Safe Haskell	Safe
Language	Haskell2010

Math.Tools.NaturalTransformation

Synopsis

newtype (f :: k -> Type) :~> (g :: k -> Type) = NatTrans {
- nattransComponent :: forall (a :: k). f a -> g a
}
type MaybeNT m = Maybe :~> m
type EitherNT a m = Either a :~> m
type PairNT a m = (,) a :~> m
type IndexNT a m = (->) a :~> m
type ListNT m = List :~> m
type IMT m = I :~> m
type IOMT m = m :~> IO
type NondetMT m = m :~> List
type FailMT m = m :~> Maybe
type AltMT a m = m :~> Either a
type FunctorMT a m = m :~> (->) a
type MonadNT m = (m :*: m) :~> m
type ComonadNT m = m :~> (m :*: m)
type TransposeNT m n = (m :*: n) :~> (n :*: m)
type (::*::) row col elem = ((:==:) row :*: (:==:) col) elem
failFailMT :: FailMT f
succeedFailMT :: FailMT I
returnMT :: Monad m => I :~> m
extractMT :: Comonad m => m :~> I
duplicateMT :: Comonad m => m :~> (m :*: m)
joinMT :: Monad m => (m :*: m) :~> m
idTrans :: f :~> f
unitTrans :: Adjunction f g => I :~> (g :*: f)
counitTrans :: Adjunction f g => (f :*: g) :~> I
vert :: (g :~> h) -> (f :~> g) -> f :~> h
horiz :: Functor h => (h :~> k) -> (f :~> g) -> (h :*: f) :~> (k :*: g)
mapNaturalMatrix :: (Functor f, Functor h) => (f :~> g) -> (h :~> i) -> (a -> b) -> (f :*: h) a -> (g :*: i) b
data Day f g c = forall a b. Day ((a, b) -> c) (f a) (g b)
convolve :: (Functor f, Functor g) => Day f g c -> (f :*: g) c
convolveTrans :: (Functor f, Functor g) => Day f g :~> (f :*: g)
mapRows :: Functor h => (f :~> g) -> (h :*: f) :~> (h :*: g)
mapColumns :: (f :~> g) -> (f :*: h) :~> (g :*: h)
mapMatrixNat :: (Functor f, Functor g) => (f :~> f') -> (g :~> g') -> (a -> b) -> (f :*: g) a -> (f' :*: g') b
recMap :: Functor f => (f :~> g) -> Rec f -> Rec g
transformMap :: (Functor f, Functor g) => Coalgebra f a -> (f :~> g) -> Algebra g b -> a -> b
tmap :: (InitialAlgebra f a, FinalCoalgebra g b) => (f :~> g) -> a -> b
tapp :: Functor f => (f :~> g) -> (a -> b) -> f a -> g b
unyoneda :: Category cat => (cat a :~> f) -> f a
yonedaFunction :: Functor t => t a -> (->) a :~> t
inverseHoriz :: CoFunctor k => (f :~> k) -> (f' :~> k') -> (f :*: k') :~> (k :*: f')
firstTrans :: (y -> x) -> (->) (x, d) :~> (->) (y, d)
reverseList :: List :~> List
concatList :: (List :*: List) :~> List
returnTrans :: Monad m => I :~> m
joinTrans :: Monad m => (m :*: m) :~> m
swapDimensions :: (Applicative g, Traversable f) => (f :*: g) :~> (g :*: f)
data f :<~>: g = NaturalIso {
- fwdNaturalIso :: f :~> g
- bckNaturalIso :: g :~> f
}
idIso :: f :<~>: f
symmetricIso :: (g :<~>: f) -> f :<~>: g
vertIso :: (g :<~>: h) -> (f :<~>: g) -> f :<~>: h
horizIso :: (Functor f, Functor f') => (f :<~>: f') -> (g :<~>: g') -> (f :*: g) :<~>: (f' :*: g')
isoMatrix :: (Functor g, Functor g') => (f :<~>: (g :*: h)) -> (g :<~>: g') -> (h :<~>: h') -> f :<~>: (g' :*: h')
isoMatrix_ :: Functor g' => (f :<~>: ((->) row :*: (->) col)) -> ((->) row :<~>: g') -> ((->) col :<~>: h') -> f :<~>: (g' :*: h')
outerIso :: Functor g' => ((->) row :<~>: g') -> ((->) col :<~>: h') -> ((->) row :*: (->) col) :<~>: (g' :*: h')
matrixFrom :: Functor g => (row -> col -> a) -> ((->) row :<~>: g) -> ((->) col :<~>: h) -> (g :*: h) a
matrixIso :: (Functor f, Functor f', Functor g, Functor g') => (f :<~>: f') -> (g :<~>: g') -> (a :==: b) -> (f :*: g) a :==: (f' :*: g') b
swapDimensionsIso :: (Traversable f, Traversable g, Applicative f, Applicative g) => (f :*: g) :<~>: (g :*: f)
runNaturalIso :: (f :<~>: g) -> f a :==: g a
newtype (f :~~> g) a = NaturalTrans {
- unNatTrans :: f a -> g a
}
transformMatrix :: Applicative f' => ((f :~~> f') :*: g) a -> (f' :*: (g :~~> g')) a -> (f :*: g) a -> (f' :*: g') a
newtype NaturalTransA (arr :: k -> k -> Type) f g a = NaturalTransA {
- unNatTransA :: arr (f a) (g a)
}
newtype NatTransA (arr :: k -> k -> Type) f g = NatTransA {
- nattransComponentA :: forall a. arr (f a) (g a)
}
idTransA :: Arrow arr => NatTransA arr f f
horizA :: (Arrow arr, FunctorArrow k arr arr) => NatTransA arr h k -> NatTransA arr f g -> NatTransA arr (h :*: f) (k :*: g)
vertA :: Arrow arr => NatTransA arr g h -> NatTransA arr f g -> NatTransA arr f h
invertNatTransA :: NatTransA arr f g -> NatTransA (OpA arr) g f
coyonedaA :: (Category cat, ArrowApply arr, ArrowChoice arr) => arr (NatTransA arr (OpA cat a) f) (f a)
unyonedaA :: (Category cat, ArrowApply arr) => arr (NatTransA arr (cat a) f) (f a)
yonedaA :: (FunctorArrow f arr arr, ArrowApply arr) => f c -> NatTransA arr (arr c) f
natTransToA :: (f :~> g) -> NatTransA (->) f g
arrowNatTrans :: NatTransA (->) f g -> f :~> g

Documentation

newtype (f :: k -> Type) :~> (g :: k -> Type) Source #

Constructors

NatTrans
Fields nattransComponent :: forall (a :: k). f a -> g a

Instances

Instances details

Category ((:~>) :: (k -> Type) -> (k -> Type) -> Type) Source #
Instance details Defined in Math.Tools.NaturalTransformation Methods id :: forall (a :: k0). a :~> a # (.) :: forall (b :: k0) (c :: k0) (a :: k0). (b :~> c) -> (a :~> b) -> a :~> c #

type MaybeNT m = Maybe :~> m Source #

type EitherNT a m = Either a :~> m Source #

type PairNT a m = (,) a :~> m Source #

type IndexNT a m = (->) a :~> m Source #

type ListNT m = List :~> m Source #

type IMT m = I :~> m Source #

type IOMT m = m :~> IO Source #

type NondetMT m = m :~> List Source #

type FailMT m = m :~> Maybe Source #

type AltMT a m = m :~> Either a Source #

type FunctorMT a m = m :~> (->) a Source #

type MonadNT m = (m :*: m) :~> m Source #

type ComonadNT m = m :~> (m :*: m) Source #

type TransposeNT m n = (m :*: n) :~> (n :*: m) Source #

type (::*::) row col elem = ((:==:) row :*: (:==:) col) elem Source #

failFailMT :: FailMT f Source #

succeedFailMT :: FailMT I Source #

returnMT :: Monad m => I :~> m Source #

extractMT :: Comonad m => m :~> I Source #

duplicateMT :: Comonad m => m :~> (m :*: m) Source #

joinMT :: Monad m => (m :*: m) :~> m Source #

idTrans :: f :~> f Source #

unitTrans :: Adjunction f g => I :~> (g :*: f) Source #

counitTrans :: Adjunction f g => (f :*: g) :~> I Source #

vert :: (g :~> h) -> (f :~> g) -> f :~> h Source #

horiz :: Functor h => (h :~> k) -> (f :~> g) -> (h :*: f) :~> (k :*: g) Source #

mapNaturalMatrix :: (Functor f, Functor h) => (f :~> g) -> (h :~> i) -> (a -> b) -> (f :*: h) a -> (g :*: i) b Source #

data Day f g c Source #

Constructors

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

convolve :: (Functor f, Functor g) => Day f g c -> (f :*: g) c Source #

convolveTrans :: (Functor f, Functor g) => Day f g :~> (f :*: g) Source #

mapRows :: Functor h => (f :~> g) -> (h :*: f) :~> (h :*: g) Source #

mapColumns :: (f :~> g) -> (f :*: h) :~> (g :*: h) Source #

mapMatrixNat :: (Functor f, Functor g) => (f :~> f') -> (g :~> g') -> (a -> b) -> (f :*: g) a -> (f' :*: g') b Source #

recMap :: Functor f => (f :~> g) -> Rec f -> Rec g Source #

transformMap :: (Functor f, Functor g) => Coalgebra f a -> (f :~> g) -> Algebra g b -> a -> b Source #

tmap :: (InitialAlgebra f a, FinalCoalgebra g b) => (f :~> g) -> a -> b Source #

tapp :: Functor f => (f :~> g) -> (a -> b) -> f a -> g b Source #

unyoneda :: Category cat => (cat a :~> f) -> f a Source #

yonedaFunction :: Functor t => t a -> (->) a :~> t Source #

inverseHoriz :: CoFunctor k => (f :~> k) -> (f' :~> k') -> (f :*: k') :~> (k :*: f') Source #

firstTrans :: (y -> x) -> (->) (x, d) :~> (->) (y, d) Source #

reverseList :: List :~> List Source #

concatList :: (List :*: List) :~> List Source #

returnTrans :: Monad m => I :~> m Source #

joinTrans :: Monad m => (m :*: m) :~> m Source #

swapDimensions :: (Applicative g, Traversable f) => (f :*: g) :~> (g :*: f) Source #

data f :<~>: g Source #

Constructors

NaturalIso
Fields fwdNaturalIso :: f :~> g bckNaturalIso :: g :~> f

Instances

Instances details

Category ((:<~>:) :: (k -> Type) -> (k -> Type) -> Type) Source #
Instance details Defined in Math.Tools.NaturalTransformation Methods id :: forall (a :: k0). a :<~>: a # (.) :: forall (b :: k0) (c :: k0) (a :: k0). (b :<~>: c) -> (a :<~>: b) -> a :<~>: c #

idIso :: f :<~>: f Source #

symmetricIso :: (g :<~>: f) -> f :<~>: g Source #

vertIso :: (g :<~>: h) -> (f :<~>: g) -> f :<~>: h Source #

horizIso :: (Functor f, Functor f') => (f :<~>: f') -> (g :<~>: g') -> (f :*: g) :<~>: (f' :*: g') Source #

isoMatrix :: (Functor g, Functor g') => (f :<~>: (g :*: h)) -> (g :<~>: g') -> (h :<~>: h') -> f :<~>: (g' :*: h') Source #

isoMatrix_ :: Functor g' => (f :<~>: ((->) row :*: (->) col)) -> ((->) row :<~>: g') -> ((->) col :<~>: h') -> f :<~>: (g' :*: h') Source #

outerIso :: Functor g' => ((->) row :<~>: g') -> ((->) col :<~>: h') -> ((->) row :*: (->) col) :<~>: (g' :*: h') Source #

matrixFrom :: Functor g => (row -> col -> a) -> ((->) row :<~>: g) -> ((->) col :<~>: h) -> (g :*: h) a Source #

matrixIso :: (Functor f, Functor f', Functor g, Functor g') => (f :<~>: f') -> (g :<~>: g') -> (a :==: b) -> (f :*: g) a :==: (f' :*: g') b Source #

swapDimensionsIso :: (Traversable f, Traversable g, Applicative f, Applicative g) => (f :*: g) :<~>: (g :*: f) Source #

runNaturalIso :: (f :<~>: g) -> f a :==: g a Source #

newtype (f :~~> g) a Source #

Constructors

NaturalTrans
Fields unNatTrans :: f a -> g a

transformMatrix :: Applicative f' => ((f :~~> f') :*: g) a -> (f' :*: (g :~~> g')) a -> (f :*: g) a -> (f' :*: g') a Source #

newtype NaturalTransA (arr :: k -> k -> Type) f g a Source #

Constructors

NaturalTransA
Fields unNatTransA :: arr (f a) (g a)

newtype NatTransA (arr :: k -> k -> Type) f g Source #

Constructors

NatTransA
Fields nattransComponentA :: forall a. arr (f a) (g a)

Instances

Instances details

Category arr => Category (NatTransA arr :: (k2 -> k1) -> (k2 -> k1) -> Type) Source #
Instance details Defined in Math.Tools.NaturalTransformation Methods id :: forall (a :: k). NatTransA arr a a # (.) :: forall (b :: k) (c :: k) (a :: k). NatTransA arr b c -> NatTransA arr a b -> NatTransA arr a c #

idTransA :: Arrow arr => NatTransA arr f f Source #

horizA :: (Arrow arr, FunctorArrow k arr arr) => NatTransA arr h k -> NatTransA arr f g -> NatTransA arr (h :*: f) (k :*: g) Source #

vertA :: Arrow arr => NatTransA arr g h -> NatTransA arr f g -> NatTransA arr f h Source #

invertNatTransA :: NatTransA arr f g -> NatTransA (OpA arr) g f Source #

coyonedaA :: (Category cat, ArrowApply arr, ArrowChoice arr) => arr (NatTransA arr (OpA cat a) f) (f a) Source #

For exposition of coyonedaA, see Bartosz Milewski's youtube videos https://www.youtube.com/watch?v=p_ydgYm9-yg

unyonedaA :: (Category cat, ArrowApply arr) => arr (NatTransA arr (cat a) f) (f a) Source #

yonedaA :: (FunctorArrow f arr arr, ArrowApply arr) => f c -> NatTransA arr (arr c) f Source #

natTransToA :: (f :~> g) -> NatTransA (->) f g Source #

arrowNatTrans :: NatTransA (->) f g -> f :~> g Source #