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

OAlg.Entity.Diagram.Transformation

Contents

Transformation
Duality

Description

natural transformations between Diagrams.

Synopsis

data Transformation t n m a = Transformation (Diagram t n m a) (Diagram t n m a) (FinList n a)
trfs :: Transformation t n m a -> FinList n a
coTransformation :: Transformation t n m a -> Dual (Transformation t n m a)

Transformation

data Transformation t n m a Source #

natural transformations between two Diagrams.

Property Let Transformation a b t be in Transformation t n m a for a Multiplicative structure a, then holds

dgQuiver a == dgQuiver b.
For all 0 <= i < n holds: orientation (t i) == p i :> q i where p = dgPoints a and q = dgPoints b.
For all 0 <= j < m holds: t (e j) * f j == g j * t (s j) where f = dgArrows a, g = dgArrows b, s j is the index of the start point of the j-th arrow and e j is the index of the end point.

                   t (s j)
   s j     p (s j) --------> q (s j)
    |         |                 |
  j |     f j |                 | g j
    |         |                 |
    v         v                 v
   e j     p (e j) --------> q (e j)
                   t (e j)

Constructors

Transformation (Diagram t n m a) (Diagram t n m a) (FinList n a)

Instances

Instances details

Oriented a => Show (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Methods showsPrec :: Int -> Transformation t n m a -> ShowS # show :: Transformation t n m a -> String # showList :: [Transformation t n m a] -> ShowS #
Oriented a => Eq (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Methods (==) :: Transformation t n m a -> Transformation t n m a -> Bool # (/=) :: Transformation t n m a -> Transformation t n m a -> Bool #
Multiplicative a => Validable (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Methods valid :: Transformation t n m a -> Statement Source #
(Multiplicative a, Typeable t, Typeable n, Typeable m) => Entity (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation
(Distributive a, Abelian a, Typeable t, Typeable n, Typeable m) => Abelian (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Methods negate :: Transformation t n m a -> Transformation t n m a Source # (-) :: Transformation t n m a -> Transformation t n m a -> Transformation t n m a Source # ztimes :: Z -> Transformation t n m a -> Transformation t n m a Source #
(Distributive a, Typeable t, Typeable n, Typeable m) => Additive (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Methods zero :: Root (Transformation t n m a) -> Transformation t n m a Source # (+) :: Transformation t n m a -> Transformation t n m a -> Transformation t n m a Source # ntimes :: N -> Transformation t n m a -> Transformation t n m a Source #
(Algebraic a, Typeable t, Typeable n, Typeable m) => Algebraic (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation
(Distributive a, Typeable t, Typeable n, Typeable m) => Distributive (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation
(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 #
(Distributive a, Typeable t, Typeable n, Typeable m) => FibredOriented (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation
(Multiplicative a, Typeable t, Typeable n, Typeable m) => Multiplicative (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Methods one :: Point (Transformation t n m a) -> Transformation t n m a Source # (*) :: Transformation t n m a -> Transformation t n m a -> Transformation t n m a Source # npower :: Transformation t n m a -> N -> Transformation t n m a Source #
(Multiplicative a, Typeable t, Typeable n, Typeable m) => Oriented (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Associated Types type Point (Transformation t n m a) Source # Methods orientation :: Transformation t n m a -> Orientation (Point (Transformation t n m a)) Source # start :: Transformation t n m a -> Point (Transformation t n m a) Source # end :: Transformation t n m a -> Point (Transformation t n m a) Source #
(Distributive a, Vectorial a, Typeable t, Typeable n, Typeable m) => Vectorial (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Associated Types type Scalar (Transformation t n m a) Source # Methods (!) :: Scalar (Transformation t n m a) -> Transformation t n m a -> Transformation t n m a Source #
type Dual (Transformation t n m a :: Type) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation type Dual (Transformation t n m a :: Type) = Transformation (Dual t) n m (Op a)
type Root (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation type Root (Transformation t n m a) = Orientation (Diagram t n m a)
type Point (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation type Point (Transformation t n m a) = Diagram t n m a
type Scalar (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation type Scalar (Transformation t n m a) = Scalar a

trfs :: Transformation t n m a -> FinList n a Source #

the underlying list of factors.

Duality

coTransformation :: Transformation t n m a -> Dual (Transformation t n m a) Source #

the dual transformation.