Data.Category.CartesianClosed

Documentation

class (HasTerminalObject k, HasBinaryProducts k) => CartesianClosed k where

A category is cartesian closed if it has all products and exponentials for all objects.

Associated Types

type Exponential k y z :: *

Methods

apply :: Obj k y -> Obj k z -> k (BinaryProduct k (Exponential k y z) y) z

tuple :: Obj k y -> Obj k z -> k z (Exponential k y (BinaryProduct k z y))

(^^^) :: k z1 z2 -> k y2 y1 -> k (Exponential k y1 z1) (Exponential k y2 z2)

Instances

CartesianClosed Cat

Exponentials in Cat are the functor categories.

Defined in Data.Category.CartesianClosed

Associated Types

type Exponential Cat y z :: Type

Methods

apply :: Obj Cat y -> Obj Cat z -> Cat (BinaryProduct Cat (Exponential Cat y z) y) z

tuple :: Obj Cat y -> Obj Cat z -> Cat z (Exponential Cat y (BinaryProduct Cat z y))

(^^^) :: Cat z1 z2 -> Cat y2 y1 -> Cat (Exponential Cat y1 z1) (Exponential Cat y2 z2)

CartesianClosed Unit

Defined in Data.Category.CartesianClosed

Associated Types

type Exponential Unit y z :: Type

Methods

apply :: Obj Unit y -> Obj Unit z -> Unit (BinaryProduct Unit (Exponential Unit y z) y) z

tuple :: Obj Unit y -> Obj Unit z -> Unit z (Exponential Unit y (BinaryProduct Unit z y))

(^^^) :: Unit z1 z2 -> Unit y2 y1 -> Unit (Exponential Unit y1 z1) (Exponential Unit y2 z2)

CartesianClosed Boolean

Implication makes the Boolean category cartesian closed.

Defined in Data.Category.Boolean

Associated Types

type Exponential Boolean y z :: Type

Methods

apply :: Obj Boolean y -> Obj Boolean z -> Boolean (BinaryProduct Boolean (Exponential Boolean y z) y) z

tuple :: Obj Boolean y -> Obj Boolean z -> Boolean z (Exponential Boolean y (BinaryProduct Boolean z y))

(^^^) :: Boolean z1 z2 -> Boolean y2 y1 -> Boolean (Exponential Boolean y1 z1) (Exponential Boolean y2 z2)

Category k => CartesianClosed (Presheaves k)

The category of presheaves on a category C is cartesian closed for any C.

Defined in Data.Category.CartesianClosed

Associated Types

type Exponential (Presheaves k) y z :: Type

Methods

apply :: Obj (Presheaves k) y -> Obj (Presheaves k) z -> Presheaves k (BinaryProduct (Presheaves k) (Exponential (Presheaves k) y z) y) z

tuple :: Obj (Presheaves k) y -> Obj (Presheaves k) z -> Presheaves k z (Exponential (Presheaves k) y (BinaryProduct (Presheaves k) z y))

(^^^) :: Presheaves k z1 z2 -> Presheaves k y2 y1 -> Presheaves k (Exponential (Presheaves k) y1 z1) (Exponential (Presheaves k) y2 z2)

CartesianClosed (f (Fix f)) => CartesianClosed (Fix f)

Defined in Data.Category.Fix

Associated Types

type Exponential (Fix f) y z :: Type

Methods

apply :: Obj (Fix f) y -> Obj (Fix f) z -> Fix f (BinaryProduct (Fix f) (Exponential (Fix f) y z) y) z

tuple :: Obj (Fix f) y -> Obj (Fix f) z -> Fix f z (Exponential (Fix f) y (BinaryProduct (Fix f) z y))

(^^^) :: Fix f z1 z2 -> Fix f y2 y1 -> Fix f (Exponential (Fix f) y1 z1) (Exponential (Fix f) y2 z2)

CartesianClosed ((->) :: Type -> Type -> Type)

Exponentials in Hask are functions.

Defined in Data.Category.CartesianClosed

Associated Types

type Exponential (->) y z :: Type

Methods

apply :: Obj (->) y -> Obj (->) z -> BinaryProduct (->) (Exponential (->) y z) y -> z

tuple :: Obj (->) y -> Obj (->) z -> z -> Exponential (->) y (BinaryProduct (->) z y)

(^^^) :: (z1 -> z2) -> (y2 -> y1) -> Exponential (->) y1 z1 -> Exponential (->) y2 z2

data ExpFunctor (k :: * -> * -> *)

Constructors

ExpFunctor

Instances

CartesianClosed k => Functor (ExpFunctor k)

The exponential as a bifunctor.

Defined in Data.Category.CartesianClosed

Associated Types

type Dom (ExpFunctor k) :: Type -> Type -> Type

type Cod (ExpFunctor k) :: Type -> Type -> Type

type (ExpFunctor k) :% a :: Type

Methods

(%) :: ExpFunctor k -> Dom (ExpFunctor k) a b -> Cod (ExpFunctor k) (ExpFunctor k :% a) (ExpFunctor k :% b)

type Dom (ExpFunctor k)

Defined in Data.Category.CartesianClosed

type Dom (ExpFunctor k) = Op k :**: k

type Cod (ExpFunctor k)

Defined in Data.Category.CartesianClosed

type Cod (ExpFunctor k) = k

type (ExpFunctor k) :% (y, z)

Defined in Data.Category.CartesianClosed

type (ExpFunctor k) :% (y, z) = Exponential k y z

flip :: CartesianClosed k => Obj k a -> Obj k b -> Obj k c -> k (Exponential k a (Exponential k b c)) (Exponential k b (Exponential k a c))

type PShExponential k y z = (Presheaves k :-*: z) :.: Opposite ((ProductFunctor (Presheaves k) :.: Tuple2 (Presheaves k) (Presheaves k) y) :.: YonedaEmbedding k)

pshExponential :: Category k => Obj (Presheaves k) y -> Obj (Presheaves k) z -> PShExponential k y z

curryAdj :: CartesianClosed k => Obj k y -> Adjunction k k (ProductFunctor k :.: Tuple2 k k y) (ExpFunctor k :.: Tuple1 (Op k) k y)

The product functor is left adjoint the the exponential functor.

curry :: CartesianClosed k => Obj k x -> Obj k y -> Obj k z -> k (BinaryProduct k x y) z -> k x (Exponential k y z)

From the adjunction between the product functor and the exponential functor we get the curry and uncurry functions, generalized to any cartesian closed category.

uncurry :: CartesianClosed k => Obj k x -> Obj k y -> Obj k z -> k x (Exponential k y z) -> k (BinaryProduct k x y) z

type State k s a = Exponential k s (BinaryProduct k a s)

From every adjunction we get a monad, in this case the State monad.

stateMonadReturn :: CartesianClosed k => Obj k s -> Obj k a -> k a (State k s a)

stateMonadJoin :: CartesianClosed k => Obj k s -> Obj k a -> k (State k s (State k s a)) (State k s a)

type Context k s a = BinaryProduct k (Exponential k s a) s

contextComonadExtract :: CartesianClosed k => Obj k s -> Obj k a -> k (Context k s a) a

contextComonadDuplicate :: CartesianClosed k => Obj k s -> Obj k a -> k (Context k s a) (Context k s (Context k s a))