Copyright	(c) Erich Gut
License	BSD3
Maintainer	zerich.gut@gmail.com
Safe Haskell	Safe-Inferred
Language	Haskell2010

OAlg.Data.Dualisable

Contents

Dual
Dualisable
Reflexive
Transposable
Site
Side
Direction

Description

data admitting a kind of duality.

Synopsis

type family Dual (x :: k) :: k
class Dualisable x where
- toDual :: x -> Dual x
- fromDual :: Dual x -> x
fromDual' :: Dualisable x => p x -> Dual x -> x
class Reflexive x where
- toBidual :: x -> Dual (Dual x)
- fromBidual :: Dual (Dual x) -> x
fromBidual' :: Reflexive x => p x -> Dual (Dual x) -> x
class Transposable x where
- transpose :: x -> x
data Site
- = From
- | To
data Side
- = LeftSide
- | RightSide
data Direction
- = LeftToRight
- | RightToLeft

Dual

type family Dual (x :: k) :: k Source #

the assigned dual kind.

Instances

Instances details

type Dual 'LeftToRight Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'LeftToRight = 'RightToLeft
type Dual 'RightToLeft Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'RightToLeft = 'LeftToRight
type Dual 'LeftSide Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'LeftSide = 'RightSide
type Dual 'RightSide Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'RightSide = 'LeftSide
type Dual 'From Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'From = 'To
type Dual 'To Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'To = 'From
type Dual 'Discrete Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual 'Discrete = 'Discrete
type Dual 'Empty Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual 'Empty = 'Empty
type Dual 'General Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual 'General = 'General
type Dual 'Injective Source #
Instance details Defined in OAlg.Limes.Perspective type Dual 'Injective = 'Projective
type Dual 'Projective Source #
Instance details Defined in OAlg.Limes.Perspective type Dual 'Projective = 'Injective
type Dual ('Chain 'From) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual ('Chain 'From) = 'Chain 'To
type Dual ('Chain 'To) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual ('Chain 'To) = 'Chain 'From
type Dual ('Parallel 'LeftToRight) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual ('Parallel 'LeftToRight) = 'Parallel 'RightToLeft
type Dual ('Parallel 'RightToLeft) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual ('Parallel 'RightToLeft) = 'Parallel 'LeftToRight
type Dual ('Star 'From) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual ('Star 'From) = 'Star 'To
type Dual ('Star 'To) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual ('Star 'To) = 'Star 'From
type Dual (SomeObjectClass m :: Type) Source #
Instance details Defined in OAlg.Category.Unify type Dual (SomeObjectClass m :: Type) = SomeObjectClass (Op2 m)
type Dual (SomePath m :: Type) Source #
Instance details Defined in OAlg.Category.Unify type Dual (SomePath m :: Type) = SomePath (Op2 m)
type Dual (SomeDiagram a :: Type) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual (SomeDiagram a :: Type) = SomeDiagram (Op a)
type Dual (Matrix x :: Type) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition type Dual (Matrix x :: Type) = Matrix (Op x)
type Dual (SomeApplMltp2 h :: Type) Source #
Instance details Defined in OAlg.Hom.Multiplicative.Proposition type Dual (SomeApplMltp2 h :: Type) = SomeApplMltp2 (OpHom h)
type Dual (SomeApplPnt h :: Type) Source #
Instance details Defined in OAlg.Hom.Multiplicative.Proposition type Dual (SomeApplPnt h :: Type) = SomeApplPnt (OpHom h)
type Dual (XHomMlt h :: Type) Source #
Instance details Defined in OAlg.Hom.Multiplicative.Proposition type Dual (XHomMlt h :: Type) = XHomMlt (OpHom h)
type Dual (Adbl2 a :: Type) Source #
Instance details Defined in OAlg.Structure.Additive.Proposition type Dual (Adbl2 a :: Type) = Adbl2 (Op a)
type Dual (Adbl3 a :: Type) Source #
Instance details Defined in OAlg.Structure.Additive.Proposition type Dual (Adbl3 a :: Type) = Adbl3 (Op a)
type Dual (Sheaf f :: Type) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Dual (Sheaf f :: Type) = Sheaf (Op f)
type Dual (Mltp2 c :: Type) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition type Dual (Mltp2 c :: Type) = Mltp2 (Op c)
type Dual (Mltp3 c :: Type) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition type Dual (Mltp3 c :: Type) = Mltp3 (Op c)
type Dual (XMlt c :: Type) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition type Dual (XMlt c :: Type) = XMlt (Op c)
type Dual (Path q :: Type) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition type Dual (Path q :: Type) = Path (Op q)
type Dual (XOrtOrientation q :: Type) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition type Dual (XOrtOrientation q :: Type) = XOrtOrientation (Op q)
type Dual (XMrphSite s m :: Type) Source #
Instance details Defined in OAlg.Category.Proposition type Dual (XMrphSite s m :: Type) = XMrphSite (Dual s) (Op2 m)
type Dual (Quiver n m :: Type) Source #
Instance details Defined in OAlg.Entity.Diagram.Quiver type Dual (Quiver n m :: Type) = Quiver n m
type Dual (DstRootSide s d :: Type) Source #
Instance details Defined in OAlg.Structure.Distributive.Proposition type Dual (DstRootSide s d :: Type) = DstRootSide (Dual s) (Op d)
type Dual (DstSide s d :: Type) Source #
Instance details Defined in OAlg.Structure.Distributive.Proposition type Dual (DstSide s d :: Type) = DstSide (Dual s) (Op d)
type Dual (XOrtSite s q :: Type) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition type Dual (XOrtSite s q :: Type) = XOrtSite (Dual s) (Op q)
type Dual (Adjunction h d c :: Type) Source #
Instance details Defined in OAlg.Adjunction.Definition type Dual (Adjunction h d c :: Type) = Adjunction (OpHom h) (Op c) (Op d)
type Dual (Path m x y :: Type) Source #
Instance details Defined in OAlg.Category.Path type Dual (Path m x y :: Type) = Path (Op2 m) y x
type Dual (SomeMorphismSite s m y :: Type) Source #
Instance details Defined in OAlg.Category.Unify type Dual (SomeMorphismSite s m y :: Type) = SomeMorphismSite (Dual s) (Op2 m) y
type Dual (SomePathSite s m y :: Type) Source #
Instance details Defined in OAlg.Category.Unify type Dual (SomePathSite s m y :: Type) = SomePathSite (Dual s) (Op2 m) y
type Dual (Slice s i c :: Type) Source #
Instance details Defined in OAlg.Entity.Slice.Definition type Dual (Slice s i c :: Type) = Slice (Dual s) i (Op c)
type Dual (SliceFactor s i c :: Type) Source #
Instance details Defined in OAlg.Entity.Slice.Definition type Dual (SliceFactor s i c :: Type) = SliceFactor (Dual s) i (Op c)
type Dual (Diagram t n m a :: Type) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual (Diagram t n m a :: Type) = Diagram (Dual t) n m (Op a)
type Dual (XDiagram t n m a :: Type) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition type Dual (XDiagram t n m a :: Type) = XDiagram (Dual t) n m (Op a)
type Dual (Transformation t n m a :: Type) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation type Dual (Transformation t n m a :: Type) = Transformation (Dual t) n m (Op a)
type Dual (Cone s p t n m a :: Type) Source #
Instance details Defined in OAlg.Limes.Cone.Definition type Dual (Cone s p t n m a :: Type) = Cone s (Dual p) (Dual t) n m (Op a)
type Dual (EligibleFactor s p t n m a :: Type) Source #
Instance details Defined in OAlg.Limes.Cone.EligibleFactor type Dual (EligibleFactor s p t n m a :: Type) = EligibleFactor s (Dual p) (Dual t) n m (Op a)
type Dual (Limes s p t n m a :: Type) Source #
Instance details Defined in OAlg.Limes.Definition type Dual (Limes s p t n m a :: Type) = Limes s (Dual p) (Dual t) n m (Op a)
type Dual (Row j (Col i x) :: TYPE LiftedRep) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries type Dual (Row j (Col i x) :: TYPE LiftedRep) = Col j (Row i (Op x))
type Dual (Entries i j x :: TYPE LiftedRep) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries type Dual (Entries i j x :: TYPE LiftedRep) = Entries j i (Op x)
type Dual (ConeZeroHead s p t n m a :: TYPE LiftedRep) Source #
Instance details Defined in OAlg.Limes.Cone.Definition type Dual (ConeZeroHead s p t n m a :: TYPE LiftedRep) = ConeZeroHead s (Dual p) (Dual t) n m (Op a)
type Dual (Limits s p t n m a :: TYPE LiftedRep) Source #
Instance details Defined in OAlg.Limes.Limits type Dual (Limits s p t n m a :: TYPE LiftedRep) = Limits s (Dual p) (Dual t) n m (Op a)

Dualisable

class Dualisable x where Source #

admitting a duality.

Property Let x be Dualisable, than holds: toDual is a bijection with its inverse fromDual.

Methods

toDual :: x -> Dual x Source #

fromDual :: Dual x -> x Source #

Instances

Instances details

Dualisable (SomeObjectClass m) Source #
Instance details Defined in OAlg.Category.Unify Methods toDual :: SomeObjectClass m -> Dual (SomeObjectClass m) Source # fromDual :: Dual (SomeObjectClass m) -> SomeObjectClass m Source #
Morphism m => Dualisable (SomePath m) Source #
Instance details Defined in OAlg.Category.Unify Methods toDual :: SomePath m -> Dual (SomePath m) Source # fromDual :: Dual (SomePath m) -> SomePath m Source #
EntityPoint x => Dualisable (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods toDual :: Matrix x -> Dual (Matrix x) Source # fromDual :: Dual (Matrix x) -> Matrix x Source #
FibredOriented a => Dualisable (Adbl2 a) Source #
Instance details Defined in OAlg.Structure.Additive.Proposition Methods toDual :: Adbl2 a -> Dual (Adbl2 a) Source # fromDual :: Dual (Adbl2 a) -> Adbl2 a Source #
FibredOriented a => Dualisable (Adbl3 a) Source #
Instance details Defined in OAlg.Structure.Additive.Proposition Methods toDual :: Adbl3 a -> Dual (Adbl3 a) Source # fromDual :: Dual (Adbl3 a) -> Adbl3 a Source #
FibredOriented f => Dualisable (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods toDual :: Sheaf f -> Dual (Sheaf f) Source # fromDual :: Dual (Sheaf f) -> Sheaf f Source #
Oriented c => Dualisable (Mltp2 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods toDual :: Mltp2 c -> Dual (Mltp2 c) Source # fromDual :: Dual (Mltp2 c) -> Mltp2 c Source #
Oriented c => Dualisable (Mltp3 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods toDual :: Mltp3 c -> Dual (Mltp3 c) Source # fromDual :: Dual (Mltp3 c) -> Mltp3 c Source #
Oriented q => Dualisable (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods toDual :: Path q -> Dual (Path q) Source # fromDual :: Dual (Path q) -> Path q Source #
Dualisable (XMrphSite 'To m) Source #
Instance details Defined in OAlg.Category.Proposition Methods toDual :: XMrphSite 'To m -> Dual (XMrphSite 'To m) Source # fromDual :: Dual (XMrphSite 'To m) -> XMrphSite 'To m Source #
Distributive d => Dualisable (DstRootSide 'RightSide d) Source #
Instance details Defined in OAlg.Structure.Distributive.Proposition Methods toDual :: DstRootSide 'RightSide d -> Dual (DstRootSide 'RightSide d) Source # fromDual :: Dual (DstRootSide 'RightSide d) -> DstRootSide 'RightSide d Source #
Distributive d => Dualisable (DstSide 'RightSide d) Source #
Instance details Defined in OAlg.Structure.Distributive.Proposition Methods toDual :: DstSide 'RightSide d -> Dual (DstSide 'RightSide d) Source # fromDual :: Dual (DstSide 'RightSide d) -> DstSide 'RightSide d Source #
Dualisable (XOrtSite 'To q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods toDual :: XOrtSite 'To q -> Dual (XOrtSite 'To q) Source # fromDual :: Dual (XOrtSite 'To q) -> XOrtSite 'To q Source #
Morphism m => Dualisable (Path m x y) Source #
Instance details Defined in OAlg.Category.Path Methods toDual :: Path m x y -> Dual (Path m x y) Source # fromDual :: Dual (Path m x y) -> Path m x y Source #
Dualisable (SomeMorphismSite 'To m y) Source #
Instance details Defined in OAlg.Category.Unify Methods toDual :: SomeMorphismSite 'To m y -> Dual (SomeMorphismSite 'To m y) Source # fromDual :: Dual (SomeMorphismSite 'To m y) -> SomeMorphismSite 'To m y Source #
Morphism m => Dualisable (SomePathSite 'To m y) Source #
Instance details Defined in OAlg.Category.Unify Methods toDual :: SomePathSite 'To m y -> Dual (SomePathSite 'To m y) Source # fromDual :: Dual (SomePathSite 'To m y) -> SomePathSite 'To m y Source #

fromDual' :: Dualisable x => p x -> Dual x -> x Source #

fromDual enriched with a parameterized type p which serves as a proxy - e.g. Proxy or Id will serve - and will not be evaluated. It serves for the type checker to pick the right fromDual.

Reflexive

class Reflexive x where Source #

admitting reflection.

Property Let x be Reflexive, than holds:

toBidual is a bijection with its inverse fromBidual.

Methods

toBidual :: x -> Dual (Dual x) Source #

fromBidual :: Dual (Dual x) -> x Source #

Instances

Instances details

Reflexive (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods toBidual :: Path q -> Dual (Dual (Path q)) Source # fromBidual :: Dual (Dual (Path q)) -> Path q Source #

fromBidual' :: Reflexive x => p x -> Dual (Dual x) -> x Source #

fromBidual enriched with a parameterized type p which serves as a proxy - e.g. Proxy or Id will serve - and will not be evaluated. It serves for the type checker to pick the right fromBidual.

Transposable

class Transposable x where Source #

transposable types..

Property Let x be a Transposable, then holds: For all x in x holds: transpose (transpose x) == x.

Methods

transpose :: x -> x Source #

Instances

Instances details

Transposable Direction Source #
Instance details Defined in OAlg.Data.Dualisable Methods transpose :: Direction -> Direction Source #
Transposable Side Source #
Instance details Defined in OAlg.Data.Dualisable Methods transpose :: Side -> Side Source #
Transposable Site Source #
Instance details Defined in OAlg.Data.Dualisable Methods transpose :: Site -> Site Source #
Transposable N Source #
Instance details Defined in OAlg.Data.Number Methods transpose :: N -> N Source #
Transposable Q Source #
Instance details Defined in OAlg.Data.Number Methods transpose :: Q -> Q Source #
Transposable Z Source #
Instance details Defined in OAlg.Data.Number Methods transpose :: Z -> Z Source #
(Distributive x, TransposableDistributive x) => Transposable (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods transpose :: Matrix x -> Matrix x Source #
(Galoisian x, TransposableDistributive x) => Transposable (GL2 x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Methods transpose :: GL2 x -> GL2 x Source #
TransposableMultiplicative c => Transposable (Inv c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods transpose :: Inv c -> Inv c Source #
Transposable (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods transpose :: Orientation p -> Orientation p Source #
(Transposable x, Ord n) => Transposable (Entries n n x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods transpose :: Entries n n x -> Entries n n x Source #

Site

data Site Source #

concept of the sites From and To.

Constructors

From
To

Instances

Instances details

Bounded Site Source #
Instance details Defined in OAlg.Data.Dualisable Methods minBound :: Site # maxBound :: Site #
Enum Site Source #
Instance details Defined in OAlg.Data.Dualisable Methods succ :: Site -> Site # pred :: Site -> Site # toEnum :: Int -> Site # fromEnum :: Site -> Int # enumFrom :: Site -> [Site] # enumFromThen :: Site -> Site -> [Site] # enumFromTo :: Site -> Site -> [Site] # enumFromThenTo :: Site -> Site -> Site -> [Site] #
Show Site Source #
Instance details Defined in OAlg.Data.Dualisable Methods showsPrec :: Int -> Site -> ShowS # show :: Site -> String # showList :: [Site] -> ShowS #
Eq Site Source #
Instance details Defined in OAlg.Data.Dualisable Methods (==) :: Site -> Site -> Bool # (/=) :: Site -> Site -> Bool #
Ord Site Source #
Instance details Defined in OAlg.Data.Dualisable Methods compare :: Site -> Site -> Ordering # (<) :: Site -> Site -> Bool # (<=) :: Site -> Site -> Bool # (>) :: Site -> Site -> Bool # (>=) :: Site -> Site -> Bool # max :: Site -> Site -> Site # min :: Site -> Site -> Site #
Transposable Site Source #
Instance details Defined in OAlg.Data.Dualisable Methods transpose :: Site -> Site Source #
type Dual 'From Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'From = 'To
type Dual 'To Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'To = 'From

Side

data Side Source #

concept of sides LeftSide and RightSide

Constructors

LeftSide
RightSide

Instances

Instances details

Bounded Side Source #
Instance details Defined in OAlg.Data.Dualisable Methods minBound :: Side # maxBound :: Side #
Enum Side Source #
Instance details Defined in OAlg.Data.Dualisable Methods succ :: Side -> Side # pred :: Side -> Side # toEnum :: Int -> Side # fromEnum :: Side -> Int # enumFrom :: Side -> [Side] # enumFromThen :: Side -> Side -> [Side] # enumFromTo :: Side -> Side -> [Side] # enumFromThenTo :: Side -> Side -> Side -> [Side] #
Show Side Source #
Instance details Defined in OAlg.Data.Dualisable Methods showsPrec :: Int -> Side -> ShowS # show :: Side -> String # showList :: [Side] -> ShowS #
Eq Side Source #
Instance details Defined in OAlg.Data.Dualisable Methods (==) :: Side -> Side -> Bool # (/=) :: Side -> Side -> Bool #
Ord Side Source #
Instance details Defined in OAlg.Data.Dualisable Methods compare :: Side -> Side -> Ordering # (<) :: Side -> Side -> Bool # (<=) :: Side -> Side -> Bool # (>) :: Side -> Side -> Bool # (>=) :: Side -> Side -> Bool # max :: Side -> Side -> Side # min :: Side -> Side -> Side #
Transposable Side Source #
Instance details Defined in OAlg.Data.Dualisable Methods transpose :: Side -> Side Source #
type Dual 'LeftSide Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'LeftSide = 'RightSide
type Dual 'RightSide Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'RightSide = 'LeftSide

Direction

data Direction Source #

concept of the directions LeftToRight and RightToLeft.

Constructors

LeftToRight
RightToLeft

Instances

Instances details

Bounded Direction Source #
Instance details Defined in OAlg.Data.Dualisable Methods minBound :: Direction # maxBound :: Direction #
Enum Direction Source #
Instance details Defined in OAlg.Data.Dualisable Methods succ :: Direction -> Direction # pred :: Direction -> Direction # toEnum :: Int -> Direction # fromEnum :: Direction -> Int # enumFrom :: Direction -> [Direction] # enumFromThen :: Direction -> Direction -> [Direction] # enumFromTo :: Direction -> Direction -> [Direction] # enumFromThenTo :: Direction -> Direction -> Direction -> [Direction] #
Show Direction Source #
Instance details Defined in OAlg.Data.Dualisable Methods showsPrec :: Int -> Direction -> ShowS # show :: Direction -> String # showList :: [Direction] -> ShowS #
Eq Direction Source #
Instance details Defined in OAlg.Data.Dualisable Methods (==) :: Direction -> Direction -> Bool # (/=) :: Direction -> Direction -> Bool #
Ord Direction Source #
Instance details Defined in OAlg.Data.Dualisable Methods compare :: Direction -> Direction -> Ordering # (<) :: Direction -> Direction -> Bool # (<=) :: Direction -> Direction -> Bool # (>) :: Direction -> Direction -> Bool # (>=) :: Direction -> Direction -> Bool # max :: Direction -> Direction -> Direction # min :: Direction -> Direction -> Direction #
Transposable Direction Source #
Instance details Defined in OAlg.Data.Dualisable Methods transpose :: Direction -> Direction Source #
type Dual 'LeftToRight Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'LeftToRight = 'RightToLeft
type Dual 'RightToLeft Source #
Instance details Defined in OAlg.Data.Dualisable type Dual 'RightToLeft = 'LeftToRight