Control.Category

base-4.20.0.0: Core data structures and operations

Copyright | (c) Ashley Yakeley 2007
License | BSD-style (see the LICENSE file in the distribution)
Maintainer | ashley@semantic.org
Stability | stable
Portability | portable
Safe Haskell | Safe
Language | Haskell2010

Control.Category

Description

Synopsis

- class Category (cat :: k -> k -> Type) where
  - id :: forall (a :: k). cat a a
  - (.) :: forall (b :: k) (c :: k) (a :: k). cat b c -> cat a b -> cat a c
- (<<<) :: forall {k} cat (b :: k) (c :: k) (a :: k). Category cat => cat b c -> cat a b -> cat a c
- (>>>) :: forall {k} cat (a :: k) (b :: k) (c :: k). Category cat => cat a b -> cat b c -> cat a c

# Documentation

class Category (cat :: k -> k -> Type) where Source #

A class for categories. Instances should satisfy the laws

Right identity
  `f `.` `id` = f`
Left identity
  ``id` `.` f = f`
Associativity
  `f `.` (g `.` h) = (f `.` g) `.` h`

Methods

id :: forall (a :: k). cat a a Source #

the identity morphism

(.) :: forall (b :: k) (c :: k) (a :: k). cat b c -> cat a b -> cat a c infixr 9 Source #

morphism composition

#### Instances

Instances details

Category Op Source # |
Instance details Defined in Data.Functor.Contravariant Methods id :: Op a a Source # (.) :: Op b c -> Op a b -> Op a c Source #
Monad m => Category (Kleisli m :: Type -> Type -> Type) | @since base-3.0
Instance details Defined in GHC.Internal.Control.Arrow Methods id :: Kleisli m a a Source # (.) :: Kleisli m b c -> Kleisli m a b -> Kleisli m a c Source #
Category (Coercion :: k -> k -> Type) | @since base-4.7.0.0
Instance details Defined in GHC.Internal.Control.Category Methods id :: forall (a :: k). Coercion a a Source # (.) :: forall (b :: k) (c :: k) (a :: k). Coercion b c -> Coercion a b -> Coercion a c Source #
Category ((:~:) :: k -> k -> Type) | @since base-4.7.0.0
Instance details Defined in GHC.Internal.Control.Category Methods id :: forall (a :: k). a :~: a Source # (.) :: forall (b :: k) (c :: k) (a :: k). (b :~: c) -> (a :~: b) -> a :~: c Source #
Category (->) | @since base-3.0
Instance details Defined in GHC.Internal.Control.Category Methods id :: a -> a Source # (.) :: (b -> c) -> (a -> b) -> a -> c Source #
Category ((:~~:) :: k -> k -> Type) | @since base-4.10.0.0
Instance details Defined in GHC.Internal.Control.Category Methods id :: forall (a :: k). a :~~: a Source # (.) :: forall (b :: k) (c :: k) (a :: k). (b :~~: c) -> (a :~~: b) -> a :~~: c Source #

(<<<) :: forall {k} cat (b :: k) (c :: k) (a :: k). Category cat => cat b c -> cat a b -> cat a c infixr 1 Source #

Right-to-left composition

(>>>) :: forall {k} cat (a :: k) (b :: k) (c :: k). Category cat => cat a b -> cat b c -> cat a c infixr 1 Source #

Left-to-right composition