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

OAlg.Limes.Cone.EligibleFactor

Contents

Eligible Factor
- Duality
- X

Description

eligible factors between Cones.

Synopsis

cnEligibleFactor :: a -> Cone s p t n m a -> Cone s p t n m a -> Bool
data EligibleFactor s p t n m a where
- EligibleFactorTo :: Cone s Projective t n m a -> a -> Cone s Projective t n m a -> EligibleFactor s Projective t n m a
- EligibleFactorFrom :: Cone s Injective t n m a -> a -> Cone s Injective t n m a -> EligibleFactor s Injective t n m a
elfFactorCone :: EligibleFactor s p t n m a -> (a, Cone s p t n m a)
elfMap :: Hom s h => h a b -> EligibleFactor s p t n m a -> EligibleFactor s p t n m b
coEligibleFactor :: EligibleFactor s p t n m a -> Dual (EligibleFactor s p t n m a)
coEligibleFactorInv :: ConeStruct s a -> (Dual (Dual p) :~: p) -> (Dual (Dual t) :~: t) -> Dual (EligibleFactor s p t n m a) -> EligibleFactor s p t n m a
elfFromOpOp :: ConeStruct s a -> EligibleFactor s p t n m (Op (Op a)) -> EligibleFactor s p t n m a
xopEligibleFactor :: ConeStruct s a -> XOrtPerspective p a -> Cone s p t n m a -> X (EligibleFactor s p t n m a)
data XOrtPerspective p a where
- XOrtProjective :: XOrtSite To a -> XOrtPerspective Projective a
- XOrtInjective :: XOrtSite From a -> XOrtPerspective Injective a
class XStandardOrtPerspective p a where
- xStandardOrtPerspective :: XOrtPerspective p a
xosEligibleFactorPrj :: XOrtSite To a -> Cone s Projective t n m a -> X (EligibleFactor s Projective t n m a)
xosEligibleFactorInj :: ConeStruct s a -> (Dual (Dual t) :~: t) -> XOrtSite From a -> Cone s Injective t n m a -> X (EligibleFactor s Injective t n m a)

Eligible Factor

cnEligibleFactor :: a -> Cone s p t n m a -> Cone s p t n m a -> Bool Source #

eligibility of a factor between two cones.

Property Let x be in a and f, t in Cone s p t n m a with cnDiagram f == cnDiagram t, then holds:

If p is equal to Projective then holds: cnEligibleFactor x f t is True if and only if
1. orientation x == tip f :> tip t.
2. ti * x == fi for all ti in shell t and fi in shell f.
If p is equal to Injective then holds: cnEligibleFactor x f t is True if and only if
1. orientation x == tip f :> tip t.
2. x * ti == fi for all ti in shell t and fi in shell f.

data EligibleFactor s p t n m a where Source #

predicate for eligible factors between cones.

Property Let e be in EligibleFactor s p t n m a for a Multiplicative structure a, then holds:

If e matches EligibleFactorTo l x c then holds: cnDiagram l == cnDiagram c and cnEligibleFactor x c l.
If e matches EligibleFactorFrom l x c then holds: cnDiagram l == cnDiagram c and cnEligibleFactor x l c.

Constructors

EligibleFactorTo :: Cone s Projective t n m a -> a -> Cone s Projective t n m a -> EligibleFactor s Projective t n m a
EligibleFactorFrom :: Cone s Injective t n m a -> a -> Cone s Injective t n m a -> EligibleFactor s Injective t n m a

Instances

Instances details

Show a => Show (EligibleFactor s p t n m a) Source #
Instance details Defined in OAlg.Limes.Cone.EligibleFactor Methods showsPrec :: Int -> EligibleFactor s p t n m a -> ShowS # show :: EligibleFactor s p t n m a -> String # showList :: [EligibleFactor s p t n m a] -> ShowS #
Oriented a => Validable (EligibleFactor s p t n m a) Source #
Instance details Defined in OAlg.Limes.Cone.EligibleFactor Methods valid :: EligibleFactor s p t n m a -> Statement Source #
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)

elfFactorCone :: EligibleFactor s p t n m a -> (a, Cone s p t n m a) Source #

the underlying factor together with its cone.

elfMap :: Hom s h => h a b -> EligibleFactor s p t n m a -> EligibleFactor s p t n m b Source #

mapping of a eligible factor.

Duality

coEligibleFactor :: EligibleFactor s p t n m a -> Dual (EligibleFactor s p t n m a) Source #

to the dual, with its inverse coEligibleFactorInv.

coEligibleFactorInv :: ConeStruct s a -> (Dual (Dual p) :~: p) -> (Dual (Dual t) :~: t) -> Dual (EligibleFactor s p t n m a) -> EligibleFactor s p t n m a Source #

from the dual, with its inverse coEligibleFactor.

elfFromOpOp :: ConeStruct s a -> EligibleFactor s p t n m (Op (Op a)) -> EligibleFactor s p t n m a Source #

from the bidual.

X

xopEligibleFactor :: ConeStruct s a -> XOrtPerspective p a -> Cone s p t n m a -> X (EligibleFactor s p t n m a) Source #

the induced random variable of eligible factors.

data XOrtPerspective p a where Source #

random variable given by a XOrtSite.

Constructors

XOrtProjective :: XOrtSite To a -> XOrtPerspective Projective a
XOrtInjective :: XOrtSite From a -> XOrtPerspective Injective a

class XStandardOrtPerspective p a where Source #

standard random variable for XOrtPerspective.

Methods

xStandardOrtPerspective :: XOrtPerspective p a Source #

Instances

Instances details

XStandardOrtSiteFrom a => XStandardOrtPerspective 'Injective a Source #
Instance details Defined in OAlg.Limes.Cone.EligibleFactor Methods xStandardOrtPerspective :: XOrtPerspective 'Injective a Source #
XStandardOrtSiteTo a => XStandardOrtPerspective 'Projective a Source #
Instance details Defined in OAlg.Limes.Cone.EligibleFactor Methods xStandardOrtPerspective :: XOrtPerspective 'Projective a Source #

xosEligibleFactorPrj :: XOrtSite To a -> Cone s Projective t n m a -> X (EligibleFactor s Projective t n m a) Source #

the induced random variable of eligible factors.

xosEligibleFactorInj :: ConeStruct s a -> (Dual (Dual t) :~: t) -> XOrtSite From a -> Cone s Injective t n m a -> X (EligibleFactor s Injective t n m a) Source #

the induced random variable of eligible factors.