Copyright	2008-2010 Edward Kmett
License	BSD
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	experimental
Portability	non-portable (class-associated types)
Safe Haskell	Trustworthy
Language	Haskell2010

Control.Category.Cartesian

Contents

(Co)Cartesian categories

Description

Synopsis

class (Symmetric k (Product k), Monoidal k (Product k)) => Cartesian (k :: Type -> Type -> Type) where
- type Product (k :: Type -> Type -> Type) :: Type -> Type -> Type
- fst :: k (Product k a b) a
- snd :: k (Product k a b) b
- diag :: k a (Product k a a)
- (&&&) :: k a b -> k a c -> k a (Product k b c)
bimapProduct :: Cartesian k => k a c -> k b d -> k (Product k a b) (Product k c d)
braidProduct :: Cartesian k => k (Product k a b) (Product k b a)
associateProduct :: Cartesian k => k (Product k (Product k a b) c) (Product k a (Product k b c))
disassociateProduct :: Cartesian k => k (Product k a (Product k b c)) (Product k (Product k a b) c)
class (Monoidal k (Sum k), Symmetric k (Sum k)) => CoCartesian (k :: Type -> Type -> Type) where
- type Sum (k :: Type -> Type -> Type) :: Type -> Type -> Type
- inl :: k a (Sum k a b)
- inr :: k b (Sum k a b)
- codiag :: k (Sum k a a) a
- (|||) :: k a c -> k b c -> k (Sum k a b) c
bimapSum :: CoCartesian k => k a c -> k b d -> k (Sum k a b) (Sum k c d)
braidSum :: CoCartesian k => k (Sum k a b) (Sum k b a)
associateSum :: CoCartesian k => k (Sum k (Sum k a b) c) (Sum k a (Sum k b c))
disassociateSum :: CoCartesian k => k (Sum k a (Sum k b c)) (Sum k (Sum k a b) c)

(Co)Cartesian categories

class (Symmetric k (Product k), Monoidal k (Product k)) => Cartesian (k :: Type -> Type -> Type) where Source #

Minimum definition:

fst, snd, diag
fst, snd, (&&&)

Minimal complete definition

fst, snd

Associated Types

type Product (k :: Type -> Type -> Type) :: Type -> Type -> Type Source #

Methods

fst :: k (Product k a b) a Source #

snd :: k (Product k a b) b Source #

diag :: k a (Product k a a) Source #

(&&&) :: k a b -> k a c -> k a (Product k b c) infixr 3 Source #

Instances

Instances details

Cartesian (->) Source #

Instance details

Defined in Control.Category.Cartesian

Associated Types

type Product (->)
Instance details Defined in Control.Category.Cartesian type Product (->) = (,)

Methods

fst :: Product (->) a b -> a Source #

snd :: Product (->) a b -> b Source #

diag :: a -> Product (->) a a Source #

(&&&) :: (a -> b) -> (a -> c) -> a -> Product (->) b c Source #

bimapProduct :: Cartesian k => k a c -> k b d -> k (Product k a b) (Product k c d) Source #

free construction of Bifunctor for the product Bifunctor Product k if (&&&) is known

braidProduct :: Cartesian k => k (Product k a b) (Product k b a) Source #

free construction of Braided for the product Bifunctor Product k

associateProduct :: Cartesian k => k (Product k (Product k a b) c) (Product k a (Product k b c)) Source #

free construction of Associative for the product Bifunctor Product k

disassociateProduct :: Cartesian k => k (Product k a (Product k b c)) (Product k (Product k a b) c) Source #

free construction of Disassociative for the product Bifunctor Product k

class (Monoidal k (Sum k), Symmetric k (Sum k)) => CoCartesian (k :: Type -> Type -> Type) where Source #

Minimal complete definition

inl, inr

Associated Types

type Sum (k :: Type -> Type -> Type) :: Type -> Type -> Type Source #

Methods

inl :: k a (Sum k a b) Source #

inr :: k b (Sum k a b) Source #

codiag :: k (Sum k a a) a Source #

(|||) :: k a c -> k b c -> k (Sum k a b) c infixr 2 Source #

Instances

Instances details

CoCartesian (->) Source #

Instance details

Defined in Control.Category.Cartesian

Associated Types

type Sum (->)
Instance details Defined in Control.Category.Cartesian type Sum (->) = Either

Methods

inl :: a -> Sum (->) a b Source #

inr :: b -> Sum (->) a b Source #

codiag :: Sum (->) a a -> a Source #

(|||) :: (a -> c) -> (b -> c) -> Sum (->) a b -> c Source #

bimapSum :: CoCartesian k => k a c -> k b d -> k (Sum k a b) (Sum k c d) Source #

free construction of Bifunctor for the coproduct Bifunctor Sum k if (|||) is known

braidSum :: CoCartesian k => k (Sum k a b) (Sum k b a) Source #

free construction of Braided for the coproduct Bifunctor Sum k

associateSum :: CoCartesian k => k (Sum k (Sum k a b) c) (Sum k a (Sum k b c)) Source #

free construction of Associative for the coproduct Bifunctor Sum k

disassociateSum :: CoCartesian k => k (Sum k a (Sum k b c)) (Sum k (Sum k a b) c) Source #

free construction of Disassociative for the coproduct Bifunctor Sum k