oalg-base-1.1.4.0: Algebraic structures on oriented entities and limits as a tool kit to solve algebraic problems.
Copyright(c) Erich Gut
LicenseBSD3
Maintainerzerich.gut@gmail.com
Safe HaskellSafe-Inferred
LanguageHaskell2010

OAlg.Entity.Slice.Adjunction

Description

Cokernel-Kernel Adjunction for Sliced structures.

Synopsis

Adjunction

slcAdjunction :: (SliceCokernelTo i c, SliceKernelFrom i c) => i c -> Adjunction (SliceCokernelKernel i c) (SliceFactor From i c) (SliceFactor To i c) Source #

the cokernel-kenrel adjunction.

Homomorphism

data SliceCokernelKernel i c x y where Source #

the left and right homomorphisms for the cokernel-kernel adjunction slcAdjunction.

Constructors

SliceCokernel :: SliceCokernelKernel i c (SliceFactor To i c) (SliceFactor From i c) 
SliceKernel :: SliceCokernelKernel i c (SliceFactor From i c) (SliceFactor To i c) 

Instances

Instances details
(Distributive c, SliceCokernelTo i c, SliceKernelFrom i c) => Applicative (SliceCokernelKernel i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

Methods

amap :: SliceCokernelKernel i c a b -> a -> b Source #

(Multiplicative c, Sliced i c) => EmbeddableMorphismTyp (SliceCokernelKernel i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

(Multiplicative c, Sliced i c) => Morphism (SliceCokernelKernel i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

Associated Types

type ObjectClass (SliceCokernelKernel i c) Source #

Methods

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

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

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

Eq2 (SliceCokernelKernel i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

Methods

eq2 :: SliceCokernelKernel i c x y -> SliceCokernelKernel i c x y -> Bool Source #

Show2 (SliceCokernelKernel i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

Methods

show2 :: SliceCokernelKernel i c a b -> String Source #

Validable2 (SliceCokernelKernel i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

Methods

valid2 :: SliceCokernelKernel i c x y -> Statement Source #

(Typeable i, Typeable c) => Entity2 (SliceCokernelKernel i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

(Distributive c, SliceCokernelTo i c, SliceKernelFrom i c) => HomMultiplicative (SliceCokernelKernel i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

(Distributive c, SliceCokernelTo i c, SliceKernelFrom i c) => HomOriented (SliceCokernelKernel i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

Methods

pmap :: SliceCokernelKernel i c a b -> Point a -> Point b Source #

(Multiplicative c, Sliced i c) => EmbeddableMorphism (SliceCokernelKernel i c) Typ Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

(Multiplicative c, Sliced i c) => EmbeddableMorphism (SliceCokernelKernel i c) Mlt Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

(Multiplicative c, Sliced i c) => EmbeddableMorphism (SliceCokernelKernel i c) Ort Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

Show (SliceCokernelKernel i c x y) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

Methods

showsPrec :: Int -> SliceCokernelKernel i c x y -> ShowS #

show :: SliceCokernelKernel i c x y -> String #

showList :: [SliceCokernelKernel i c x y] -> ShowS #

Eq (SliceCokernelKernel i c x y) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

Methods

(==) :: SliceCokernelKernel i c x y -> SliceCokernelKernel i c x y -> Bool #

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

Validable (SliceCokernelKernel i c x y) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

Methods

valid :: SliceCokernelKernel i c x y -> Statement Source #

(Typeable i, Typeable c, Typeable x, Typeable y) => Entity (SliceCokernelKernel i c x y) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

type ObjectClass (SliceCokernelKernel i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Adjunction

type ObjectClass (SliceCokernelKernel i c) = Mlt

class (Distributive c, Sliced i c) => SliceCokernelTo i c where Source #

Distributive structures c having to each Slice To i c a Cokernel.

Property Let h = SliceTo _ h' be in Slice To i c for a i sliced, Distributive structure c, then holds:

diagram (universalCone coker) == cokernelDiagram h' where coker = sliceCokernelTo h.

Methods

sliceCokernelTo :: Slice To i c -> Cokernel N1 c Source #

class (Distributive c, Sliced i c) => SliceKernelFrom i c where Source #

Distributive structures c having to each Slice From i c a Kernel.

Property Let h = SliceFrom _ h' be in Slice From i c for a i sliced, Distributive structure c, then holds:

diagram (universalCone ker) == kernelDiagram h' where coker = sliceKernelFrom h.

Methods

sliceKernelFrom :: Slice From i c -> Kernel N1 c Source #

Unit

slcCokerKer :: (SliceCokernelTo i c, SliceKernelFrom i c) => Slice To i c -> SliceFactor To i c Source #

the right unit of the cokernel-kernel adjunction slcAdjunction.

slcKerCoker :: (SliceCokernelTo i c, SliceKernelFrom i c) => Slice From i c -> SliceFactor From i c Source #

the left unit of the cokernel-kenrel adjunction slcAdjunction.

X

xSliceFactorTo :: (Multiplicative c, Sliced i c) => XOrtSite To c -> i c -> X (SliceFactor To i c) Source #

random variable for SliceFactor To i c.

xSliceFactorFrom :: (Multiplicative c, Sliced i c) => XOrtSite From c -> i c -> X (SliceFactor From i c) Source #

random variable for SliceFactor From i c.

Proposition

prpHomMltSliceCokernelKernel :: (SliceCokernelTo i c, SliceKernelFrom i c) => XOrtSite To c -> XOrtSite From c -> i c -> Statement Source #

validity for the values of SliceCokernelKernel to be HomMultiplicative.