invertible-0.2.0.5: bidirectional arrows, bijective functions, and invariant functors

Control.Invertible.BiArrow

Contents

Description

Bidirectional arrows. Taken directly from

Synopsis

# Documentation

class (Groupoid a, Category a) => BiArrow a where Source #

The bidirectional arrow class.

Instances should satisfy the following laws:

• f1 <-> g2 >>> g1 <-> f2 = (f1 >>> g1) <-> (f2 >>> g2)
• invert (invert f) = f
• invert (f <-> g) = g <-> f
• first (f <-> g) = f *** id <-> g *** id
• first h >>> id *** f <-> id *** g = id *** f <-> id *** g >>> first h
• first (first f) >>> assoc = assoc >>> first f

where assoc = [biCase|((x,y),z) <-> (x,(y,z))|]

Minimal complete definition

(<->)

Methods

(<->) :: (b -> c) -> (c -> b) -> a b c infix 2 Source #

Take two functions and lift them into a bidirectional arrow. The intention is that these functions are each other's inverse.

invert :: a b c -> a c b Source #

Inverse: reverse the direction of a bidirectional arrow.

Instances

 Source # Methods(<->) :: (b -> c) -> (c -> b) -> Iso b c Source #invert :: Iso b c -> Iso c b Source # (Semigroupoid * a, Arrow a) => BiArrow (Bijection a) Source # Methods(<->) :: (b -> c) -> (c -> b) -> Bijection a b c Source #invert :: Bijection a b c -> Bijection a c b Source # (Semigroupoid * a, Arrow a) => BiArrow (Iso * a) Source # Methods(<->) :: (b -> c) -> (c -> b) -> Iso * a b c Source #invert :: Iso * a b c -> Iso * a c b Source # (BiArrow a, Monad m) => BiArrow (MonadArrow a m) Source # Methods(<->) :: (b -> c) -> (c -> b) -> MonadArrow a m b c Source #invert :: MonadArrow a m b c -> MonadArrow a m c b Source #

class (BiArrow a, Arrow a) => BiArrow' a Source #

Bidirectional arrows under Arrow.

Although BiArrow should not, strictly speaking, be a subclass of Arrow (as it is often impossible to define arr), this is done because (as the paper says) "conceptually bi-arrows form an extension of the arrow class. Moreover, it allows us to use bi-arrows as normal arrows." This class exists to register this confound.

Instances

 (Semigroupoid * a, Arrow a) => BiArrow' (Bijection a) Source # Monad m => BiArrow' (MonadArrow (<->) m) Source #

biarr :: BiArrow a => (b <-> c) -> a b c Source #

Lift a bidirectional function to an arbitrary arrow using <->.

involve :: BiArrow a => (b -> b) -> a b b Source #

Construct an involution (a biarrow where the function and inverse are the same).

(^^>>) :: BiArrow a => (b <-> c) -> a c d -> a b d infixr 1 Source #

Precomposition with a pure bijection.

(>>^^) :: BiArrow a => a b c -> (c <-> d) -> a b d infixr 1 Source #

Postcomposition with a pure bijection.

(<<^^) :: BiArrow a => a c d -> (b <-> c) -> a b d infixr 1 Source #

Precomposition with a pure bijection (right-to-left variant).

(^^<<) :: BiArrow a => (c <-> d) -> a b c -> a b d infixr 1 Source #

Postcomposition with a pure bijection (right-to-left variant).

type BiKleisli m = Bijection (Kleisli m) Source #

Bidirectional Kleisli monad arrow transformer.

# Orphan instances

 Source # Poor orphans. Please will someone adopt us? Methodsinv :: k1 a b -> k1 b a # Source # Poor orphans. Please will someone adopt us? Methodso :: c j k1 -> c i j -> c i k1 #