Math.QuantumAlgebra.TensorCategory

A module defining classes and example instances of categories, monoidal categories and braided categories

Documentation

class MCategory c where

Associated Types

data Ob c :: *

data Ar c :: *

Methods

id_ :: Ob c -> Ar c

source :: Ar c -> Ob c

target :: Ar c -> Ob c

(>>>) :: Ar c -> Ar c -> Ar c

Instances

MCategory Cob2

Defined in Math.QuantumAlgebra.TensorCategory

Associated Types

data Ob Cob2 :: Type

data Ar Cob2 :: Type

Methods

id_ :: Ob Cob2 -> Ar Cob2

source :: Ar Cob2 -> Ob Cob2

target :: Ar Cob2 -> Ob Cob2

(>>>) :: Ar Cob2 -> Ar Cob2 -> Ar Cob2

MCategory Braid

Defined in Math.QuantumAlgebra.TensorCategory

Associated Types

data Ob Braid :: Type

data Ar Braid :: Type

Methods

id_ :: Ob Braid -> Ar Braid

source :: Ar Braid -> Ob Braid

target :: Ar Braid -> Ob Braid

(>>>) :: Ar Braid -> Ar Braid -> Ar Braid

MCategory FinCard

Defined in Math.QuantumAlgebra.TensorCategory

Associated Types

data Ob FinCard :: Type

data Ar FinCard :: Type

Methods

id_ :: Ob FinCard -> Ar FinCard

source :: Ar FinCard -> Ob FinCard

target :: Ar FinCard -> Ob FinCard

(>>>) :: Ar FinCard -> Ar FinCard -> Ar FinCard

MCategory FinOrd

Defined in Math.QuantumAlgebra.TensorCategory

Associated Types

data Ob FinOrd :: Type

data Ar FinOrd :: Type

Methods

id_ :: Ob FinOrd -> Ar FinOrd

source :: Ar FinOrd -> Ob FinOrd

target :: Ar FinOrd -> Ob FinOrd

(>>>) :: Ar FinOrd -> Ar FinOrd -> Ar FinOrd

MCategory Tangle

Defined in Math.QuantumAlgebra.Tangle

Associated Types

data Ob Tangle :: Type

data Ar Tangle :: Type

Methods

id_ :: Ob Tangle -> Ar Tangle

source :: Ar Tangle -> Ob Tangle

target :: Ar Tangle -> Ob Tangle

(>>>) :: Ar Tangle -> Ar Tangle -> Ar Tangle

MCategory OrientedTangle

Defined in Math.QuantumAlgebra.OrientedTangle

Associated Types

data Ob OrientedTangle :: Type

data Ar OrientedTangle :: Type

Methods

id_ :: Ob OrientedTangle -> Ar OrientedTangle

source :: Ar OrientedTangle -> Ob OrientedTangle

target :: Ar OrientedTangle -> Ob OrientedTangle

(>>>) :: Ar OrientedTangle -> Ar OrientedTangle -> Ar OrientedTangle

Num k => MCategory (Vect k)

Defined in Math.QuantumAlgebra.TensorCategory

Associated Types

data Ob (Vect k) :: Type

data Ar (Vect k) :: Type

Methods

id_ :: Ob (Vect k) -> Ar (Vect k)

source :: Ar (Vect k) -> Ob (Vect k)

target :: Ar (Vect k) -> Ob (Vect k)

(>>>) :: Ar (Vect k) -> Ar (Vect k) -> Ar (Vect k)

class (MCategory a, MCategory b) => MFunctor a b where

Methods

fob :: Ob a -> Ob b

far :: Ar a -> Ar b

Instances

MFunctor Braid FinCard

Defined in Math.QuantumAlgebra.TensorCategory

Methods

fob :: Ob Braid -> Ob FinCard

far :: Ar Braid -> Ar FinCard

MFunctor FinOrd FinCard

Defined in Math.QuantumAlgebra.TensorCategory

Methods

fob :: Ob FinOrd -> Ob FinCard

far :: Ar FinOrd -> Ar FinCard

class MCategory c => Monoidal c where

Methods

tunit :: Ob c

tob :: Ob c -> Ob c -> Ob c

tar :: Ar c -> Ar c -> Ar c

Instances

Monoidal Cob2

Defined in Math.QuantumAlgebra.TensorCategory

Methods

tunit :: Ob Cob2

tob :: Ob Cob2 -> Ob Cob2 -> Ob Cob2

tar :: Ar Cob2 -> Ar Cob2 -> Ar Cob2

Monoidal Braid

Defined in Math.QuantumAlgebra.TensorCategory

Methods

tunit :: Ob Braid

tob :: Ob Braid -> Ob Braid -> Ob Braid

tar :: Ar Braid -> Ar Braid -> Ar Braid

Monoidal FinCard

Defined in Math.QuantumAlgebra.TensorCategory

Methods

tunit :: Ob FinCard

tob :: Ob FinCard -> Ob FinCard -> Ob FinCard

tar :: Ar FinCard -> Ar FinCard -> Ar FinCard

Monoidal FinOrd

Defined in Math.QuantumAlgebra.TensorCategory

Methods

tunit :: Ob FinOrd

tob :: Ob FinOrd -> Ob FinOrd -> Ob FinOrd

tar :: Ar FinOrd -> Ar FinOrd -> Ar FinOrd

Monoidal Tangle

Defined in Math.QuantumAlgebra.Tangle

Methods

tunit :: Ob Tangle

tob :: Ob Tangle -> Ob Tangle -> Ob Tangle

tar :: Ar Tangle -> Ar Tangle -> Ar Tangle

Monoidal OrientedTangle

Defined in Math.QuantumAlgebra.OrientedTangle

Methods

tunit :: Ob OrientedTangle

tob :: Ob OrientedTangle -> Ob OrientedTangle -> Ob OrientedTangle

tar :: Ar OrientedTangle -> Ar OrientedTangle -> Ar OrientedTangle

class Monoidal c => StrictMonoidal c

class Monoidal c => WeakMonoidal c where

Methods

assoc :: Ob c -> Ob c -> Ob c -> Ar c

lunit :: Ob c -> Ar c

runit :: Ob c -> Ar c

class Monoidal c => Braided c where

Methods

twist :: Ob c -> Ob c -> Ar c

Instances

Braided Braid

Defined in Math.QuantumAlgebra.TensorCategory

Methods

twist :: Ob Braid -> Ob Braid -> Ar Braid

class Braided c => Symmetric c

data FinOrd

Instances

Monoidal FinOrd

Defined in Math.QuantumAlgebra.TensorCategory

Methods

tunit :: Ob FinOrd

tob :: Ob FinOrd -> Ob FinOrd -> Ob FinOrd

tar :: Ar FinOrd -> Ar FinOrd -> Ar FinOrd

MCategory FinOrd

Defined in Math.QuantumAlgebra.TensorCategory

Associated Types

data Ob FinOrd :: Type

data Ar FinOrd :: Type

Methods

id_ :: Ob FinOrd -> Ar FinOrd

source :: Ar FinOrd -> Ob FinOrd

target :: Ar FinOrd -> Ob FinOrd

(>>>) :: Ar FinOrd -> Ar FinOrd -> Ar FinOrd

MFunctor FinOrd FinCard

Defined in Math.QuantumAlgebra.TensorCategory

Methods

fob :: Ob FinOrd -> Ob FinCard

far :: Ar FinOrd -> Ar FinCard

Eq (Ob FinOrd)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

(==) :: Ob FinOrd -> Ob FinOrd -> Bool

(/=) :: Ob FinOrd -> Ob FinOrd -> Bool

Eq (Ar FinOrd)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

(==) :: Ar FinOrd -> Ar FinOrd -> Bool

(/=) :: Ar FinOrd -> Ar FinOrd -> Bool

Ord (Ob FinOrd)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

compare :: Ob FinOrd -> Ob FinOrd -> Ordering

(<) :: Ob FinOrd -> Ob FinOrd -> Bool

(<=) :: Ob FinOrd -> Ob FinOrd -> Bool

(>) :: Ob FinOrd -> Ob FinOrd -> Bool

(>=) :: Ob FinOrd -> Ob FinOrd -> Bool

max :: Ob FinOrd -> Ob FinOrd -> Ob FinOrd

min :: Ob FinOrd -> Ob FinOrd -> Ob FinOrd

Ord (Ar FinOrd)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

compare :: Ar FinOrd -> Ar FinOrd -> Ordering

(<) :: Ar FinOrd -> Ar FinOrd -> Bool

(<=) :: Ar FinOrd -> Ar FinOrd -> Bool

(>) :: Ar FinOrd -> Ar FinOrd -> Bool

(>=) :: Ar FinOrd -> Ar FinOrd -> Bool

max :: Ar FinOrd -> Ar FinOrd -> Ar FinOrd

min :: Ar FinOrd -> Ar FinOrd -> Ar FinOrd

Show (Ob FinOrd)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

showsPrec :: Int -> Ob FinOrd -> ShowS

show :: Ob FinOrd -> String

showList :: [Ob FinOrd] -> ShowS

Show (Ar FinOrd)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

showsPrec :: Int -> Ar FinOrd -> ShowS

show :: Ar FinOrd -> String

showList :: [Ar FinOrd] -> ShowS

data Ob FinOrd

Defined in Math.QuantumAlgebra.TensorCategory

data Ob FinOrd = FinOrdOb Int

data Ar FinOrd

Defined in Math.QuantumAlgebra.TensorCategory

data Ar FinOrd = FinOrdAr Int Int [Int]

finOrdAr :: Int -> Int -> [Int] -> Ar FinOrd

data FinCard

Instances

Monoidal FinCard

Defined in Math.QuantumAlgebra.TensorCategory

Methods

tunit :: Ob FinCard

tob :: Ob FinCard -> Ob FinCard -> Ob FinCard

tar :: Ar FinCard -> Ar FinCard -> Ar FinCard

MCategory FinCard

Defined in Math.QuantumAlgebra.TensorCategory

Associated Types

data Ob FinCard :: Type

data Ar FinCard :: Type

Methods

id_ :: Ob FinCard -> Ar FinCard

source :: Ar FinCard -> Ob FinCard

target :: Ar FinCard -> Ob FinCard

(>>>) :: Ar FinCard -> Ar FinCard -> Ar FinCard

MFunctor Braid FinCard

Defined in Math.QuantumAlgebra.TensorCategory

Methods

fob :: Ob Braid -> Ob FinCard

far :: Ar Braid -> Ar FinCard

MFunctor FinOrd FinCard

Defined in Math.QuantumAlgebra.TensorCategory

Methods

fob :: Ob FinOrd -> Ob FinCard

far :: Ar FinOrd -> Ar FinCard

Eq (Ob FinCard)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

(==) :: Ob FinCard -> Ob FinCard -> Bool

(/=) :: Ob FinCard -> Ob FinCard -> Bool

Eq (Ar FinCard)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

(==) :: Ar FinCard -> Ar FinCard -> Bool

(/=) :: Ar FinCard -> Ar FinCard -> Bool

Ord (Ob FinCard)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

compare :: Ob FinCard -> Ob FinCard -> Ordering

(<) :: Ob FinCard -> Ob FinCard -> Bool

(<=) :: Ob FinCard -> Ob FinCard -> Bool

(>) :: Ob FinCard -> Ob FinCard -> Bool

(>=) :: Ob FinCard -> Ob FinCard -> Bool

max :: Ob FinCard -> Ob FinCard -> Ob FinCard

min :: Ob FinCard -> Ob FinCard -> Ob FinCard

Ord (Ar FinCard)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

compare :: Ar FinCard -> Ar FinCard -> Ordering

(<) :: Ar FinCard -> Ar FinCard -> Bool

(<=) :: Ar FinCard -> Ar FinCard -> Bool

(>) :: Ar FinCard -> Ar FinCard -> Bool

(>=) :: Ar FinCard -> Ar FinCard -> Bool

max :: Ar FinCard -> Ar FinCard -> Ar FinCard

min :: Ar FinCard -> Ar FinCard -> Ar FinCard

Show (Ob FinCard)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

showsPrec :: Int -> Ob FinCard -> ShowS

show :: Ob FinCard -> String

showList :: [Ob FinCard] -> ShowS

Show (Ar FinCard)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

showsPrec :: Int -> Ar FinCard -> ShowS

show :: Ar FinCard -> String

showList :: [Ar FinCard] -> ShowS

data Ob FinCard

Defined in Math.QuantumAlgebra.TensorCategory

data Ob FinCard = FinCardOb Int

data Ar FinCard

Defined in Math.QuantumAlgebra.TensorCategory

data Ar FinCard = FinCardAr Int Int [Int]

finCardAr :: Int -> Int -> [Int] -> Ar FinCard

finPerm :: [Int] -> Ar FinCard

data Braid

Instances

Braided Braid

Defined in Math.QuantumAlgebra.TensorCategory

Methods

twist :: Ob Braid -> Ob Braid -> Ar Braid

Monoidal Braid

Defined in Math.QuantumAlgebra.TensorCategory

Methods

tunit :: Ob Braid

tob :: Ob Braid -> Ob Braid -> Ob Braid

tar :: Ar Braid -> Ar Braid -> Ar Braid

MCategory Braid

Defined in Math.QuantumAlgebra.TensorCategory

Associated Types

data Ob Braid :: Type

data Ar Braid :: Type

Methods

id_ :: Ob Braid -> Ar Braid

source :: Ar Braid -> Ob Braid

target :: Ar Braid -> Ob Braid

(>>>) :: Ar Braid -> Ar Braid -> Ar Braid

MFunctor Braid FinCard

Defined in Math.QuantumAlgebra.TensorCategory

Methods

fob :: Ob Braid -> Ob FinCard

far :: Ar Braid -> Ar FinCard

Eq (Ob Braid)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

(==) :: Ob Braid -> Ob Braid -> Bool

(/=) :: Ob Braid -> Ob Braid -> Bool

Eq (Ar Braid)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

(==) :: Ar Braid -> Ar Braid -> Bool

(/=) :: Ar Braid -> Ar Braid -> Bool

Ord (Ob Braid)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

compare :: Ob Braid -> Ob Braid -> Ordering

(<) :: Ob Braid -> Ob Braid -> Bool

(<=) :: Ob Braid -> Ob Braid -> Bool

(>) :: Ob Braid -> Ob Braid -> Bool

(>=) :: Ob Braid -> Ob Braid -> Bool

max :: Ob Braid -> Ob Braid -> Ob Braid

min :: Ob Braid -> Ob Braid -> Ob Braid

Ord (Ar Braid)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

compare :: Ar Braid -> Ar Braid -> Ordering

(<) :: Ar Braid -> Ar Braid -> Bool

(<=) :: Ar Braid -> Ar Braid -> Bool

(>) :: Ar Braid -> Ar Braid -> Bool

(>=) :: Ar Braid -> Ar Braid -> Bool

max :: Ar Braid -> Ar Braid -> Ar Braid

min :: Ar Braid -> Ar Braid -> Ar Braid

Show (Ob Braid)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

showsPrec :: Int -> Ob Braid -> ShowS

show :: Ob Braid -> String

showList :: [Ob Braid] -> ShowS

Show (Ar Braid)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

showsPrec :: Int -> Ar Braid -> ShowS

show :: Ar Braid -> String

showList :: [Ar Braid] -> ShowS

data Ob Braid

Defined in Math.QuantumAlgebra.TensorCategory

data Ob Braid = BraidOb Int

data Ar Braid

Defined in Math.QuantumAlgebra.TensorCategory

data Ar Braid = BraidAr Int [Int]

t :: Int -> Int -> Ar Braid

t' :: Int -> Int -> Ar Braid

data Vect k

Instances

Eq (Ob (Vect k))

Defined in Math.QuantumAlgebra.TensorCategory

Methods

(==) :: Ob (Vect k) -> Ob (Vect k) -> Bool

(/=) :: Ob (Vect k) -> Ob (Vect k) -> Bool

Eq (Ar (Vect k))

Defined in Math.QuantumAlgebra.TensorCategory

Methods

(==) :: Ar (Vect k) -> Ar (Vect k) -> Bool

(/=) :: Ar (Vect k) -> Ar (Vect k) -> Bool

Ord (Ob (Vect k))

Defined in Math.QuantumAlgebra.TensorCategory

Methods

compare :: Ob (Vect k) -> Ob (Vect k) -> Ordering

(<) :: Ob (Vect k) -> Ob (Vect k) -> Bool

(<=) :: Ob (Vect k) -> Ob (Vect k) -> Bool

(>) :: Ob (Vect k) -> Ob (Vect k) -> Bool

(>=) :: Ob (Vect k) -> Ob (Vect k) -> Bool

max :: Ob (Vect k) -> Ob (Vect k) -> Ob (Vect k)

min :: Ob (Vect k) -> Ob (Vect k) -> Ob (Vect k)

Ord (Ar (Vect k))

Defined in Math.QuantumAlgebra.TensorCategory

Methods

compare :: Ar (Vect k) -> Ar (Vect k) -> Ordering

(<) :: Ar (Vect k) -> Ar (Vect k) -> Bool

(<=) :: Ar (Vect k) -> Ar (Vect k) -> Bool

(>) :: Ar (Vect k) -> Ar (Vect k) -> Bool

(>=) :: Ar (Vect k) -> Ar (Vect k) -> Bool

max :: Ar (Vect k) -> Ar (Vect k) -> Ar (Vect k)

min :: Ar (Vect k) -> Ar (Vect k) -> Ar (Vect k)

Show (Ob (Vect k))

Defined in Math.QuantumAlgebra.TensorCategory

Methods

showsPrec :: Int -> Ob (Vect k) -> ShowS

show :: Ob (Vect k) -> String

showList :: [Ob (Vect k)] -> ShowS

Show (Ar (Vect k))

Defined in Math.QuantumAlgebra.TensorCategory

Methods

showsPrec :: Int -> Ar (Vect k) -> ShowS

show :: Ar (Vect k) -> String

showList :: [Ar (Vect k)] -> ShowS

Num k => MCategory (Vect k)

Defined in Math.QuantumAlgebra.TensorCategory

Associated Types

data Ob (Vect k) :: Type

data Ar (Vect k) :: Type

Methods

id_ :: Ob (Vect k) -> Ar (Vect k)

source :: Ar (Vect k) -> Ob (Vect k)

target :: Ar (Vect k) -> Ob (Vect k)

(>>>) :: Ar (Vect k) -> Ar (Vect k) -> Ar (Vect k)

data Ob (Vect k)

Defined in Math.QuantumAlgebra.TensorCategory

data Ob (Vect k) = VectOb Int

data Ar (Vect k)

Defined in Math.QuantumAlgebra.TensorCategory

data Ar (Vect k) = VectAr Int Int [[Int]]

data Cob2

Instances

Monoidal Cob2

Defined in Math.QuantumAlgebra.TensorCategory

Methods

tunit :: Ob Cob2

tob :: Ob Cob2 -> Ob Cob2 -> Ob Cob2

tar :: Ar Cob2 -> Ar Cob2 -> Ar Cob2

MCategory Cob2

Defined in Math.QuantumAlgebra.TensorCategory

Associated Types

data Ob Cob2 :: Type

data Ar Cob2 :: Type

Methods

id_ :: Ob Cob2 -> Ar Cob2

source :: Ar Cob2 -> Ob Cob2

target :: Ar Cob2 -> Ob Cob2

(>>>) :: Ar Cob2 -> Ar Cob2 -> Ar Cob2

Eq (Ob Cob2)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

(==) :: Ob Cob2 -> Ob Cob2 -> Bool

(/=) :: Ob Cob2 -> Ob Cob2 -> Bool

Eq (Ar Cob2)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

(==) :: Ar Cob2 -> Ar Cob2 -> Bool

(/=) :: Ar Cob2 -> Ar Cob2 -> Bool

Ord (Ob Cob2)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

compare :: Ob Cob2 -> Ob Cob2 -> Ordering

(<) :: Ob Cob2 -> Ob Cob2 -> Bool

(<=) :: Ob Cob2 -> Ob Cob2 -> Bool

(>) :: Ob Cob2 -> Ob Cob2 -> Bool

(>=) :: Ob Cob2 -> Ob Cob2 -> Bool

max :: Ob Cob2 -> Ob Cob2 -> Ob Cob2

min :: Ob Cob2 -> Ob Cob2 -> Ob Cob2

Ord (Ar Cob2)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

compare :: Ar Cob2 -> Ar Cob2 -> Ordering

(<) :: Ar Cob2 -> Ar Cob2 -> Bool

(<=) :: Ar Cob2 -> Ar Cob2 -> Bool

(>) :: Ar Cob2 -> Ar Cob2 -> Bool

(>=) :: Ar Cob2 -> Ar Cob2 -> Bool

max :: Ar Cob2 -> Ar Cob2 -> Ar Cob2

min :: Ar Cob2 -> Ar Cob2 -> Ar Cob2

Show (Ob Cob2)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

showsPrec :: Int -> Ob Cob2 -> ShowS

show :: Ob Cob2 -> String

showList :: [Ob Cob2] -> ShowS

Show (Ar Cob2)

Defined in Math.QuantumAlgebra.TensorCategory

Methods

showsPrec :: Int -> Ar Cob2 -> ShowS

show :: Ar Cob2 -> String

showList :: [Ar Cob2] -> ShowS

data Ob Cob2

Defined in Math.QuantumAlgebra.TensorCategory

data Ob Cob2 = O Int

data Ar Cob2

Defined in Math.QuantumAlgebra.TensorCategory

data Ar Cob2

• = Id Int
• | Unit
• | Mult
• | Counit
• | Comult
• | Par (Ar Cob2) (Ar Cob2)
• | Seq (Ar Cob2) (Ar Cob2)

rewrite :: Ar Cob2 -> Ar Cob2