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

OAlg.Limes.Cone.ZeroHead

Contents

Duality

Description

cones having a zero for its first arrow.

Synopsis

data ConeZeroHead s (p :: Perspective) (d :: DiagramType -> N' -> N' -> Type -> Type) (t :: DiagramType) (n :: N') (m :: N') x where
- ConeZeroHead :: forall x s (p :: Perspective) (d :: DiagramType -> N' -> N' -> Type -> Type) (t :: DiagramType) (n :: N') (m1 :: N'). Distributive x => Cone s p d t n (m1 + 1) x -> ConeZeroHead s p d t n (m1 + 1) x
cnZeroHead :: forall (p :: Perspective) (t :: DiagramType) (n :: N') (m :: N') a. Cone Dst p Diagram t n m a -> ConeZeroHead Mlt p Diagram t n (m + 1) a
cnKernel :: forall (p :: Perspective) (t :: DiagramType) (n :: N') (m :: N') a. (p ~ 'Projective, t ~ 'Parallel 'LeftToRight) => ConeZeroHead Mlt p Diagram t n (m + 1) a -> Cone Dst p Diagram t n m a
cnCokernel :: forall (p :: Perspective) (t :: DiagramType) (n :: N') (m :: N') a. (p ~ 'Injective, t ~ 'Parallel 'RightToLeft, n ~ N2) => ConeZeroHead Mlt p Diagram t n (m + 1) a -> Cone Dst p Diagram t n m a
cnDiffHead :: forall a (p :: Perspective) (d :: Direction) (n :: N') (m :: N'). (Distributive a, Abelian a) => Cone Mlt p Diagram ('Parallel d) n (m + 1) a -> ConeZeroHead Mlt p Diagram ('Parallel d) n (m + 1) a
module OAlg.Limes.Cone.ZeroHead.Duality

Documentation

data ConeZeroHead s (p :: Perspective) (d :: DiagramType -> N' -> N' -> Type -> Type) (t :: DiagramType) (n :: N') (m :: N') x where Source #

predicate for cones where the first arrow of its underlying diagram is equal to zero.

Constructors

ConeZeroHead :: forall x s (p :: Perspective) (d :: DiagramType -> N' -> N' -> Type -> Type) (t :: DiagramType) (n :: N') (m1 :: N'). Distributive x => Cone s p d t n (m1 + 1) x -> ConeZeroHead s p d t n (m1 + 1) x

Instances

Instances details

Conic ConeZeroHead Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Core Methods cone :: forall s (p :: Perspective) (d :: DiagramType -> N' -> N' -> Type -> Type) (t :: DiagramType) (n :: N') (m :: N') x. ConeZeroHead s p d t n m x -> Cone s p d t n m x Source #
(HomDistributiveDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => NaturalConic h ConeZeroHead Mlt p d t n m Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Duality
(HomDistributiveDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => NaturalTransformable h (->) (SDualBi (ConeG ConeZeroHead Mlt p d t n m)) (SDualBi (ConeG Cone Mlt p d t n m)) Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Duality
(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 (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, 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 (ConeZeroHead s p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Duality
Show (d t n ('S m) x) => Show (ConeZeroHead s p d t n ('S m) x) Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Core Methods showsPrec :: Int -> ConeZeroHead s p d t n ('S m) x -> ShowS # show :: ConeZeroHead s p d t n ('S m) x -> String # showList :: [ConeZeroHead s p d t n ('S m) x] -> ShowS #
Eq (d t n ('S m) x) => Eq (ConeZeroHead s p d t n ('S m) x) Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Core Methods (==) :: ConeZeroHead s p d t n ('S m) x -> ConeZeroHead s p d t n ('S m) x -> Bool # (/=) :: ConeZeroHead s p d t n ('S m) x -> ConeZeroHead s p d t n ('S m) x -> Bool #
(Diagrammatic d, Validable (d t n ('S m) x)) => Validable (ConeZeroHead s p d t n ('S m) x) Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Core Methods valid :: ConeZeroHead s p d t n ('S m) x -> Statement Source #
type Dual1 (ConeZeroHead s p d t n m :: Type -> Type) Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Duality type Dual1 (ConeZeroHead s p d t n m :: Type -> Type) = ConeZeroHead s (Dual p) d (Dual t) n m

cnZeroHead :: forall (p :: Perspective) (t :: DiagramType) (n :: N') (m :: N') a. Cone Dst p Diagram t n m a -> ConeZeroHead Mlt p Diagram t n (m + 1) a Source #

embedding of a cone in a distributive structure to its multiplicative cone.

cnKernel :: forall (p :: Perspective) (t :: DiagramType) (n :: N') (m :: N') a. (p ~ 'Projective, t ~ 'Parallel 'LeftToRight) => ConeZeroHead Mlt p Diagram t n (m + 1) a -> Cone Dst p Diagram t n m a Source #

the kernel cone of a zero headed parallel cone, i.e. the inverse of cnZeroHead.

cnCokernel :: forall (p :: Perspective) (t :: DiagramType) (n :: N') (m :: N') a. (p ~ 'Injective, t ~ 'Parallel 'RightToLeft, n ~ N2) => ConeZeroHead Mlt p Diagram t n (m + 1) a -> Cone Dst p Diagram t n m a Source #

the cokernel cone of a zero headed parallel cone, i.e. the inverse of cnZeroHead.

cnDiffHead :: forall a (p :: Perspective) (d :: Direction) (n :: N') (m :: N'). (Distributive a, Abelian a) => Cone Mlt p Diagram ('Parallel d) n (m + 1) a -> ConeZeroHead Mlt p Diagram ('Parallel d) n (m + 1) a Source #

subtracts to every arrow of the underlying parallel diagram the first arrow and adapts the shell accordingly.

Duality

module OAlg.Limes.Cone.ZeroHead.Duality