OAlg.Structure.Fibred.Definition

fibred structures, i.e. type f with an associated root type Root f such that every value in f has a root.

Fibred

class (Entity f, Entity (Root f)) => Fibred f where

types with a Fibred structure. An entity of a Fibred structure will be called a stalk.

Note

1. On should accept the default for root only for FibredOriented structures!
2. For Distributive structures the only thing to be implemented is the Root type and should be defined as Root d = Orientation p where-- p = Point d (see the default implementation of root).

Minimal complete definition

Nothing

Associated Types

type Root f

the type of roots.

Methods

root :: f -> Root f

the root of a stalk in f.

default root :: (Root f ~ Orientation (Point f), Oriented f) => f -> Root f

Instances

Fibred N

Defined in OAlg.Structure.Fibred.Definition

Associated Types

type Root N

Methods

root :: N -> Root N

Fibred Q

Defined in OAlg.Structure.Fibred.Definition

Associated Types

type Root Q

Methods

root :: Q -> Root Q

Fibred Z

Defined in OAlg.Structure.Fibred.Definition

Associated Types

type Root Z

Methods

root :: Z -> Root Z

Fibred N'

Defined in OAlg.Entity.Natural

Associated Types

type Root N'

Methods

root :: N' -> Root N'

Fibred W'

Defined in OAlg.Entity.Natural

Associated Types

type Root W'

Methods

root :: W' -> Root W'

Fibred Integer

Defined in OAlg.Structure.Fibred.Definition

Associated Types

type Root Integer

Methods

root :: Integer -> Root Integer

Fibred ()

Defined in OAlg.Structure.Fibred.Definition

Associated Types

type Root ()

Methods

root :: () -> Root ()

Fibred Int

Defined in OAlg.Structure.Fibred.Definition

Associated Types

type Root Int

Methods

root :: Int -> Root Int

FibredOriented f => Fibred (Op f)

Defined in OAlg.Structure.Fibred.Definition

Associated Types

type Root (Op f)

Methods

root :: Op f -> Root (Op f)

(Additive x, FibredOriented x) => Fibred (Matrix x)

Defined in OAlg.Entity.Matrix.Definition

Associated Types

type Root (Matrix x)

Methods

root :: Matrix x -> Root (Matrix x)

Semiring r => Fibred (Vector r)

Defined in OAlg.Entity.Matrix.Vector

Associated Types

type Root (Vector r)

Methods

root :: Vector r -> Root (Vector r)

Entity a => Fibred (R a)

Defined in OAlg.Entity.Sum.SumSymbol

Associated Types

type Root (R a)

Methods

root :: R a -> Root (R a)

Fibred f => Fibred (Sheaf f)

Defined in OAlg.Structure.Fibred.Definition

Associated Types

type Root (Sheaf f)

Methods

root :: Sheaf f -> Root (Sheaf f)

Entity p => Fibred (Orientation p)

Defined in OAlg.Structure.Fibred.Definition

Associated Types

type Root (Orientation p)

Methods

root :: Orientation p -> Root (Orientation p)

(Fibred a, Semiring r, Commutative r) => Fibred (Sum r a)

Defined in OAlg.Entity.Sum.Definition

Associated Types

type Root (Sum r a)

Methods

root :: Sum r a -> Root (Sum r a)

(Fibred a, Semiring r, Commutative r) => Fibred (SumForm r a)

Defined in OAlg.Entity.Sum.Definition

Associated Types

type Root (SumForm r a)

Methods

root :: SumForm r a -> Root (SumForm r a)

(Semiring r, Commutative r, Entity a) => Fibred (SumSymbol r a)

Defined in OAlg.Entity.Sum.SumSymbol

Associated Types

type Root (SumSymbol r a)

Methods

root :: SumSymbol r a -> Root (SumSymbol r a)

(Distributive a, Typeable t, Typeable n, Typeable m) => Fibred (Transformation t n m a)

Defined in OAlg.Entity.Diagram.Transformation

Associated Types

type Root (Transformation t n m a)

Methods

root :: Transformation t n m a -> Root (Transformation t n m a)

data Fbr

type representing the class of Fibred structures.

Instances

ForgetfulTyp Fbr

Defined in OAlg.Structure.Fibred.Definition

ForgetfulFbr Fbr

Defined in OAlg.Structure.Fibred.Definition

Transformable Abl Fbr

Defined in OAlg.Structure.Additive.Definition

Methods

tau :: Struct Abl x -> Struct Fbr x

Transformable Add Fbr

Defined in OAlg.Structure.Additive.Definition

Methods

tau :: Struct Add x -> Struct Fbr x

Transformable Dst Fbr

Defined in OAlg.Structure.Distributive.Definition

Methods

tau :: Struct Dst x -> Struct Fbr x

Transformable Fbr Ent

Defined in OAlg.Structure.Fibred.Definition

Methods

tau :: Struct Fbr x -> Struct Ent x

Transformable Fbr Typ

Defined in OAlg.Structure.Fibred.Definition

Methods

tau :: Struct Fbr x -> Struct Typ x

Transformable FbrOrt Fbr

Defined in OAlg.Structure.Fibred.Definition

Methods

tau :: Struct FbrOrt x -> Struct Fbr x

(Semiring r, Commutative r) => EmbeddableMorphism (HomSymbol r) Fbr

Defined in OAlg.Entity.Matrix.Vector

EmbeddableMorphism h Fbr => EmbeddableMorphism (OpHom h) Fbr

Defined in OAlg.Hom.Oriented.Definition

Transformable (Alg k) Fbr

Defined in OAlg.Structure.Algebraic.Definition

Methods

tau :: Struct (Alg k) x -> Struct Fbr x

Transformable (Vec k) Fbr

Defined in OAlg.Structure.Vectorial.Definition

Methods

tau :: Struct (Vec k) x -> Struct Fbr x

type Hom Fbr h

Defined in OAlg.Hom.Fibred

type Hom Fbr h = HomFibred h

type Structure Fbr x

Defined in OAlg.Structure.Fibred.Definition

type Structure Fbr x = Fibred x

class Transformable s Fbr => ForgetfulFbr s

transformable to Fibred structure.

Instances

ForgetfulFbr Abl

Defined in OAlg.Structure.Additive.Definition

ForgetfulFbr Add

Defined in OAlg.Structure.Additive.Definition

ForgetfulFbr Dst

Defined in OAlg.Structure.Distributive.Definition

ForgetfulFbr Fbr

Defined in OAlg.Structure.Fibred.Definition

ForgetfulFbr FbrOrt

Defined in OAlg.Structure.Fibred.Definition

ForgetfulFbr (Alg k)

Defined in OAlg.Structure.Algebraic.Definition

ForgetfulFbr (Vec k)

Defined in OAlg.Structure.Vectorial.Definition

Fibred Oriented

class (Fibred d, Oriented d, Root d ~ Orientation (Point d)) => FibredOriented d

Fibred and Oriented structure with matching root and orientation.

Property Let d be a FibredOriented structure, then holds: For all s in d holds: root s == orientation s.

Note FibredOriented structures are required for Distributive structures.

Instances

FibredOriented N

Defined in OAlg.Structure.Fibred.Definition

FibredOriented Q

Defined in OAlg.Structure.Fibred.Definition

FibredOriented Z

Defined in OAlg.Structure.Fibred.Definition

FibredOriented N'

Defined in OAlg.Entity.Natural

FibredOriented W'

Defined in OAlg.Entity.Natural

FibredOriented Integer

Defined in OAlg.Structure.Fibred.Definition

FibredOriented ()

Defined in OAlg.Structure.Fibred.Definition

FibredOriented Int

Defined in OAlg.Structure.Fibred.Definition

FibredOriented f => FibredOriented (Op f)

Defined in OAlg.Structure.Fibred.Definition

(Additive x, FibredOriented x) => FibredOriented (Matrix x)

Defined in OAlg.Entity.Matrix.Definition

Entity p => FibredOriented (Orientation p)

Defined in OAlg.Structure.Fibred.Definition

(Distributive a, Typeable t, Typeable n, Typeable m) => FibredOriented (Transformation t n m a)

Defined in OAlg.Entity.Diagram.Transformation

data FbrOrt

type representing the class of FibredOriented structures.

Instances

ForgetfulTyp FbrOrt

Defined in OAlg.Structure.Fibred.Definition

ForgetfulFbr FbrOrt

Defined in OAlg.Structure.Fibred.Definition

ForgetfulFbrOrt FbrOrt

Defined in OAlg.Structure.Fibred.Definition

ForgetfulOrt FbrOrt

Defined in OAlg.Structure.Fibred.Definition

Transformable Dst FbrOrt

Defined in OAlg.Structure.Distributive.Definition

Methods

tau :: Struct Dst x -> Struct FbrOrt x

Transformable FbrOrt Ent

Defined in OAlg.Structure.Fibred.Definition

Methods

tau :: Struct FbrOrt x -> Struct Ent x

Transformable FbrOrt Typ

Defined in OAlg.Structure.Fibred.Definition

Methods

tau :: Struct FbrOrt x -> Struct Typ x

Transformable FbrOrt Fbr

Defined in OAlg.Structure.Fibred.Definition

Methods

tau :: Struct FbrOrt x -> Struct Fbr x

Transformable FbrOrt Ort

Defined in OAlg.Structure.Fibred.Definition

Methods

tau :: Struct FbrOrt x -> Struct Ort x

EmbeddableMorphism h FbrOrt => EmbeddableMorphism (OpHom h) FbrOrt

Defined in OAlg.Hom.Oriented.Definition

Transformable (Alg k) FbrOrt

Defined in OAlg.Structure.Algebraic.Definition

Methods

tau :: Struct (Alg k) x -> Struct FbrOrt x

type Hom FbrOrt h

Defined in OAlg.Hom.Fibred

type Hom FbrOrt h = HomFibredOriented h

type Structure FbrOrt x

Defined in OAlg.Structure.Fibred.Definition

type Structure FbrOrt x = FibredOriented x

class (ForgetfulFbr s, ForgetfulOrt s, Transformable s FbrOrt) => ForgetfulFbrOrt s

transformable to FibredOriented structure.

Instances

ForgetfulFbrOrt Dst

Defined in OAlg.Structure.Distributive.Definition

ForgetfulFbrOrt FbrOrt

Defined in OAlg.Structure.Fibred.Definition

ForgetfulFbrOrt (Alg k)

Defined in OAlg.Structure.Algebraic.Definition

Spezial classes

class Ord (Root f) => OrdRoot f

type where the associated root type is ordered.

Note Helper class to circumvent undecidable instances.

Instances

OrdRoot (R a)

Defined in OAlg.Entity.Sum.SumSymbol

class Singleton (Root f) => TotalRoot f

type where the associated root type is a singleton.

Sheaf

data Sheaf f

a list in a Fibred structure having all the same root.

Definition Let f be a Fibred structure and s = Sheaf r [t 0 .. t (n-1)] a sheaf in Sheaf f, then s is valid if and only if

1. r is valid and t i are valid for all i = 0..n-1.
2. root (t i) == r for all i = 0..n-1.

furthermore n is called the length of s.

If two sheafs have the same root then there stalks can be composed - via (++) - to a new sheaf having the same root. But as (++) is not commutative they are equipped with a Multiplicative structure.

Constructors

Sheaf (Root f) [f]

Instances

Foldable Sheaf

Defined in OAlg.Structure.Fibred.Definition

Methods

fold :: Monoid m => Sheaf m -> m

foldMap :: Monoid m => (a -> m) -> Sheaf a -> m

foldMap' :: Monoid m => (a -> m) -> Sheaf a -> m

foldr :: (a -> b -> b) -> b -> Sheaf a -> b

foldr' :: (a -> b -> b) -> b -> Sheaf a -> b

foldl :: (b -> a -> b) -> b -> Sheaf a -> b

foldl' :: (b -> a -> b) -> b -> Sheaf a -> b

foldr1 :: (a -> a -> a) -> Sheaf a -> a

foldl1 :: (a -> a -> a) -> Sheaf a -> a

toList :: Sheaf a -> [a]

null :: Sheaf a -> Bool

length :: Sheaf a -> Int

elem :: Eq a => a -> Sheaf a -> Bool

maximum :: Ord a => Sheaf a -> a

minimum :: Ord a => Sheaf a -> a

sum :: Num a => Sheaf a -> a

product :: Num a => Sheaf a -> a

Fibred f => Embeddable f (Sheaf f)

Defined in OAlg.Structure.Fibred.Definition

Methods

inj :: f -> Sheaf f

Additive a => Projectible a (Sheaf a)

Defined in OAlg.Structure.Additive.Definition

Methods

prj :: Sheaf a -> a

Fibred f => Show (Sheaf f)

Defined in OAlg.Structure.Fibred.Definition

Methods

showsPrec :: Int -> Sheaf f -> ShowS

show :: Sheaf f -> String

showList :: [Sheaf f] -> ShowS

Fibred f => Eq (Sheaf f)

Defined in OAlg.Structure.Fibred.Definition

Methods

(==) :: Sheaf f -> Sheaf f -> Bool

(/=) :: Sheaf f -> Sheaf f -> Bool

FibredOriented f => Dualisable (Sheaf f)

Defined in OAlg.Structure.Fibred.Definition

Methods

toDual :: Sheaf f -> Dual (Sheaf f)

fromDual :: Dual (Sheaf f) -> Sheaf f

Fibred f => Validable (Sheaf f)

Defined in OAlg.Structure.Fibred.Definition

Methods

valid :: Sheaf f -> Statement

Fibred f => Entity (Sheaf f)

Defined in OAlg.Structure.Fibred.Definition

Fibred f => Fibred (Sheaf f)

Defined in OAlg.Structure.Fibred.Definition

Associated Types

type Root (Sheaf f)

Methods

root :: Sheaf f -> Root (Sheaf f)

Fibred f => Multiplicative (Sheaf f)

'Data.List.(++)' is not commutative!

Defined in OAlg.Structure.Fibred.Definition

Methods

one :: Point (Sheaf f) -> Sheaf f

(*) :: Sheaf f -> Sheaf f -> Sheaf f

npower :: Sheaf f -> N -> Sheaf f

Fibred f => Oriented (Sheaf f)

Defined in OAlg.Structure.Fibred.Definition

Associated Types

type Point (Sheaf f)

Methods

orientation :: Sheaf f -> Orientation (Point (Sheaf f))

start :: Sheaf f -> Point (Sheaf f)

end :: Sheaf f -> Point (Sheaf f)

type Dual (Sheaf f :: Type)

Defined in OAlg.Structure.Fibred.Definition

type Dual (Sheaf f :: Type) = Sheaf (Op f)

type Root (Sheaf f)

Defined in OAlg.Structure.Fibred.Definition

type Root (Sheaf f) = Root f

type Point (Sheaf f)

Defined in OAlg.Structure.Fibred.Definition

type Point (Sheaf f) = Root f