Copyright	(c) Erich Gut
License	BSD3
Maintainer	zerich.gut@gmail.com
Safe Haskell	None
Language	Haskell2010

OAlg.Category.Definition

Contents

Category
Cayleyan
Morphism
Applicative
Functorial
Transformables

Description

categories of morphisms. We adapted the concept of categories form Category to better cover our needs.

Synopsis

class Morphism c => Category (c :: Type -> Type -> Type) where
- cOne :: Struct (ObjectClass c) x -> c x x
- (.) :: c y z -> c x y -> c x z
cOne' :: Category c => p c -> Struct (ObjectClass c) x -> c x x
data Sub s (c :: Type -> Type -> Type) x y where
- Sub :: forall s x y (c :: Type -> Type -> Type). (Structure s x, Structure s y) => c x y -> Sub s c x y
cOneSub :: forall (c :: Type -> Type -> Type) t s x. (Category c, t ~ ObjectClass c) => Struct s x -> Struct t x -> Sub s c x x
sub :: (Morphism h, Transformable (ObjectClass h) s) => h x y -> Sub s h x y
sub' :: Homomorphous s x y -> h x y -> Sub s h x y
subG :: forall d (a :: Type -> Type -> Type) (b :: Type -> Type -> Type) s t x y. (ApplicativeG d a b, TransformableG d s t) => Sub s a x y -> Sub t b (d x) (d y)
newtype Op2 (h :: Type -> Type -> Type) x y = Op2 (h y x)
id :: x -> x
const :: b -> a -> b
curry :: ((a, b) -> c) -> a -> b -> c
uncurry :: (a -> b -> c) -> (a, b) -> c
fst :: (a, b) -> a
snd :: (a, b) -> b
curry3 :: ((a, b, c) -> d) -> a -> b -> c -> d
uncurry3 :: (a -> b -> c -> d) -> (a, b, c) -> d
class (Category c, Eq2 c) => Cayleyan2 (c :: Type -> Type -> Type) where
- invert2 :: c x y -> c y x
data Inv2 (c :: Type -> Type -> Type) x y = Inv2 (c x y) (c y x)
inv2 :: forall (c :: Type -> Type -> Type) x y. Inv2 c x y -> Inv2 c y x
inv2Forget :: forall s (h :: Type -> Type -> Type) x y. Inv2 (Sub s h) x y -> Inv2 h x y
class Morphism (m :: Type -> Type -> Type) where
- type ObjectClass (m :: Type -> Type -> Type)
- homomorphous :: m x y -> Homomorphous (ObjectClass m) x y
- domain :: m x y -> Struct (ObjectClass m) x
- range :: m x y -> Struct (ObjectClass m) y
data Homomorphous s x y = (Struct s x) :>: (Struct s y)
tauHom :: Transformable s t => Homomorphous s x y -> Homomorphous t x y
tauHomG :: TransformableG t u v => Homomorphous u x y -> Homomorphous v (t x) (t y)
tau1Hom :: Transformable1 f s => Homomorphous s x y -> Homomorphous s (f x) (f y)
eqlDomain :: Struct Typ x -> Struct Typ x' -> m x y -> m x' y -> Maybe (x :~: x')
eqlRange :: Struct Typ y -> Struct Typ y' -> m x y -> m x y' -> Maybe (y :~: y')
eqlEndo :: Struct Typ x -> Struct Typ y -> h x y -> Maybe (x :~: y)
eqlMorphism :: Typeable m => Struct Typ x -> Struct Typ x' -> Struct Typ y -> Struct Typ y' -> m x y -> m x' y' -> Maybe (m x y :~: m x' y')
class ApplicativeG (t :: Type -> Type) (a :: Type -> Type -> Type) (b :: Type -> Type -> Type) where
- amapG :: a x y -> b (t x) (t y)
type Applicative1 (h :: Type -> Type -> Type) (f :: Type -> Type) = ApplicativeG f h (->)
amap1 :: Applicative1 h f => h x y -> f x -> f y
amapF :: FunctorialG t a b => a x y -> b (t x) (t y)
type Functorial1 (c :: Type -> Type -> Type) (f :: Type -> Type) = FunctorialG f c (->)
data Functor1 (c :: Type -> Type -> Type) (f :: Type -> Type) where
- Functor1 :: forall (c :: Type -> Type -> Type) (f :: Type -> Type). Functorial1 c f => Functor1 c f
class (Category a, Category b, ApplicativeG t a b, TransformableG t (ObjectClass a) (ObjectClass b)) => FunctorialG (t :: Type -> Type) (a :: Type -> Type -> Type) (b :: Type -> Type -> Type)
data FunctorG (t :: Type -> Type) (a :: Type -> Type -> Type) (b :: Type -> Type -> Type) where
- FunctorG :: forall (t :: Type -> Type) (a :: Type -> Type -> Type) (b :: Type -> Type -> Type). FunctorialG t a b => FunctorG t a b
class Transformable s (ObjectClass c) => TransformableObjectClass s (c :: Type -> Type -> Type)
class Transformable (ObjectClass m) Typ => TransformableObjectClassTyp (m :: Type -> Type -> Type)
class TransformableG t (ObjectClass a) (ObjectClass b) => TransformableGObjectClass (t :: Type -> Type) (a :: Type -> Type -> Type) (b :: Type -> Type -> Type)
class TransformableG d (ObjectClass a) t => TransformableGObjectClassDomain (d :: Type -> Type) (a :: Type -> Type -> Type) t
class TransformableG d s (ObjectClass c) => TransformableGObjectClassRange (d :: Type -> Type) s (c :: Type -> Type -> Type)

Cayleyan

class (Category c, Eq2 c) => Cayleyan2 (c :: Type -> Type -> Type) where Source #

category of isomorphisms.

Property Let c be a type instance of Cayleyan2, then holds: For all types x, y and f in c x y holds:

(invert2 f . f) == cOne (domain f) and (f . invert2 f) == cOne (range f) where (==) = eq2.

Methods

invert2 :: c x y -> c y x Source #

Instances

Instances details

Cayleyan2 (Homomorphous m) Source #
Instance details Defined in OAlg.Category.Definition Methods invert2 :: Homomorphous m x y -> Homomorphous m y x Source #
(Category c, Eq2 c) => Cayleyan2 (Inv2 c) Source #
Instance details Defined in OAlg.Category.Definition Methods invert2 :: Inv2 c x y -> Inv2 c y x Source #
Cayleyan2 c => Cayleyan2 (Op2 c) Source #
Instance details Defined in OAlg.Category.Definition Methods invert2 :: Op2 c x y -> Op2 c y x Source #
(Cayleyan2 m, TransformableObjectClassTyp m) => Cayleyan2 (Path m) Source #
Instance details Defined in OAlg.Category.Path Methods invert2 :: Path m x y -> Path m y x Source #

data Inv2 (c :: Type -> Type -> Type) x y Source #

predicate for invertible morphisms within a category c.

Property Let Inv2 f f' be in Inv2 c x y for a Category c with Eq2 c, then holds:

f' . f .=. cOne (domain f).
f . f' .=. cOne (range f).

Constructors

Inv2 (c x y) (c y x)

Instances

Instances details

CategoryDisjunctive h => Disjunctive2 (Inv2 h :: Type -> Type -> Type) Source #

Instance details

Defined in OAlg.Data.Variant

Methods

variant2 :: Inv2 h x y -> Variant Source #

(CategoryDisjunctive h, HomSlicedOriented i h) => HomSlicedOriented i (Inv2 h) Source #

Instance details

Defined in OAlg.Entity.Slice.Sliced

ApplicativeG Id h c => ApplicativeG Id (Inv2 h) c Source #

Instance details

Defined in OAlg.Data.Identity

Methods

amapG :: Inv2 h x y -> c (Id x) (Id y) Source #

ApplicativeG Rt h c => ApplicativeG Rt (Inv2 h) c Source #

Instance details

Defined in OAlg.Structure.Fibred.Root

Methods

amapG :: Inv2 h x y -> c (Rt x) (Rt y) Source #

ApplicativeG Pnt h c => ApplicativeG Pnt (Inv2 h) c Source #

Instance details

Defined in OAlg.Structure.Oriented.Point

Methods

amapG :: Inv2 h x y -> c (Pnt x) (Pnt y) Source #

FunctorialG Id h c => FunctorialG Id (Inv2 h) c Source #

Instance details

Defined in OAlg.Data.Identity

FunctorialG Pnt h c => FunctorialG Pnt (Inv2 h) c Source #

Instance details

Defined in OAlg.Structure.Oriented.Point

(TransformableOrt s, TransformableType s, TransformableOp s) => HomSlicedOriented i (Sub (s, Sld i) (IsoO s Op)) Source #

Instance details

Defined in OAlg.Entity.Slice.Sliced

Category c => Category (Inv2 c) Source #

Instance details

Defined in OAlg.Category.Definition

Methods

cOne :: Struct (ObjectClass (Inv2 c)) x -> Inv2 c x x Source #

(.) :: Inv2 c y z -> Inv2 c x y -> Inv2 c x z Source #

(Category c, Eq2 c) => Cayleyan2 (Inv2 c) Source #

Instance details

Defined in OAlg.Category.Definition

Methods

invert2 :: Inv2 c x y -> Inv2 c y x Source #

Morphism c => Morphism (Inv2 c) Source #

Instance details

Defined in OAlg.Category.Definition

Associated Types

type ObjectClass (Inv2 c)
Instance details Defined in OAlg.Category.Definition type ObjectClass (Inv2 c) = ObjectClass c

Methods

homomorphous :: Inv2 c x y -> Homomorphous (ObjectClass (Inv2 c)) x y Source #

domain :: Inv2 c x y -> Struct (ObjectClass (Inv2 c)) x Source #

range :: Inv2 c x y -> Struct (ObjectClass (Inv2 c)) y Source #

Eq2 c => Eq2 (Inv2 c) Source #

Instance details

Defined in OAlg.Category.Definition

Methods

eq2 :: Inv2 c x y -> Inv2 c x y -> Bool Source #

(Category c, EqExt c) => Validable2 (Inv2 c) Source #

Instance details

Defined in OAlg.Data.Validable

Methods

valid2 :: Inv2 c x y -> Statement Source #

CategoryDisjunctive c => CategoryDisjunctive (Inv2 c) Source #

Instance details

Defined in OAlg.Data.Variant

HomOrientedSlicedFree (Inv2 (HomFree Dst)) Source #

Instance details

Defined in OAlg.Entity.Slice.Free

HomOrientedSlicedFree (Inv2 (HomFree Mlt)) Source #

Instance details

Defined in OAlg.Entity.Slice.Free

HomAdditive h => HomAdditive (Inv2 h) Source #

Instance details

Defined in OAlg.Hom.Additive

HomDistributive h => HomDistributive (Inv2 h) Source #

Instance details

Defined in OAlg.Hom.Distributive

(CategoryDisjunctive h, HomDistributiveDisjunctive h) => HomDistributiveDisjunctive (Inv2 h) Source #

Instance details

Defined in OAlg.Hom.Distributive

HomFibred h => HomFibred (Inv2 h) Source #

Instance details

Defined in OAlg.Hom.Fibred

HomFibredOriented h => HomFibredOriented (Inv2 h) Source #

Instance details

Defined in OAlg.Hom.FibredOriented

(CategoryDisjunctive h, HomFibredOrientedDisjunctive h) => HomFibredOrientedDisjunctive (Inv2 h) Source #

Instance details

Defined in OAlg.Hom.FibredOriented

HomMultiplicative h => HomMultiplicative (Inv2 h) Source #

Instance details

Defined in OAlg.Hom.Multiplicative

(CategoryDisjunctive h, HomMultiplicativeDisjunctive h) => HomMultiplicativeDisjunctive (Inv2 h) Source #

Instance details

Defined in OAlg.Hom.Multiplicative

FunctorialOriented h => FunctorialOriented (Inv2 h) Source #

Instance details

Defined in OAlg.Hom.Oriented.Definition

HomOriented h => HomOriented (Inv2 h) Source #

Instance details

Defined in OAlg.Hom.Oriented.Definition

(CategoryDisjunctive h, HomOrientedDisjunctive h) => HomOrientedDisjunctive (Inv2 h) Source #

Instance details

Defined in OAlg.Hom.Oriented.Definition

(CategoryDisjunctive h, HomSlicedDistributive i h, FunctorialOriented h, p ~ Dual (Dual p), t ~ Dual (Dual t), s ~ Dst) => NaturalConic (Inv2 h) (LiftableCone i) s p Diagram t n m Source #

Instance details

Defined in OAlg.Entity.Slice.Liftable

(NaturalDiagrammatic (Inv2 (HomFree s)) d ('Parallel 'LeftToRight) N2 N1, NaturalDiagrammatic (Inv2 (HomFree s)) d ('Parallel 'RightToLeft) N2 N1, p ~ Dual (Dual p), t ~ Dual (Dual t)) => ApplicativeG (SDualBi (ConeLiftable s p d t n m)) (Inv2 (HomFree s)) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Free

Methods

amapG :: Inv2 (HomFree s) x y -> SDualBi (ConeLiftable s p d t n m) x -> SDualBi (ConeLiftable s p d t n m) y Source #

p ~ Dual (Dual p) => ApplicativeG (SDualBi (LiftableFree p)) (Inv2 (HomFree Dst)) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Free

Methods

amapG :: Inv2 (HomFree Dst) x y -> SDualBi (LiftableFree p) x -> SDualBi (LiftableFree p) y Source #

p ~ Dual (Dual p) => ApplicativeG (SDualBi (LiftableFree p)) (Inv2 (HomFree Mlt)) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Free

Methods

amapG :: Inv2 (HomFree Mlt) x y -> SDualBi (LiftableFree p) x -> SDualBi (LiftableFree p) y Source #

(CategoryDisjunctive h, HomSlicedMultiplicative i h, p ~ Dual (Dual p)) => ApplicativeG (SDualBi (Liftable p i)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Liftable

Methods

amapG :: Inv2 h x y -> SDualBi (Liftable p i) x -> SDualBi (Liftable p i) y Source #

(CategoryDisjunctive h, HomSlicedDistributive i h, FunctorialOriented h, p ~ Dual (Dual p), t ~ Dual (Dual t)) => ApplicativeG (SDualBi (LiftableCone i s p d t n m)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Liftable

Methods

amapG :: Inv2 h x y -> SDualBi (LiftableCone i s p d t n m) x -> SDualBi (LiftableCone i s p d t n m) y Source #

(CategoryDisjunctive h, HomSlicedDistributive i h, FunctorialOriented h, p ~ Dual (Dual p), t ~ Dual (Dual t)) => ApplicativeG (SDualBi (ConeG (LiftableCone i) s p d t n m)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Liftable

Methods

amapG :: Inv2 h x y -> SDualBi (ConeG (LiftableCone i) s p d t n m) x -> SDualBi (ConeG (LiftableCone i) s p d t n m) y Source #

NaturalConicBi (Inv2 h) c s p d t n m => ApplicativeG (SDualBi (LimesG c s p d t n m)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Limes.Definition.Duality

Methods

amapG :: Inv2 h x y -> SDualBi (LimesG c s p d t n m) x -> SDualBi (LimesG c s p d t n m) y Source #

(HomDistributiveDisjunctive h, CategoryDisjunctive h, NaturalKernelCokernel (Inv2 h) k c d, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (VarianceG t k c d n)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Limes.Exact.Deviation

Methods

amapG :: Inv2 h x y -> SDualBi (VarianceG t k c d n) x -> SDualBi (VarianceG t k c d n) y Source #

(HomDistributiveDisjunctive h, CategoryDisjunctive h, NaturalKernelCokernel (Inv2 h) k c d, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (VarianceGHom t k c d n)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Limes.Exact.Deviation

Methods

amapG :: Inv2 h x y -> SDualBi (VarianceGHom t k c d n) x -> SDualBi (VarianceGHom t k c d n) y Source #

NaturalConicBi (Inv2 h) c s p d t n m => ApplicativeG (SDualBi (LimitsG c s p d t n m)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Limes.Limits.Duality

Methods

amapG :: Inv2 h x y -> SDualBi (LimitsG c s p d t n m) x -> SDualBi (LimitsG c s p d t n m) y Source #

(NaturalDiagrammatic (Inv2 (HomFree s)) d ('Parallel 'LeftToRight) N2 N1, NaturalDiagrammatic (Inv2 (HomFree s)) d ('Parallel 'RightToLeft) N2 N1, p ~ Dual (Dual p), t ~ Dual (Dual t)) => FunctorialG (SDualBi (ConeLiftable s p d t n m)) (Inv2 (HomFree s)) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Free

p ~ Dual (Dual p) => FunctorialG (SDualBi (LiftableFree p)) (Inv2 (HomFree Dst)) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Free

p ~ Dual (Dual p) => FunctorialG (SDualBi (LiftableFree p)) (Inv2 (HomFree Mlt)) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Free

(CategoryDisjunctive h, HomSlicedMultiplicative i h, p ~ Dual (Dual p)) => FunctorialG (SDualBi (Liftable p i)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Liftable

(CategoryDisjunctive h, HomSlicedDistributive i h, FunctorialOriented h, p ~ Dual (Dual p), t ~ Dual (Dual t)) => FunctorialG (SDualBi (LiftableCone i s p d t n m)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Liftable

(CategoryDisjunctive h, HomSlicedDistributive i h, FunctorialOriented h, p ~ Dual (Dual p), t ~ Dual (Dual t)) => FunctorialG (SDualBi (ConeG (LiftableCone i) s p d t n m)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Entity.Slice.Liftable

NaturalConicBi (Inv2 h) c s p d t n m => FunctorialG (SDualBi (LimesG c s p d t n m)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Limes.Definition.Duality

(HomDistributiveDisjunctive h, CategoryDisjunctive h, NaturalKernelCokernel (Inv2 h) k c d, t ~ Dual (Dual t)) => FunctorialG (SDualBi (VarianceG t k c d n)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Limes.Exact.Deviation

(HomDistributiveDisjunctive h, CategoryDisjunctive h, NaturalKernelCokernel (Inv2 h) k c d, t ~ Dual (Dual t)) => FunctorialG (SDualBi (VarianceGHom t k c d n)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Limes.Exact.Deviation

NaturalConicBi (Inv2 h) c s p d t n m => FunctorialG (SDualBi (LimitsG c s p d t n m)) (Inv2 h) (->) Source #

Instance details

Defined in OAlg.Limes.Limits.Duality

(CategoryDisjunctive h, HomSlicedDistributive i h, FunctorialOriented h, p ~ Dual (Dual p), t ~ Dual (Dual t), s ~ Dst) => NaturalTransformable (Inv2 h) (->) (SDualBi (ConeG (LiftableCone i) s p Diagram t n m)) (SDualBi (ConeG Cone s p Diagram t n m)) Source #

Instance details

Defined in OAlg.Entity.Slice.Liftable

(Show (c x y), Show (c y x)) => Show (Inv2 c x y) Source #

Instance details

Defined in OAlg.Category.Definition

Methods

showsPrec :: Int -> Inv2 c x y -> ShowS #

show :: Inv2 c x y -> String #

showList :: [Inv2 c x y] -> ShowS #

(Eq (c x y), Eq (c y x)) => Eq (Inv2 c x y) Source #

Instance details

Defined in OAlg.Category.Definition

Methods

(==) :: Inv2 c x y -> Inv2 c x y -> Bool #

(/=) :: Inv2 c x y -> Inv2 c x y -> Bool #

(Category c, EqExt c) => Validable (Inv2 c x y) Source #

Instance details

Defined in OAlg.Data.Validable

Methods

valid :: Inv2 c x y -> Statement Source #

type ObjectClass (Inv2 c) Source #

Instance details

Defined in OAlg.Category.Definition

type ObjectClass (Inv2 c) = ObjectClass c

inv2 :: forall (c :: Type -> Type -> Type) x y. Inv2 c x y -> Inv2 c y x Source #

the inverse.

inv2Forget :: forall s (h :: Type -> Type -> Type) x y. Inv2 (Sub s h) x y -> Inv2 h x y Source #

forgetting the restriction to Sub.

Morphism

class Morphism (m :: Type -> Type -> Type) where Source #

morphism.

Minimal complete definition

homomorphous | domain, range

Associated Types

type ObjectClass (m :: Type -> Type -> Type) Source #

the object class.

Methods

homomorphous :: m x y -> Homomorphous (ObjectClass m) x y Source #

attests, that the types x and y fulfill the constraints given by Homomorphous (ObjectClass m) x y, i.e both fulfill the constraints given by Structure (ObjectClass m) x and Structure (ObjectClass m) y respectively.

domain :: m x y -> Struct (ObjectClass m) x Source #

attests that the domain type x fulfills the constraints given by Structure (ObjectClass m) x.

range :: m x y -> Struct (ObjectClass m) y Source #

attests that the range type y fulfills the constraints given by Structure (ObjectClass m) y.

Instances

Instances details

Morphism GLApp Source #

Instance details

Defined in OAlg.Entity.Matrix.GeneralLinearGroup

Associated Types

type ObjectClass GLApp
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup type ObjectClass GLApp = Mlt

Methods

homomorphous :: GLApp x y -> Homomorphous (ObjectClass GLApp) x y Source #

domain :: GLApp x y -> Struct (ObjectClass GLApp) x Source #

range :: GLApp x y -> Struct (ObjectClass GLApp) y Source #

Morphism TrApp Source #

Instance details

Defined in OAlg.Entity.Matrix.GeneralLinearGroup

Associated Types

type ObjectClass TrApp
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup type ObjectClass TrApp = Ort

Methods

homomorphous :: TrApp x y -> Homomorphous (ObjectClass TrApp) x y Source #

domain :: TrApp x y -> Struct (ObjectClass TrApp) x Source #

range :: TrApp x y -> Struct (ObjectClass TrApp) y Source #

Morphism (Homomorphous s) Source #

Instance details

Defined in OAlg.Category.Definition

Associated Types

type ObjectClass (Homomorphous s)
Instance details Defined in OAlg.Category.Definition type ObjectClass (Homomorphous s) = s

Methods

homomorphous :: Homomorphous s x y -> Homomorphous (ObjectClass (Homomorphous s)) x y Source #

domain :: Homomorphous s x y -> Struct (ObjectClass (Homomorphous s)) x Source #

range :: Homomorphous s x y -> Struct (ObjectClass (Homomorphous s)) y Source #

Morphism c => Morphism (Inv2 c) Source #

Instance details

Defined in OAlg.Category.Definition

Associated Types

type ObjectClass (Inv2 c)
Instance details Defined in OAlg.Category.Definition type ObjectClass (Inv2 c) = ObjectClass c

Methods

homomorphous :: Inv2 c x y -> Homomorphous (ObjectClass (Inv2 c)) x y Source #

domain :: Inv2 c x y -> Struct (ObjectClass (Inv2 c)) x Source #

range :: Inv2 c x y -> Struct (ObjectClass (Inv2 c)) y Source #

Morphism h => Morphism (Op2 h) Source #

Instance details

Defined in OAlg.Category.Definition

Associated Types

type ObjectClass (Op2 h)
Instance details Defined in OAlg.Category.Definition type ObjectClass (Op2 h) = ObjectClass h

Methods

homomorphous :: Op2 h x y -> Homomorphous (ObjectClass (Op2 h)) x y Source #

domain :: Op2 h x y -> Struct (ObjectClass (Op2 h)) x Source #

range :: Op2 h x y -> Struct (ObjectClass (Op2 h)) y Source #

Morphism (Map s) Source #

Instance details

Defined in OAlg.Category.Map

Associated Types

type ObjectClass (Map s)
Instance details Defined in OAlg.Category.Map type ObjectClass (Map s) = s

Methods

homomorphous :: Map s x y -> Homomorphous (ObjectClass (Map s)) x y Source #

domain :: Map s x y -> Struct (ObjectClass (Map s)) x Source #

range :: Map s x y -> Struct (ObjectClass (Map s)) y Source #

Morphism m => Morphism (Path m) Source #

Instance details

Defined in OAlg.Category.Path

Associated Types

type ObjectClass (Path m)
Instance details Defined in OAlg.Category.Path type ObjectClass (Path m) = ObjectClass m

Methods

homomorphous :: Path m x y -> Homomorphous (ObjectClass (Path m)) x y Source #

domain :: Path m x y -> Struct (ObjectClass (Path m)) x Source #

range :: Path m x y -> Struct (ObjectClass (Path m)) y Source #

Morphism h => Morphism (Id2 h) Source #

Instance details

Defined in OAlg.Data.Identity

Associated Types

type ObjectClass (Id2 h)
Instance details Defined in OAlg.Data.Identity type ObjectClass (Id2 h) = ObjectClass h

Methods

homomorphous :: Id2 h x y -> Homomorphous (ObjectClass (Id2 h)) x y Source #

domain :: Id2 h x y -> Struct (ObjectClass (Id2 h)) x Source #

range :: Id2 h x y -> Struct (ObjectClass (Id2 h)) y Source #

Morphism (Ornt s) Source #

Instance details

Defined in OAlg.Data.Ornt

Associated Types

type ObjectClass (Ornt s)
Instance details Defined in OAlg.Data.Ornt type ObjectClass (Ornt s) = s

Methods

homomorphous :: Ornt s x y -> Homomorphous (ObjectClass (Ornt s)) x y Source #

domain :: Ornt s x y -> Struct (ObjectClass (Ornt s)) x Source #

range :: Ornt s x y -> Struct (ObjectClass (Ornt s)) y Source #

(Semiring r, Commutative r) => Morphism (HomSymbol r) Source #

Instance details

Defined in OAlg.Entity.Matrix.Vector

Associated Types

type ObjectClass (HomSymbol r)
Instance details Defined in OAlg.Entity.Matrix.Vector type ObjectClass (HomSymbol r) = Vec r

Methods

homomorphous :: HomSymbol r x y -> Homomorphous (ObjectClass (HomSymbol r)) x y Source #

domain :: HomSymbol r x y -> Struct (ObjectClass (HomSymbol r)) x Source #

range :: HomSymbol r x y -> Struct (ObjectClass (HomSymbol r)) y Source #

Morphism (SliceFactorDrop s) Source #

Instance details

Defined in OAlg.Entity.Slice.Definition

Associated Types

type ObjectClass (SliceFactorDrop s)
Instance details Defined in OAlg.Entity.Slice.Definition type ObjectClass (SliceFactorDrop s) = Mlt

Methods

homomorphous :: SliceFactorDrop s x y -> Homomorphous (ObjectClass (SliceFactorDrop s)) x y Source #

domain :: SliceFactorDrop s x y -> Struct (ObjectClass (SliceFactorDrop s)) x Source #

range :: SliceFactorDrop s x y -> Struct (ObjectClass (SliceFactorDrop s)) y Source #

Morphism (HomEmpty s) Source #

Instance details

Defined in OAlg.Hom.Definition

Associated Types

type ObjectClass (HomEmpty s)
Instance details Defined in OAlg.Hom.Definition type ObjectClass (HomEmpty s) = s

Methods

homomorphous :: HomEmpty s x y -> Homomorphous (ObjectClass (HomEmpty s)) x y Source #

domain :: HomEmpty s x y -> Struct (ObjectClass (HomEmpty s)) x Source #

range :: HomEmpty s x y -> Struct (ObjectClass (HomEmpty s)) y Source #

Morphism (HomId s) Source #

Instance details

Defined in OAlg.Hom.Definition

Associated Types

type ObjectClass (HomId s)
Instance details Defined in OAlg.Hom.Definition type ObjectClass (HomId s) = s

Methods

homomorphous :: HomId s x y -> Homomorphous (ObjectClass (HomId s)) x y Source #

domain :: HomId s x y -> Struct (ObjectClass (HomId s)) x Source #

range :: HomId s x y -> Struct (ObjectClass (HomId s)) y Source #

Morphism (Sub s c) Source #

Instance details

Defined in OAlg.Category.Definition

Associated Types

type ObjectClass (Sub s c)
Instance details Defined in OAlg.Category.Definition type ObjectClass (Sub s c) = s

Methods

homomorphous :: Sub s c x y -> Homomorphous (ObjectClass (Sub s c)) x y Source #

domain :: Sub s c x y -> Struct (ObjectClass (Sub s c)) x Source #

range :: Sub s c x y -> Struct (ObjectClass (Sub s c)) y Source #

(Morphism f, Morphism g, ObjectClass f ~ ObjectClass g) => Morphism (Either2 f g) Source #

Instance details

Defined in OAlg.Category.Definition

Associated Types

type ObjectClass (Either2 f g)
Instance details Defined in OAlg.Category.Definition type ObjectClass (Either2 f g) = ObjectClass f

Methods

homomorphous :: Either2 f g x y -> Homomorphous (ObjectClass (Either2 f g)) x y Source #

domain :: Either2 f g x y -> Struct (ObjectClass (Either2 f g)) x Source #

range :: Either2 f g x y -> Struct (ObjectClass (Either2 f g)) y Source #

Morphism (->) Source #

Instance details

Defined in OAlg.Category.Definition

Associated Types

type ObjectClass (->)
Instance details Defined in OAlg.Category.Definition type ObjectClass (->) = Type

Methods

homomorphous :: (x -> y) -> Homomorphous (ObjectClass (->)) x y Source #

domain :: (x -> y) -> Struct (ObjectClass (->)) x Source #

range :: (x -> y) -> Struct (ObjectClass (->)) y Source #

Morphism (HomCo m s o) Source #

Instance details

Defined in OAlg.Data.HomCo

Associated Types

type ObjectClass (HomCo m s o)
Instance details Defined in OAlg.Data.HomCo type ObjectClass (HomCo m s o) = s

Methods

homomorphous :: HomCo m s o x y -> Homomorphous (ObjectClass (HomCo m s o)) x y Source #

domain :: HomCo m s o x y -> Struct (ObjectClass (HomCo m s o)) x Source #

range :: HomCo m s o x y -> Struct (ObjectClass (HomCo m s o)) y Source #

Morphism (MorCo m s o) Source #

Instance details

Defined in OAlg.Data.HomCo

Associated Types

type ObjectClass (MorCo m s o)
Instance details Defined in OAlg.Data.HomCo type ObjectClass (MorCo m s o) = s

Methods

homomorphous :: MorCo m s o x y -> Homomorphous (ObjectClass (MorCo m s o)) x y Source #

domain :: MorCo m s o x y -> Struct (ObjectClass (MorCo m s o)) x Source #

range :: MorCo m s o x y -> Struct (ObjectClass (MorCo m s o)) y Source #

(Multiplicative d, Sliced i d) => Morphism (SliceAdjunction i c d) Source #

Instance details

Defined in OAlg.Entity.Slice.Adjunction

Associated Types

type ObjectClass (SliceAdjunction i c d)
Instance details Defined in OAlg.Entity.Slice.Adjunction type ObjectClass (SliceAdjunction i c d) = Mlt

Methods

homomorphous :: SliceAdjunction i c d x y -> Homomorphous (ObjectClass (SliceAdjunction i c d)) x y Source #

domain :: SliceAdjunction i c d x y -> Struct (ObjectClass (SliceAdjunction i c d)) x Source #

range :: SliceAdjunction i c d x y -> Struct (ObjectClass (SliceAdjunction i c d)) y Source #

Morphism h => Morphism (HomDisj s o h) Source #

Instance details

Defined in OAlg.Hom.Definition

Associated Types

type ObjectClass (HomDisj s o h)
Instance details Defined in OAlg.Hom.Definition type ObjectClass (HomDisj s o h) = s

Methods

homomorphous :: HomDisj s o h x y -> Homomorphous (ObjectClass (HomDisj s o h)) x y Source #

domain :: HomDisj s o h x y -> Struct (ObjectClass (HomDisj s o h)) x Source #

range :: HomDisj s o h x y -> Struct (ObjectClass (HomDisj s o h)) y Source #

Morphism h => Morphism (SHom r s o h) Source #

Instance details

Defined in OAlg.Category.SDuality

Associated Types

type ObjectClass (SHom r s o h)
Instance details Defined in OAlg.Category.SDuality type ObjectClass (SHom r s o h) = s

Methods

homomorphous :: SHom r s o h x y -> Homomorphous (ObjectClass (SHom r s o h)) x y Source #

domain :: SHom r s o h x y -> Struct (ObjectClass (SHom r s o h)) x Source #

range :: SHom r s o h x y -> Struct (ObjectClass (SHom r s o h)) y Source #

Morphism h => Morphism (SMorphism r s o h) Source #

Instance details

Defined in OAlg.Category.SDuality

Associated Types

type ObjectClass (SMorphism r s o h)
Instance details Defined in OAlg.Category.SDuality type ObjectClass (SMorphism r s o h) = s

Methods

homomorphous :: SMorphism r s o h x y -> Homomorphous (ObjectClass (SMorphism r s o h)) x y Source #

domain :: SMorphism r s o h x y -> Struct (ObjectClass (SMorphism r s o h)) x Source #

range :: SMorphism r s o h x y -> Struct (ObjectClass (SMorphism r s o h)) y Source #

Morphism h => Morphism (Variant2 v h) Source #

Instance details

Defined in OAlg.Data.Variant

Associated Types

type ObjectClass (Variant2 v h)
Instance details Defined in OAlg.Data.Variant type ObjectClass (Variant2 v h) = ObjectClass h

Methods

homomorphous :: Variant2 v h x y -> Homomorphous (ObjectClass (Variant2 v h)) x y Source #

domain :: Variant2 v h x y -> Struct (ObjectClass (Variant2 v h)) x Source #

range :: Variant2 v h x y -> Struct (ObjectClass (Variant2 v h)) y Source #

data Homomorphous s x y Source #

attest that both x and y have homomorphous structures, i.e. both admit the same constraints given by the parameter s.

Constructors

(Struct s x) :>: (Struct s y) infix 5

Instances

Instances details

Category (Homomorphous s) Source #

Instance details

Defined in OAlg.Category.Definition

Methods

cOne :: Struct (ObjectClass (Homomorphous s)) x -> Homomorphous s x x Source #

(.) :: Homomorphous s y z -> Homomorphous s x y -> Homomorphous s x z Source #

Cayleyan2 (Homomorphous m) Source #

Instance details

Defined in OAlg.Category.Definition

Methods

invert2 :: Homomorphous m x y -> Homomorphous m y x Source #

Morphism (Homomorphous s) Source #

Instance details

Defined in OAlg.Category.Definition

Associated Types

type ObjectClass (Homomorphous s)
Instance details Defined in OAlg.Category.Definition type ObjectClass (Homomorphous s) = s

Methods

homomorphous :: Homomorphous s x y -> Homomorphous (ObjectClass (Homomorphous s)) x y Source #

domain :: Homomorphous s x y -> Struct (ObjectClass (Homomorphous s)) x Source #

range :: Homomorphous s x y -> Struct (ObjectClass (Homomorphous s)) y Source #

Eq2 (Homomorphous m) Source #

Instance details

Defined in OAlg.Category.Definition

Methods

eq2 :: Homomorphous m x y -> Homomorphous m x y -> Bool Source #

Show2 (Homomorphous m) Source #

Instance details

Defined in OAlg.Category.Definition

Methods

show2 :: Homomorphous m a b -> String Source #

Show (Homomorphous s x y) Source #

Instance details

Defined in OAlg.Category.Definition

Methods

showsPrec :: Int -> Homomorphous s x y -> ShowS #

show :: Homomorphous s x y -> String #

showList :: [Homomorphous s x y] -> ShowS #

Eq (Homomorphous s x y) Source #

Instance details

Defined in OAlg.Category.Definition

Methods

(==) :: Homomorphous s x y -> Homomorphous s x y -> Bool #

(/=) :: Homomorphous s x y -> Homomorphous s x y -> Bool #

type ObjectClass (Homomorphous s) Source #

Instance details

Defined in OAlg.Category.Definition

type ObjectClass (Homomorphous s) = s

tauHom :: Transformable s t => Homomorphous s x y -> Homomorphous t x y Source #

transforming homomorphous structural attests.

tauHomG :: TransformableG t u v => Homomorphous u x y -> Homomorphous v (t x) (t y) Source #

transforming homomorphous structural attests.

tau1Hom :: Transformable1 f s => Homomorphous s x y -> Homomorphous s (f x) (f y) Source #

transforming homomorphous structural attests.

eqlDomain :: Struct Typ x -> Struct Typ x' -> m x y -> m x' y -> Maybe (x :~: x') Source #

gets for two Typeable types x and x' and for two parameterized types maybe an attest that the domain types are equal.

eqlRange :: Struct Typ y -> Struct Typ y' -> m x y -> m x y' -> Maybe (y :~: y') Source #

gets for two Typeable types y and y' and for two parameterized types maybe an attest that the range types are equal.

eqlEndo :: Struct Typ x -> Struct Typ y -> h x y -> Maybe (x :~: y) Source #

maybe endomorphism.

eqlMorphism :: Typeable m => Struct Typ x -> Struct Typ x' -> Struct Typ y -> Struct Typ y' -> m x y -> m x' y' -> Maybe (m x y :~: m x' y') Source #

gets maybe an attest that the two given morphisms types are equal.

Applicative

class ApplicativeG (t :: Type -> Type) (a :: Type -> Type -> Type) (b :: Type -> Type -> Type) where Source #

generalized application.

Methods

amapG :: a x y -> b (t x) (t y) Source #

application.

Instances

Instances details

ApplicativeG t EntEmpty2 b Source #
Instance details Defined in OAlg.Entity.Definition Methods amapG :: EntEmpty2 x y -> b (t x) (t y) Source #
ApplicativeG Id GLApp (->) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Methods amapG :: GLApp x y -> Id x -> Id y Source #
ApplicativeG Id TrApp (->) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Methods amapG :: TrApp x y -> Id x -> Id y Source #
ApplicativeG Pnt GLApp (->) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Methods amapG :: GLApp x y -> Pnt x -> Pnt y Source #
ApplicativeG Pnt TrApp (->) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Methods amapG :: TrApp x y -> Pnt x -> Pnt y Source #
ApplicativeG Id h c => ApplicativeG Id (Inv2 h) c Source #
Instance details Defined in OAlg.Data.Identity Methods amapG :: Inv2 h x y -> c (Id x) (Id y) Source #
ApplicativeG Id h c => ApplicativeG Id (Id2 h) c Source #
Instance details Defined in OAlg.Data.Identity Methods amapG :: Id2 h x y -> c (Id x) (Id y) Source #
ApplicativeG Id (HomEmpty s) c Source #
Instance details Defined in OAlg.Hom.Definition Methods amapG :: HomEmpty s x y -> c (Id x) (Id y) Source #
ApplicativeG Rt h c => ApplicativeG Rt (Inv2 h) c Source #
Instance details Defined in OAlg.Structure.Fibred.Root Methods amapG :: Inv2 h x y -> c (Rt x) (Rt y) Source #
ApplicativeG Rt (HomEmpty s) c Source #
Instance details Defined in OAlg.Hom.Definition Methods amapG :: HomEmpty s x y -> c (Rt x) (Rt y) Source #
ApplicativeG Pnt h c => ApplicativeG Pnt (Inv2 h) c Source #
Instance details Defined in OAlg.Structure.Oriented.Point Methods amapG :: Inv2 h x y -> c (Pnt x) (Pnt y) Source #
ApplicativeG Pnt h c => ApplicativeG Pnt (Id2 h) c Source #
Instance details Defined in OAlg.Structure.Oriented.Point Methods amapG :: Id2 h x y -> c (Pnt x) (Pnt y) Source #
ApplicativeG Pnt (HomEmpty s) c Source #
Instance details Defined in OAlg.Hom.Definition Methods amapG :: HomEmpty s x y -> c (Pnt x) (Pnt y) Source #
(Category c, ApplicativeG t m c, TransformableGObjectClass t m c) => ApplicativeG t (Path m) c Source #
Instance details Defined in OAlg.Category.Path Methods amapG :: Path m x y -> c (t x) (t y) Source #
TransformableOrt s => ApplicativeG Id (Ornt s) (->) Source #
Instance details Defined in OAlg.Data.Ornt Methods amapG :: Ornt s x y -> Id x -> Id y Source #
(Semiring r, Commutative r) => ApplicativeG Id (HomSymbol r) (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Vector Methods amapG :: HomSymbol r x y -> Id x -> Id y Source #
ApplicativeG Id (SliceFactorDrop s) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Definition Methods amapG :: SliceFactorDrop s x y -> Id x -> Id y Source #
ApplicativeG Id (HomId s) (->) Source #
Instance details Defined in OAlg.Hom.Definition Methods amapG :: HomId s x y -> Id x -> Id y Source #
ApplicativeG Set (Map EntOrd) (->) Source #
Instance details Defined in OAlg.Entity.Sequence.Set Methods amapG :: Map EntOrd x y -> Set x -> Set y Source #
ApplicativeG Set (Map Ord') (->) Source #
Instance details Defined in OAlg.Entity.Sequence.Set Methods amapG :: Map Ord' x y -> Set x -> Set y Source #
Transformable s FbrOrt => ApplicativeG Rt (Ornt s) (->) Source #
Instance details Defined in OAlg.Data.Ornt Methods amapG :: Ornt s x y -> Rt x -> Rt y Source #
ApplicativeG Rt (HomSymbol r) (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Vector Methods amapG :: HomSymbol r x y -> Rt x -> Rt y Source #
ApplicativeG Pnt (Ornt s) (->) Source #
Instance details Defined in OAlg.Data.Ornt Methods amapG :: Ornt s x y -> Pnt x -> Pnt y Source #
ApplicativeG Pnt (SliceFactorDrop s) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Definition Methods amapG :: SliceFactorDrop s x y -> Pnt x -> Pnt y Source #
ApplicativeG Pnt (HomId s) (->) Source #
Instance details Defined in OAlg.Hom.Definition Methods amapG :: HomId s x y -> Pnt x -> Pnt y Source #
ApplicativeG [] (Map s) (->) Source #
Instance details Defined in OAlg.Category.Map Methods amapG :: Map s x y -> [x] -> [y] Source #
(ApplicativeG t f c, ApplicativeG t g c) => ApplicativeG t (Either2 f g) c Source #
Instance details Defined in OAlg.Category.Applicative Methods amapG :: Either2 f g x y -> c (t x) (t y) Source #
ApplicativeG Id (->) (->) Source #
Instance details Defined in OAlg.Data.Identity Methods amapG :: (x -> y) -> Id x -> Id y Source #
ApplicativeG X (->) (->) Source #
Instance details Defined in OAlg.Category.Applicative Methods amapG :: (x -> y) -> X x -> X y Source #
ApplicativeG SomeFinList (->) (->) Source #
Instance details Defined in OAlg.Entity.FinList Methods amapG :: (x -> y) -> SomeFinList x -> SomeFinList y Source #
ApplicativeG Orientation (->) (->) Source #
Instance details Defined in OAlg.Structure.Oriented.Orientation Methods amapG :: (x -> y) -> Orientation x -> Orientation y Source #
ApplicativeG Maybe (->) (->) Source #
Instance details Defined in OAlg.Category.Applicative Methods amapG :: (x -> y) -> Maybe x -> Maybe y Source #
ApplicativeG [] (->) (->) Source #
Instance details Defined in OAlg.Category.Applicative Methods amapG :: (x -> y) -> [x] -> [y] Source #
(ApplicativeG d a b, TransformableG d s t) => ApplicativeG d (Sub s a) (Sub t b) Source #
Instance details Defined in OAlg.Category.Definition Methods amapG :: Sub s a x y -> Sub t b (d x) (d y) Source #
ApplicativeG f h (->) => ApplicativeG f (Sub t h) (->) Source #
Instance details Defined in OAlg.Category.Definition Methods amapG :: Sub t h x y -> f x -> f y Source #
(Morphism h, ApplicativeG Id h c, DualisableG s c o Id, c ~ (->)) => ApplicativeG Id (HomDisj s o h) c Source #
Instance details Defined in OAlg.Hom.Definition Methods amapG :: HomDisj s o h x y -> c (Id x) (Id y) Source #
(TransformableDst s, TransformableGRefl o s, DualisableDistributive s o, TransformableGRefl Matrix s) => ApplicativeG Id (MorCo Matrix s o) (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods amapG :: MorCo Matrix s o x y -> Id x -> Id y Source #
(Distributive d, Sliced i d, Conic c) => ApplicativeG Id (SliceAdjunction i c d) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Adjunction Methods amapG :: SliceAdjunction i c d x y -> Id x -> Id y Source #
(HomOriented h, DualisableOriented s o) => ApplicativeG SomeDiagram (HomDisj s o h) (->) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition Methods amapG :: HomDisj s o h x y -> SomeDiagram x -> SomeDiagram y Source #
(TransformableGRefl o s, DualisableDistributive s o, TransformableGRefl Matrix s) => ApplicativeG Rt (MorCo Matrix s o) (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods amapG :: MorCo Matrix s o x y -> Rt x -> Rt y Source #
(Morphism h, ApplicativeRoot h, DualisableG s (->) o Rt) => ApplicativeG Rt (HomDisj s o h) (->) Source #
Instance details Defined in OAlg.Hom.Definition Methods amapG :: HomDisj s o h x y -> Rt x -> Rt y Source #
(TransformableGRefl o s, DualisableDistributive s o, TransformableGRefl Matrix s) => ApplicativeG Pnt (MorCo Matrix s o) (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods amapG :: MorCo Matrix s o x y -> Pnt x -> Pnt y Source #
(Distributive d, Sliced i d, Conic c) => ApplicativeG Pnt (SliceAdjunction i c d) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Adjunction Methods amapG :: SliceAdjunction i c d x y -> Pnt x -> Pnt y Source #
(Morphism h, ApplicativePoint h, DualisableG s (->) o Pnt) => ApplicativeG Pnt (HomDisj s o h) (->) Source #
Instance details Defined in OAlg.Hom.Definition Methods amapG :: HomDisj s o h x y -> Pnt x -> Pnt y Source #
(ApplicativeMorCo d m s o (->), DualisableG s (->) o d) => ApplicativeG d (HomCo m s o) (->) Source #
Instance details Defined in OAlg.Data.HomCo Methods amapG :: HomCo m s o x y -> d x -> d y Source #
ApplicativeG Id h c => ApplicativeG Id (Variant2 v h) c Source #
Instance details Defined in OAlg.Data.Variant Methods amapG :: Variant2 v h x y -> c (Id x) (Id y) Source #
ApplicativeG Rt h c => ApplicativeG Rt (Variant2 v h) c Source #
Instance details Defined in OAlg.Data.Variant Methods amapG :: Variant2 v h x y -> c (Rt x) (Rt y) Source #
ApplicativeG Pnt h c => ApplicativeG Pnt (Variant2 v h) c Source #
Instance details Defined in OAlg.Data.Variant Methods amapG :: Variant2 v h x y -> c (Pnt x) (Pnt y) Source #
(HomOrientedDisjunctive h, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (Diagram t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition Methods amapG :: h x y -> SDualBi (Diagram t n m) x -> SDualBi (Diagram t n m) y Source #
(HomOrientedDisjunctive h, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (DiagramG Diagram t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Diagram.Diagrammatic Methods amapG :: h x y -> SDualBi (DiagramG Diagram t n m) x -> SDualBi (DiagramG Diagram t n m) y Source #
(HomSlicedOriented i h, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (DiagramG (SliceDiagram i) t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Adjunction Methods amapG :: h x y -> SDualBi (DiagramG (SliceDiagram i) t n m) x -> SDualBi (DiagramG (SliceDiagram i) t n m) y Source #
(HomOrientedSlicedFree h, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (DiagramG DiagramFree t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free Methods amapG :: h x y -> SDualBi (DiagramG DiagramFree t n m) x -> SDualBi (DiagramG DiagramFree t n m) y Source #
(HomMultiplicativeDisjunctive h, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (DiagramTrafo t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Methods amapG :: h x y -> SDualBi (DiagramTrafo t n m) x -> SDualBi (DiagramTrafo t n m) y Source #
HomDistributiveDisjunctive h => ApplicativeG (SDualBi Matrix) h (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods amapG :: h x y -> SDualBi Matrix x -> SDualBi Matrix y Source #
(HomSlicedOriented i h, s ~ Dual (Dual s)) => ApplicativeG (SDualBi (Slice s i)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Definition Methods amapG :: h x y -> SDualBi (Slice s i) x -> SDualBi (Slice s i) y Source #
(HomSlicedMultiplicative i h, s ~ Dual (Dual s)) => ApplicativeG (SDualBi (SliceFactor s i)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Definition Methods amapG :: h x y -> SDualBi (SliceFactor s i) x -> SDualBi (SliceFactor s i) y Source #
(HomOrientedSlicedFree h, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (DiagramFree t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free Methods amapG :: h x y -> SDualBi (DiagramFree t n m) x -> SDualBi (DiagramFree t n m) y Source #
(HomOrientedSlicedFree h, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (SomeFreeSliceDiagram t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free Methods amapG :: h x y -> SDualBi (SomeFreeSliceDiagram t n m) x -> SDualBi (SomeFreeSliceDiagram t n m) y Source #
(HomDistributiveDisjunctive h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p), t ~ Dual (Dual t)) => ApplicativeG (SDualBi (ConeG Cone Dst p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Conic.Duality Methods amapG :: h x y -> SDualBi (ConeG Cone Dst p d t n m) x -> SDualBi (ConeG Cone Dst p d t n m) y Source #
(HomMultiplicativeDisjunctive h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p), t ~ Dual (Dual t)) => ApplicativeG (SDualBi (ConeG Cone Mlt p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Conic.Duality Methods amapG :: h x y -> SDualBi (ConeG Cone Mlt p d t n m) x -> SDualBi (ConeG Cone Mlt p d t n m) y Source #
(HomDistributiveDisjunctive h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => ApplicativeG (SDualBi (ConeG ConeZeroHead s p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Duality Methods amapG :: h x y -> SDualBi (ConeG ConeZeroHead s p d t n m) x -> SDualBi (ConeG ConeZeroHead s p d t n m) y Source #
(HomDistributiveDisjunctive h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => ApplicativeG (SDualBi (Cone Dst p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Duality Methods amapG :: h x y -> SDualBi (Cone Dst p d t n m) x -> SDualBi (Cone Dst p d t n m) y Source #
(HomMultiplicativeDisjunctive h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => ApplicativeG (SDualBi (Cone Mlt p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Duality Methods amapG :: h x y -> SDualBi (Cone Mlt p d t n m) x -> SDualBi (Cone Mlt p d t n m) y Source #
(HomDistributiveDisjunctive h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => ApplicativeG (SDualBi (ConeZeroHead s p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Duality Methods amapG :: h x y -> SDualBi (ConeZeroHead s p d t n m) x -> SDualBi (ConeZeroHead s p d t n m) y Source #
(HomDistributiveDisjunctive h, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (ConsecutiveZero t n)) h (->) Source #
Instance details Defined in OAlg.Limes.Exact.ConsecutiveZero Methods amapG :: h x y -> SDualBi (ConsecutiveZero t n) x -> SDualBi (ConsecutiveZero t n) y Source #
(HomDistributiveDisjunctive h, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (ConsecutiveZeroHom t n)) h (->) Source #
Instance details Defined in OAlg.Limes.Exact.ConsecutiveZero Methods amapG :: h x y -> SDualBi (ConsecutiveZeroHom t n) x -> SDualBi (ConsecutiveZeroHom t n) y Source #
(HomDistributiveDisjunctive h, HomOrientedSlicedFree h, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (ConsecutiveZeroFree t n)) h (->) Source #
Instance details Defined in OAlg.Limes.Exact.Free Methods amapG :: h x y -> SDualBi (ConsecutiveZeroFree t n) x -> SDualBi (ConsecutiveZeroFree t n) y Source #
(NaturalDiagrammatic (Inv2 (HomFree s)) d ('Parallel 'LeftToRight) N2 N1, NaturalDiagrammatic (Inv2 (HomFree s)) d ('Parallel 'RightToLeft) N2 N1, p ~ Dual (Dual p), t ~ Dual (Dual t)) => ApplicativeG (SDualBi (ConeLiftable s p d t n m)) (Inv2 (HomFree s)) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free Methods amapG :: Inv2 (HomFree s) x y -> SDualBi (ConeLiftable s p d t n m) x -> SDualBi (ConeLiftable s p d t n m) y Source #
p ~ Dual (Dual p) => ApplicativeG (SDualBi (LiftableFree p)) (Inv2 (HomFree Dst)) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free Methods amapG :: Inv2 (HomFree Dst) x y -> SDualBi (LiftableFree p) x -> SDualBi (LiftableFree p) y Source #
p ~ Dual (Dual p) => ApplicativeG (SDualBi (LiftableFree p)) (Inv2 (HomFree Mlt)) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free Methods amapG :: Inv2 (HomFree Mlt) x y -> SDualBi (LiftableFree p) x -> SDualBi (LiftableFree p) y Source #
(CategoryDisjunctive h, HomSlicedMultiplicative i h, p ~ Dual (Dual p)) => ApplicativeG (SDualBi (Liftable p i)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Liftable Methods amapG :: Inv2 h x y -> SDualBi (Liftable p i) x -> SDualBi (Liftable p i) y Source #
(CategoryDisjunctive h, HomSlicedDistributive i h, FunctorialOriented h, p ~ Dual (Dual p), t ~ Dual (Dual t)) => ApplicativeG (SDualBi (LiftableCone i s p d t n m)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Liftable Methods amapG :: Inv2 h x y -> SDualBi (LiftableCone i s p d t n m) x -> SDualBi (LiftableCone i s p d t n m) y Source #
(CategoryDisjunctive h, HomSlicedDistributive i h, FunctorialOriented h, p ~ Dual (Dual p), t ~ Dual (Dual t)) => ApplicativeG (SDualBi (ConeG (LiftableCone i) s p d t n m)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Liftable Methods amapG :: Inv2 h x y -> SDualBi (ConeG (LiftableCone i) s p d t n m) x -> SDualBi (ConeG (LiftableCone i) s p d t n m) y Source #
NaturalConicBi (Inv2 h) c s p d t n m => ApplicativeG (SDualBi (LimesG c s p d t n m)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Limes.Definition.Duality Methods amapG :: Inv2 h x y -> SDualBi (LimesG c s p d t n m) x -> SDualBi (LimesG c s p d t n m) y Source #
(HomDistributiveDisjunctive h, CategoryDisjunctive h, NaturalKernelCokernel (Inv2 h) k c d, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (VarianceG t k c d n)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Limes.Exact.Deviation Methods amapG :: Inv2 h x y -> SDualBi (VarianceG t k c d n) x -> SDualBi (VarianceG t k c d n) y Source #
(HomDistributiveDisjunctive h, CategoryDisjunctive h, NaturalKernelCokernel (Inv2 h) k c d, t ~ Dual (Dual t)) => ApplicativeG (SDualBi (VarianceGHom t k c d n)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Limes.Exact.Deviation Methods amapG :: Inv2 h x y -> SDualBi (VarianceGHom t k c d n) x -> SDualBi (VarianceGHom t k c d n) y Source #
NaturalConicBi (Inv2 h) c s p d t n m => ApplicativeG (SDualBi (LimitsG c s p d t n m)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Limes.Limits.Duality Methods amapG :: Inv2 h x y -> SDualBi (LimitsG c s p d t n m) x -> SDualBi (LimitsG c s p d t n m) y Source #
ApplicativeG (FinList n) (->) (->) Source #
Instance details Defined in OAlg.Entity.FinList Methods amapG :: (x -> y) -> FinList n x -> FinList n y Source #
ApplicativeG (Col i) (->) (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods amapG :: (x -> y) -> Col i x -> Col i y Source #
ApplicativeG (Row i) (->) (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods amapG :: (x -> y) -> Row i x -> Row i y Source #
ApplicativeG (Word r) (->) (->) Source #
Instance details Defined in OAlg.Entity.Product.Definition Methods amapG :: (x -> y) -> Word r x -> Word r y Source #
ApplicativeG (PSequence i) (->) (->) Source #
Instance details Defined in OAlg.Entity.Sequence.PSequence Methods amapG :: (x -> y) -> PSequence i x -> PSequence i y Source #
ApplicativeG (LinearCombination r) (->) (->) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods amapG :: (x -> y) -> LinearCombination r x -> LinearCombination r y Source #
(Morphism h, ApplicativeG d h c, DualisableG r c o d, Transformable s r, c ~ (->)) => ApplicativeG (SVal d) (SHom r s o h) c Source #
Instance details Defined in OAlg.Category.SDuality Methods amapG :: SHom r s o h x y -> c (SVal d x) (SVal d y) Source #
(Morphism h, ApplicativeG d h c, DualisableG r c o d, Transformable s r, c ~ (->)) => ApplicativeG (SVal d) (SMorphism r s o h) c Source #
Instance details Defined in OAlg.Category.SDuality Methods amapG :: SMorphism r s o h x y -> c (SVal d x) (SVal d y) Source #
(Morphism h, ApplicativeGBi d h (->), DualisableGBi r (->) o d, Transformable s r) => ApplicativeG (SDualBi d) (SHom r s o h) (->) Source #
Instance details Defined in OAlg.Category.SDuality Methods amapG :: SHom r s o h x y -> SDualBi d x -> SDualBi d y Source #
ApplicativeG (Entries i j) (->) (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods amapG :: (x -> y) -> Entries i j x -> Entries i j y Source #
HomOriented h => ApplicativeG (Diagram t n m) h (->) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition Methods amapG :: h x y -> Diagram t n m x -> Diagram t n m y Source #

type Applicative1 (h :: Type -> Type -> Type) (f :: Type -> Type) = ApplicativeG f h (->) Source #

representable hs according to f.

amap1 :: Applicative1 h f => h x y -> f x -> f y Source #

representation of h in (->) according to f.

Functorial

amapF :: FunctorialG t a b => a x y -> b (t x) (t y) Source #

functorial application.

type Functorial1 (c :: Type -> Type -> Type) (f :: Type -> Type) = FunctorialG f c (->) Source #

functorials form c to (->) according to f.

data Functor1 (c :: Type -> Type -> Type) (f :: Type -> Type) where Source #

attest of being Functorial1 for the Category c to the Category (->) according to f.

Constructors

Functor1 :: forall (c :: Type -> Type -> Type) (f :: Type -> Type). Functorial1 c f => Functor1 c f

class (Category a, Category b, ApplicativeG t a b, TransformableG t (ObjectClass a) (ObjectClass b)) => FunctorialG (t :: Type -> Type) (a :: Type -> Type -> Type) (b :: Type -> Type -> Type) Source #

functorials from Category a to Category b according to the type function t.

Properties Let FunctorialG f a b, the holdst:

For all x and s in Struct (ObjectClass a) x holds: amapG (cOne s) .=. cOne (tauG s).
For all x, y, z and f in c y z, g in c x y holds: amapG (f . g) .=. amapG f . amapG g.

Instances

Instances details

FunctorialG Id h c => FunctorialG Id (Inv2 h) c Source #
Instance details Defined in OAlg.Data.Identity
FunctorialG Pnt h c => FunctorialG Pnt (Inv2 h) c Source #
Instance details Defined in OAlg.Structure.Oriented.Point
(Morphism m, Category c, ApplicativeG t m c, TransformableGObjectClass t m c) => FunctorialG t (Path m) c Source #
Instance details Defined in OAlg.Category.Path
FunctorialG Set (Map EntOrd) (->) Source #
Instance details Defined in OAlg.Entity.Sequence.Set
FunctorialG Set (Map Ord') (->) Source #
Instance details Defined in OAlg.Entity.Sequence.Set
FunctorialG [] (Map s) (->) Source #
Instance details Defined in OAlg.Category.Map
(FunctorialG d a b, TransformableObjectClass s a, TransformableObjectClass t b, TransformableG d s t) => FunctorialG d (Sub s a) (Sub t b) Source #
Instance details Defined in OAlg.Category.Definition
(FunctorialG f c (->), TransformableObjectClass s c) => FunctorialG f (Sub s c) (->) Source #
Instance details Defined in OAlg.Category.Definition
(Morphism h, ApplicativeG Id h c, DualisableG s c o Id, c ~ (->)) => FunctorialG Id (HomDisj s o h) c Source #
Instance details Defined in OAlg.Hom.Definition
(HomOriented h, DualisableOriented s o) => FunctorialG SomeDiagram (HomDisj s o h) (->) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition
(Morphism h, ApplicativeRoot h, DualisableG s (->) o Rt) => FunctorialG Rt (HomDisj s o h) (->) Source #
Instance details Defined in OAlg.Hom.Definition
(Morphism h, ApplicativePoint h, DualisableG s (->) o Pnt) => FunctorialG Pnt (HomDisj s o h) (->) Source #
Instance details Defined in OAlg.Hom.Definition
(FunctorialHomCo d m s o (->), DualisableG s (->) o d) => FunctorialG d (HomCo m s o) (->) Source #
Instance details Defined in OAlg.Data.HomCo
(HomOrientedDisjunctive h, FunctorialOriented h, t ~ Dual (Dual t)) => FunctorialG (SDualBi (Diagram t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition
(HomOrientedDisjunctive h, FunctorialOriented h, t ~ Dual (Dual t)) => FunctorialG (SDualBi (DiagramG Diagram t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Diagram.Diagrammatic
(CategoryDisjunctive h, HomSlicedOriented i h, FunctorialOriented h, t ~ Dual (Dual t)) => FunctorialG (SDualBi (DiagramG (SliceDiagram i) t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Adjunction
(HomOrientedSlicedFree h, FunctorialOriented h, t ~ Dual (Dual t)) => FunctorialG (SDualBi (DiagramG DiagramFree t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free
(HomMultiplicativeDisjunctive h, FunctorialOriented h, t ~ Dual (Dual t)) => FunctorialG (SDualBi (DiagramTrafo t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation
(HomDistributiveDisjunctive h, FunctorialOriented h) => FunctorialG (SDualBi Matrix) h (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
(HomSlicedOriented i h, FunctorialOriented h, s ~ Dual (Dual s)) => FunctorialG (SDualBi (Slice s i)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Definition
(HomSlicedMultiplicative i h, FunctorialOriented h, s ~ Dual (Dual s)) => FunctorialG (SDualBi (SliceFactor s i)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Definition
(HomOrientedSlicedFree h, FunctorialOriented h, t ~ Dual (Dual t)) => FunctorialG (SDualBi (DiagramFree t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free
(HomOrientedSlicedFree h, FunctorialOriented h, t ~ Dual (Dual t)) => FunctorialG (SDualBi (SomeFreeSliceDiagram t n m)) h (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free
(HomDistributiveDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p), t ~ Dual (Dual t)) => FunctorialG (SDualBi (ConeG Cone Dst p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Conic.Duality
(HomMultiplicativeDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p), t ~ Dual (Dual t)) => FunctorialG (SDualBi (ConeG Cone Mlt p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Conic.Duality
(HomDistributiveDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => FunctorialG (SDualBi (ConeG ConeZeroHead s p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Duality
(HomDistributiveDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => FunctorialG (SDualBi (Cone Dst p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Duality
(HomMultiplicativeDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => FunctorialG (SDualBi (Cone Mlt p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Duality
(HomDistributiveDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => FunctorialG (SDualBi (ConeZeroHead s p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Duality
(HomDistributiveDisjunctive h, t ~ Dual (Dual t), FunctorialOriented h) => FunctorialG (SDualBi (ConsecutiveZero t n)) h (->) Source #
Instance details Defined in OAlg.Limes.Exact.ConsecutiveZero
(HomDistributiveDisjunctive h, FunctorialOriented h, t ~ Dual (Dual t)) => FunctorialG (SDualBi (ConsecutiveZeroHom t n)) h (->) Source #
Instance details Defined in OAlg.Limes.Exact.ConsecutiveZero
(HomDistributiveDisjunctive h, FunctorialOriented h, HomOrientedSlicedFree h, t ~ Dual (Dual t)) => FunctorialG (SDualBi (ConsecutiveZeroFree t n)) h (->) Source #
Instance details Defined in OAlg.Limes.Exact.Free
(NaturalDiagrammatic (Inv2 (HomFree s)) d ('Parallel 'LeftToRight) N2 N1, NaturalDiagrammatic (Inv2 (HomFree s)) d ('Parallel 'RightToLeft) N2 N1, p ~ Dual (Dual p), t ~ Dual (Dual t)) => FunctorialG (SDualBi (ConeLiftable s p d t n m)) (Inv2 (HomFree s)) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free
p ~ Dual (Dual p) => FunctorialG (SDualBi (LiftableFree p)) (Inv2 (HomFree Dst)) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free
p ~ Dual (Dual p) => FunctorialG (SDualBi (LiftableFree p)) (Inv2 (HomFree Mlt)) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free
(CategoryDisjunctive h, HomSlicedMultiplicative i h, p ~ Dual (Dual p)) => FunctorialG (SDualBi (Liftable p i)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Liftable
(CategoryDisjunctive h, HomSlicedDistributive i h, FunctorialOriented h, p ~ Dual (Dual p), t ~ Dual (Dual t)) => FunctorialG (SDualBi (LiftableCone i s p d t n m)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Liftable
(CategoryDisjunctive h, HomSlicedDistributive i h, FunctorialOriented h, p ~ Dual (Dual p), t ~ Dual (Dual t)) => FunctorialG (SDualBi (ConeG (LiftableCone i) s p d t n m)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Liftable
NaturalConicBi (Inv2 h) c s p d t n m => FunctorialG (SDualBi (LimesG c s p d t n m)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Limes.Definition.Duality
(HomDistributiveDisjunctive h, CategoryDisjunctive h, NaturalKernelCokernel (Inv2 h) k c d, t ~ Dual (Dual t)) => FunctorialG (SDualBi (VarianceG t k c d n)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Limes.Exact.Deviation
(HomDistributiveDisjunctive h, CategoryDisjunctive h, NaturalKernelCokernel (Inv2 h) k c d, t ~ Dual (Dual t)) => FunctorialG (SDualBi (VarianceGHom t k c d n)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Limes.Exact.Deviation
NaturalConicBi (Inv2 h) c s p d t n m => FunctorialG (SDualBi (LimitsG c s p d t n m)) (Inv2 h) (->) Source #
Instance details Defined in OAlg.Limes.Limits.Duality
(Morphism h, ApplicativeG d h c, DualisableG r c o d, Transformable s r, c ~ (->)) => FunctorialG (SVal d) (SHom r s o h) c Source #
Instance details Defined in OAlg.Category.SDuality
(Morphism h, ApplicativeGBi d h (->), DualisableGBi r (->) o d, Transformable s r) => FunctorialG (SDualBi d) (SHom r s o h) (->) Source #
Instance details Defined in OAlg.Category.SDuality
(HomOriented h, FunctorialOriented h) => FunctorialG (Diagram t n m) h (->) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition

data FunctorG (t :: Type -> Type) (a :: Type -> Type -> Type) (b :: Type -> Type -> Type) where Source #

attest of being FunctorialG.

Constructors

FunctorG :: forall (t :: Type -> Type) (a :: Type -> Type -> Type) (b :: Type -> Type -> Type). FunctorialG t a b => FunctorG t a b

Transformables

class Transformable s (ObjectClass c) => TransformableObjectClass s (c :: Type -> Type -> Type) Source #

helper class to avoid undecided instances.

Instances

Instances details

TransformableObjectClass OrtX EqualExtOrt Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Transformable s Type => TransformableObjectClass s (->) Source #
Instance details Defined in OAlg.Category.Definition
TransformableObjectClass OrtX (HomDisj OrtX Op (HomEmpty OrtX)) Source #
Instance details Defined in OAlg.Hom.Definition
TransformableObjectClass s (SHom r s o h) Source #
Instance details Defined in OAlg.Category.SDuality
Transformable (s, Sld i) s => TransformableObjectClass (s, Sld i) (HomDisjEmpty s Op) Source #
Instance details Defined in OAlg.Entity.Slice.Sliced
TransformableObjectClass (Dst, SldFr) (HomDisj Dst Op (HomEmpty Dst)) Source #
Instance details Defined in OAlg.Entity.Slice.Free
TransformableObjectClass (Mlt, SldFr) (HomDisj Mlt Op (HomEmpty Mlt)) Source #
Instance details Defined in OAlg.Entity.Slice.Free

class Transformable (ObjectClass m) Typ => TransformableObjectClassTyp (m :: Type -> Type -> Type) Source #

helper class to avoid undecided instances.

Example By declaring an instance instance (..,Transformable (ObjectClass m) Typ,..) => C m for a Morphism m and a class C one will get the compiler error:

   • Illegal use of type family ‘ObjectClass’
       in the constraint ‘Transformable (ObjectClass m) Typ’
     (Use UndecidableInstances to permit this)

To avoid this error use this instance declaration: instance (..,TransformableObjectClassTyp m),..) => C m which will solve the problem!

Instances

Instances details

TransformableObjectClassTyp GLApp Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup
TransformableObjectClassTyp TrApp Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup
TransformableObjectClassTyp m => TransformableObjectClassTyp (Path m) Source #
Instance details Defined in OAlg.Category.Path
TransformableTyp s => TransformableObjectClassTyp (Ornt s) Source #
Instance details Defined in OAlg.Data.Ornt
TransformableObjectClassTyp (SliceAdjunction i c d) Source #
Instance details Defined in OAlg.Entity.Slice.Adjunction
Transformable s Typ => TransformableObjectClassTyp (SMorphism r s o h) Source #
Instance details Defined in OAlg.Category.SDuality

class TransformableG t (ObjectClass a) (ObjectClass b) => TransformableGObjectClass (t :: Type -> Type) (a :: Type -> Type -> Type) (b :: Type -> Type -> Type) Source #

helper class to avoid undecided instances.

Instances

Instances details

TransformableGObjectClass t a (->) Source #
Instance details Defined in OAlg.Category.Definition
TransformableGObjectClass t m c => TransformableGObjectClass t (Path m) c Source #
Instance details Defined in OAlg.Category.Path

class TransformableG d (ObjectClass a) t => TransformableGObjectClassDomain (d :: Type -> Type) (a :: Type -> Type -> Type) t Source #

helper class to avoid undecided instances.

Instances

Instances details

TransformableGObjectClassDomain Id (HomDisj OrtX Op (HomEmpty OrtX)) EqEOrt Source #
Instance details Defined in OAlg.Hom.Definition
TransformableGObjectClassDomain Pnt (HomDisj OrtX Op (HomEmpty OrtX)) EqEOrt Source #
Instance details Defined in OAlg.Hom.Definition
TransformableG d s t => TransformableGObjectClassDomain d (SHom r s o h) t Source #
Instance details Defined in OAlg.Category.SDuality

class TransformableG d s (ObjectClass c) => TransformableGObjectClassRange (d :: Type -> Type) s (c :: Type -> Type -> Type) Source #

helper class to avoid undecided instances.

Instances

Instances details

TransformableGObjectClassRange Id OrtX EqualExtOrt Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
TransformableGObjectClassRange Pnt OrtX EqualExtOrt Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
TransformableGObjectClassRange d s (->) Source #
Instance details Defined in OAlg.Category.Definition

Category (Homomorphous s) Source #
Instance details Defined in OAlg.Category.Definition Methods cOne :: Struct (ObjectClass (Homomorphous s)) x -> Homomorphous s x x Source # (.) :: Homomorphous s y z -> Homomorphous s x y -> Homomorphous s x z Source #
Category c => Category (Inv2 c) Source #
Instance details Defined in OAlg.Category.Definition Methods cOne :: Struct (ObjectClass (Inv2 c)) x -> Inv2 c x x Source # (.) :: Inv2 c y z -> Inv2 c x y -> Inv2 c x z Source #
Category c => Category (Op2 c) Source #
Instance details Defined in OAlg.Category.Definition Methods cOne :: Struct (ObjectClass (Op2 c)) x -> Op2 c x x Source # (.) :: Op2 c y z -> Op2 c x y -> Op2 c x z Source #
Category (Map s) Source #
Instance details Defined in OAlg.Category.Map Methods cOne :: Struct (ObjectClass (Map s)) x -> Map s x x Source # (.) :: Map s y z -> Map s x y -> Map s x z Source #
Morphism m => Category (Path m) Source #
Instance details Defined in OAlg.Category.Path Methods cOne :: Struct (ObjectClass (Path m)) x -> Path m x x Source # (.) :: Path m y z -> Path m x y -> Path m x z Source #
(Category c, TransformableObjectClass s c) => Category (Sub s c) Source #
Instance details Defined in OAlg.Category.Definition Methods cOne :: Struct (ObjectClass (Sub s c)) x -> Sub s c x x Source # (.) :: Sub s c y z -> Sub s c x y -> Sub s c x z Source #
Category (->) Source #
Instance details Defined in OAlg.Category.Definition Methods cOne :: Struct (ObjectClass (->)) x -> x -> x Source # (.) :: (y -> z) -> (x -> y) -> x -> z Source #
Category (HomCo m s o) Source #
Instance details Defined in OAlg.Data.HomCo Methods cOne :: Struct (ObjectClass (HomCo m s o)) x -> HomCo m s o x x Source # (.) :: HomCo m s o y z -> HomCo m s o x y -> HomCo m s o x z Source #
Morphism h => Category (HomDisj s o h) Source #
Instance details Defined in OAlg.Hom.Definition Methods cOne :: Struct (ObjectClass (HomDisj s o h)) x -> HomDisj s o h x x Source # (.) :: HomDisj s o h y z -> HomDisj s o h x y -> HomDisj s o h x z Source #
Morphism h => Category (SHom r s o h) Source #
Instance details Defined in OAlg.Category.SDuality Methods cOne :: Struct (ObjectClass (SHom r s o h)) x -> SHom r s o h x x Source # (.) :: SHom r s o h y z -> SHom r s o h x y -> SHom r s o h x z Source #
CategoryDisjunctive h => Category (Variant2 'Covariant h) Source #
Instance details Defined in OAlg.Data.Variant Methods cOne :: Struct (ObjectClass (Variant2 'Covariant h)) x -> Variant2 'Covariant h x x Source # (.) :: Variant2 'Covariant h y z -> Variant2 'Covariant h x y -> Variant2 'Covariant h x z Source #