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

OAlg.Structure.Fibred.Definition

Contents

Fibred
Fibred Oriented
Spezial classes
Sheaf

Description

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

Synopsis

class (Entity f, Entity (Root f)) => Fibred f where
- type Root f
- root :: f -> Root f
data Fbr
class Transformable s Fbr => ForgetfulFbr s
class (Fibred d, Oriented d, Root d ~ Orientation (Point d)) => FibredOriented d
data FbrOrt
class (ForgetfulFbr s, ForgetfulOrt s, Transformable s FbrOrt) => ForgetfulFbrOrt s
class Ord (Root f) => OrdRoot f
class Singleton (Root f) => TotalRoot f
data Sheaf f = Sheaf (Root f) [f]

Fibred

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

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

Note

On should accept the default for root only for FibredOriented structures!
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 Source #

the type of roots.

Methods

root :: f -> Root f Source #

the root of a stalk in f.

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

Instances

Instances details

Fibred N Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Root N Source # Methods root :: N -> Root N Source #
Fibred Q Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Root Q Source # Methods root :: Q -> Root Q Source #
Fibred Z Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Root Z Source # Methods root :: Z -> Root Z Source #
Fibred N' Source #
Instance details Defined in OAlg.Entity.Natural Associated Types type Root N' Source # Methods root :: N' -> Root N' Source #
Fibred W' Source #
Instance details Defined in OAlg.Entity.Natural Associated Types type Root W' Source # Methods root :: W' -> Root W' Source #
Fibred Integer Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Root Integer Source # Methods root :: Integer -> Root Integer Source #
Fibred () Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Root () Source # Methods root :: () -> Root () Source #
Fibred Int Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Root Int Source # Methods root :: Int -> Root Int Source #
FibredOriented f => Fibred (Op f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Root (Op f) Source # Methods root :: Op f -> Root (Op f) Source #
(Additive x, FibredOriented x) => Fibred (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Associated Types type Root (Matrix x) Source # Methods root :: Matrix x -> Root (Matrix x) Source #
Semiring r => Fibred (Vector r) Source #
Instance details Defined in OAlg.Entity.Matrix.Vector Associated Types type Root (Vector r) Source # Methods root :: Vector r -> Root (Vector r) Source #
Entity a => Fibred (R a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol Associated Types type Root (R a) Source # Methods root :: R a -> Root (R a) Source #
Fibred f => Fibred (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Root (Sheaf f) Source # Methods root :: Sheaf f -> Root (Sheaf f) Source #
Entity p => Fibred (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Root (Orientation p) Source # Methods root :: Orientation p -> Root (Orientation p) Source #
(Fibred a, Semiring r, Commutative r) => Fibred (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Associated Types type Root (Sum r a) Source # Methods root :: Sum r a -> Root (Sum r a) Source #
(Fibred a, Semiring r, Commutative r) => Fibred (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Associated Types type Root (SumForm r a) Source # Methods root :: SumForm r a -> Root (SumForm r a) Source #
(Semiring r, Commutative r, Entity a) => Fibred (SumSymbol r a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol Associated Types type Root (SumSymbol r a) Source # Methods root :: SumSymbol r a -> Root (SumSymbol r a) Source #
(Distributive a, Typeable t, Typeable n, Typeable m) => Fibred (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Associated Types type Root (Transformation t n m a) Source # Methods root :: Transformation t n m a -> Root (Transformation t n m a) Source #

data Fbr Source #

type representing the class of Fibred structures.

Instances

Instances details

ForgetfulTyp Fbr Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ForgetfulFbr Fbr Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
Transformable Abl Fbr Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Abl x -> Struct Fbr x Source #
Transformable Add Fbr Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Add x -> Struct Fbr x Source #
Transformable Dst Fbr Source #
Instance details Defined in OAlg.Structure.Distributive.Definition Methods tau :: Struct Dst x -> Struct Fbr x Source #
Transformable Fbr Ent Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods tau :: Struct Fbr x -> Struct Ent x Source #
Transformable Fbr Typ Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods tau :: Struct Fbr x -> Struct Typ x Source #
Transformable FbrOrt Fbr Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods tau :: Struct FbrOrt x -> Struct Fbr x Source #
(Semiring r, Commutative r) => EmbeddableMorphism (HomSymbol r) Fbr Source #
Instance details Defined in OAlg.Entity.Matrix.Vector
EmbeddableMorphism h Fbr => EmbeddableMorphism (OpHom h) Fbr Source #
Instance details Defined in OAlg.Hom.Oriented.Definition
Transformable (Alg k) Fbr Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Fbr x Source #
Transformable (Vec k) Fbr Source #
Instance details Defined in OAlg.Structure.Vectorial.Definition Methods tau :: Struct (Vec k) x -> Struct Fbr x Source #
type Hom Fbr h Source #
Instance details Defined in OAlg.Hom.Fibred type Hom Fbr h = HomFibred h
type Structure Fbr x Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Structure Fbr x = Fibred x

class Transformable s Fbr => ForgetfulFbr s Source #

transformable to Fibred structure.

Instances

Instances details

ForgetfulFbr Abl Source #
Instance details Defined in OAlg.Structure.Additive.Definition
ForgetfulFbr Add Source #
Instance details Defined in OAlg.Structure.Additive.Definition
ForgetfulFbr Dst Source #
Instance details Defined in OAlg.Structure.Distributive.Definition
ForgetfulFbr Fbr Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ForgetfulFbr FbrOrt Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ForgetfulFbr (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
ForgetfulFbr (Vec k) Source #
Instance details Defined in OAlg.Structure.Vectorial.Definition

Fibred Oriented

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

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

Instances details

FibredOriented N Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
FibredOriented Q Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
FibredOriented Z Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
FibredOriented N' Source #
Instance details Defined in OAlg.Entity.Natural
FibredOriented W' Source #
Instance details Defined in OAlg.Entity.Natural
FibredOriented Integer Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
FibredOriented () Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
FibredOriented Int Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
FibredOriented f => FibredOriented (Op f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
(Additive x, FibredOriented x) => FibredOriented (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
Entity p => FibredOriented (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
(Distributive a, Typeable t, Typeable n, Typeable m) => FibredOriented (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation

data FbrOrt Source #

type representing the class of FibredOriented structures.

Instances

Instances details

ForgetfulTyp FbrOrt Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ForgetfulFbr FbrOrt Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ForgetfulFbrOrt FbrOrt Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ForgetfulOrt FbrOrt Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
Transformable Dst FbrOrt Source #
Instance details Defined in OAlg.Structure.Distributive.Definition Methods tau :: Struct Dst x -> Struct FbrOrt x Source #
Transformable FbrOrt Ent Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods tau :: Struct FbrOrt x -> Struct Ent x Source #
Transformable FbrOrt Typ Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods tau :: Struct FbrOrt x -> Struct Typ x Source #
Transformable FbrOrt Fbr Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods tau :: Struct FbrOrt x -> Struct Fbr x Source #
Transformable FbrOrt Ort Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods tau :: Struct FbrOrt x -> Struct Ort x Source #
EmbeddableMorphism h FbrOrt => EmbeddableMorphism (OpHom h) FbrOrt Source #
Instance details Defined in OAlg.Hom.Oriented.Definition
Transformable (Alg k) FbrOrt Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct FbrOrt x Source #
type Hom FbrOrt h Source #
Instance details Defined in OAlg.Hom.Fibred type Hom FbrOrt h = HomFibredOriented h
type Structure FbrOrt x Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Structure FbrOrt x = FibredOriented x

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

transformable to FibredOriented structure.

Instances

Instances details

ForgetfulFbrOrt Dst Source #
Instance details Defined in OAlg.Structure.Distributive.Definition
ForgetfulFbrOrt FbrOrt Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ForgetfulFbrOrt (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition

Spezial classes

class Ord (Root f) => OrdRoot f Source #

type where the associated root type is ordered.

Note Helper class to circumvent undecidable instances.

Instances

Instances details

OrdRoot (R a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol

class Singleton (Root f) => TotalRoot f Source #

type where the associated root type is a singleton.

Sheaf

data Sheaf f Source #

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

r is valid and t i are valid for all i = 0..n-1.
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

Instances details

Foldable Sheaf Source #
Instance details 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) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods inj :: f -> Sheaf f Source #
Additive a => Projectible a (Sheaf a) Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods prj :: Sheaf a -> a Source #
Fibred f => Show (Sheaf f) Source #
Instance details 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) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods (==) :: Sheaf f -> Sheaf f -> Bool # (/=) :: Sheaf f -> Sheaf f -> Bool #
FibredOriented f => Dualisable (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods toDual :: Sheaf f -> Dual (Sheaf f) Source # fromDual :: Dual (Sheaf f) -> Sheaf f Source #
Fibred f => Validable (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods valid :: Sheaf f -> Statement Source #
Fibred f => Entity (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
Fibred f => Fibred (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Root (Sheaf f) Source # Methods root :: Sheaf f -> Root (Sheaf f) Source #
Fibred f => Multiplicative (Sheaf f) Source #	`'Data.List.(++)'` is not commutative!
Instance details Defined in OAlg.Structure.Fibred.Definition Methods one :: Point (Sheaf f) -> Sheaf f Source # (*) :: Sheaf f -> Sheaf f -> Sheaf f Source # npower :: Sheaf f -> N -> Sheaf f Source #
Fibred f => Oriented (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Point (Sheaf f) Source # Methods orientation :: Sheaf f -> Orientation (Point (Sheaf f)) Source # start :: Sheaf f -> Point (Sheaf f) Source # end :: Sheaf f -> Point (Sheaf f) Source #
type Dual (Sheaf f :: Type) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Dual (Sheaf f :: Type) = Sheaf (Op f)
type Root (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Root (Sheaf f) = Root f
type Point (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Point (Sheaf f) = Root f