupoid * (Bijection a)

Semigroupoid * a => Semigroupoid * (Bijection a)

Arrow a => Arrow (Bijection a)

In order to use all the Arrow functions, we make a partially broken instance, where arr creates a bijection with a broken biFrom. See note on BiArrow'. &&& is first-biased, and uses only the left argument's biFrom.

ArrowChoice a => ArrowChoice (Bijection a)

||| is Left-biased, and uses only the left argument's biFrom.

ArrowZero a => ArrowZero (Bijection a)

Invariant2 (Bijection (->))

(Semigroupoid * a, Arrow a) => BiArrow' (Bijection a)

(Semigroupoid * a, Arrow a) => BiArrow (Bijection a)

Monad m => Arrow (MonadArrow (<->) m)

Monad m => ArrowChoice (MonadArrow (<->) m)

MonadPlus m => ArrowZero (MonadArrow (<->) m)

MonadPlus m => ArrowPlus (MonadArrow (<->) m)

Invariant (Bijection (->) b)

Monad m => BiArrow' (MonadArrow (<->) m)

(Semigroupoid * a, Arrow a) => Functor (Bijection a b)

Monoidal (Bijection (->) ())

type (<->) = Bijection (->) infix 2

Specialization of Bijection to function arrows. Represents both a function, f, and its (presumed) inverse, g, represented as f :<->: g.