Copyright	(C) 2013-2016 Edward Kmett Gershom Bazerman and Derek Elkins
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	portable
Safe Haskell	None
Language	Haskell2010

Data.Functor.Contravariant.Day

Description

The Day convolution of two contravariant functors is a contravariant functor.

http://ncatlab.org/nlab/show/Day+convolution

Synopsis

data Day (f :: Type -> Type) (g :: Type -> Type) a = Day (f b) (g c) (a -> (b, c))
day :: f a -> g b -> Day f g (a, b)
runDay :: (Contravariant f, Contravariant g) => Day f g a -> (f a, g a)
assoc :: forall (f :: Type -> Type) (g :: Type -> Type) (h :: Type -> Type) a. Day f (Day g h) a -> Day (Day f g) h a
disassoc :: forall (f :: Type -> Type) (g :: Type -> Type) (h :: Type -> Type) a. Day (Day f g) h a -> Day f (Day g h) a
swapped :: forall (f :: Type -> Type) (g :: Type -> Type) a. Day f g a -> Day g f a
intro1 :: f a -> Day (Proxy :: Type -> Type) f a
intro2 :: f a -> Day f (Proxy :: Type -> Type) a
day1 :: forall f (g :: Type -> Type) a. Contravariant f => Day f g a -> f a
day2 :: forall g (f :: Type -> Type) a. Contravariant g => Day f g a -> g a
diag :: f a -> Day f f a
trans1 :: forall f g (h :: Type -> Type) a. (forall x. f x -> g x) -> Day f h a -> Day g h a
trans2 :: forall g h (f :: Type -> Type) a. (forall x. g x -> h x) -> Day f g a -> Day f h a

Documentation

data Day (f :: Type -> Type) (g :: Type -> Type) a Source #

The Day convolution of two contravariant functors.

Constructors

Day (f b) (g c) (a -> (b, c))

Instances

Instances details

(Representable f, Representable g) => Representable (Day f g) Source #

Instance details

Defined in Data.Functor.Contravariant.Day

Associated Types

type Rep (Day f g)
Instance details Defined in Data.Functor.Contravariant.Day type Rep (Day f g) = (Rep f, Rep g)

Methods

tabulate :: (a -> Rep (Day f g)) -> Day f g a #

index :: Day f g a -> a -> Rep (Day f g) #

contramapWithRep :: (b -> Either a (Rep (Day f g))) -> Day f g a -> Day f g b #

Contravariant (Day f g) Source #

Instance details

Defined in Data.Functor.Contravariant.Day

Methods

contramap :: (a' -> a) -> Day f g a -> Day f g a' #

(>$) :: b -> Day f g b -> Day f g a #

(Divisible f, Divisible g) => Divisible (Day f g) Source #

Instance details

Defined in Data.Functor.Contravariant.Day

Methods

divide :: (a -> (b, c)) -> Day f g b -> Day f g c -> Day f g a #

conquer :: Day f g a #

(Adjunction f u, Adjunction f' u') => Adjunction (Day f f') (Day u u') Source #

Instance details

Defined in Data.Functor.Contravariant.Day

Methods

unit :: a -> Day u u' (Day f f' a) #

counit :: a -> Day f f' (Day u u' a) #

leftAdjunct :: (b -> Day f f' a) -> a -> Day u u' b #

rightAdjunct :: (a -> Day u u' b) -> b -> Day f f' a #

type Rep (Day f g) Source #

Instance details

Defined in Data.Functor.Contravariant.Day

type Rep (Day f g) = (Rep f, Rep g)

day :: f a -> g b -> Day f g (a, b) Source #

Construct the Day convolution

day1 (day f g) = f
day2 (day f g) = g

runDay :: (Contravariant f, Contravariant g) => Day f g a -> (f a, g a) Source #

Break apart the Day convolution of two contravariant functors.

assoc :: forall (f :: Type -> Type) (g :: Type -> Type) (h :: Type -> Type) a. Day f (Day g h) a -> Day (Day f g) h a Source #

Day convolution provides a monoidal product. The associativity of this monoid is witnessed by assoc and disassoc.

assoc . disassoc = id
disassoc . assoc = id
contramap f . assoc = assoc . contramap f

disassoc :: forall (f :: Type -> Type) (g :: Type -> Type) (h :: Type -> Type) a. Day (Day f g) h a -> Day f (Day g h) a Source #

Day convolution provides a monoidal product. The associativity of this monoid is witnessed by assoc and disassoc.

assoc . disassoc = id
disassoc . assoc = id
contramap f . disassoc = disassoc . contramap f

swapped :: forall (f :: Type -> Type) (g :: Type -> Type) a. Day f g a -> Day g f a Source #

The monoid for Day convolution in Haskell is symmetric.

contramap f . swapped = swapped . contramap f

intro1 :: f a -> Day (Proxy :: Type -> Type) f a Source #

Proxy serves as the unit of Day convolution.

day1 . intro1 = id
contramap f . intro1 = intro1 . contramap f

intro2 :: f a -> Day f (Proxy :: Type -> Type) a Source #

Proxy serves as the unit of Day convolution.

day2 . intro2 = id
contramap f . intro2 = intro2 . contramap f

day1 :: forall f (g :: Type -> Type) a. Contravariant f => Day f g a -> f a Source #

In Haskell we can do general purpose elimination, but in a more general setting it is only possible to eliminate the unit.

day1 . intro1 = id
day1 = fst . runDay
contramap f . day1 = day1 . contramap f

day2 :: forall g (f :: Type -> Type) a. Contravariant g => Day f g a -> g a Source #

In Haskell we can do general purpose elimination, but in a more general setting it is only possible to eliminate the unit. day2 . intro2 = id day2 = snd . runDay contramap f . day2 = day2 . contramap f

diag :: f a -> Day f f a Source #

Diagonalize the Day convolution:

day1 . diag = id
day2 . diag = id
runDay . diag = a -> (a,a)
contramap f . diag = diag . contramap f

trans1 :: forall f g (h :: Type -> Type) a. (forall x. f x -> g x) -> Day f h a -> Day g h a Source #

Apply a natural transformation to the left-hand side of a Day convolution.

This respects the naturality of the natural transformation you supplied:

contramap f . trans1 fg = trans1 fg . contramap f

trans2 :: forall g h (f :: Type -> Type) a. (forall x. g x -> h x) -> Day f g a -> Day f h a Source #

Apply a natural transformation to the right-hand side of a Day convolution.

This respects the naturality of the natural transformation you supplied:

contramap f . trans2 fg = trans2 fg . contramap f