Copyright	2008-2012 Edward Kmett
License	BSD
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	experimental
Portability	portable
Safe Haskell	Trustworthy
Language	Haskell2010

Control.Category.Braided

Description

Synopsis

class Associative k p => Braided (k :: Type -> Type -> Type) (p :: Type -> Type -> Type) where
- braid :: k (p a b) (p b a)
class Braided k p => Symmetric (k :: Type -> Type -> Type) (p :: Type -> Type -> Type)
swap :: Symmetric k p => k (p a b) (p b a)

Documentation

class Associative k p => Braided (k :: Type -> Type -> Type) (p :: Type -> Type -> Type) where Source #

A braided (co)(monoidal or associative) category can commute the arguments of its bi-endofunctor. Obeys the laws:

associate . braid . associate = second braid . associate . first braid
disassociate . braid . disassociate = first braid . disassociate . second braid

If the category is Monoidal the following laws should be satisfied

idr . braid = idl
idl . braid = idr

If the category is Comonoidal the following laws should be satisfied

braid . coidr = coidl
braid . coidl = coidr

Methods

braid :: k (p a b) (p b a) Source #

Instances

Instances details

Braided (->) Either Source #
Instance details Defined in Control.Category.Braided Methods braid :: Either a b -> Either b a Source #
Braided (->) (,) Source #
Instance details Defined in Control.Category.Braided Methods braid :: (a, b) -> (b, a) Source #

class Braided k p => Symmetric (k :: Type -> Type -> Type) (p :: Type -> Type -> Type) Source #

If we have a symmetric (co)Monoidal category, you get the additional law:

swap . swap = id

Instances

Instances details

Symmetric (->) Either Source #
Instance details Defined in Control.Category.Braided
Symmetric (->) (,) Source #
Instance details Defined in Control.Category.Braided

swap :: Symmetric k p => k (p a b) (p b a) Source #