License	BSD-style (see the file LICENSE)
Maintainer	sjoerd@w3future.com
Stability	experimental
Portability	non-portable
Safe Haskell	Safe-Inferred
Language	Haskell2010

Data.Category

Contents

Category
Opposite category

Description

Synopsis

class Category k where
- src :: k a b -> Obj k a
- tgt :: k a b -> Obj k b
- (.) :: k b c -> k a b -> k a c
type Obj k a = k a a
type Kind (cat :: k -> k -> *) = k
newtype Op k a b = Op {
- unOp :: k b a
}

Category

class Category k where #

An instance of Category k declares the arrow k as a category.

Methods

src :: k a b -> Obj k a #

tgt :: k a b -> Obj k b #

(.) :: k b c -> k a b -> k a c infixr 8 #

Instances

Instances details

Category k2 => Category (Op k2 :: k1 -> k1 -> Type) #	`Op k` is opposite category of the category `k`.
Instance details Defined in Data.Category Methods src :: forall (a :: k) (b :: k). Op k2 a b -> Obj (Op k2) a # tgt :: forall (a :: k) (b :: k). Op k2 a b -> Obj (Op k2) b # (.) :: forall (b :: k) (c :: k) (a :: k). Op k2 b c -> Op k2 a b -> Op k2 a c #
Category Unit #	`Unit` is the category with one object.
Instance details Defined in Data.Category.Unit Methods src :: forall (a :: k) (b :: k). Unit a b -> Obj Unit a # tgt :: forall (a :: k) (b :: k). Unit a b -> Obj Unit b # (.) :: forall (b :: k) (c :: k) (a :: k). Unit b c -> Unit a b -> Unit a c #
Category Void #	`Void` is the category with no objects.
Instance details Defined in Data.Category.Void Methods src :: forall (a :: k) (b :: k). Void a b -> Obj Void a # tgt :: forall (a :: k) (b :: k). Void a b -> Obj Void b # (.) :: forall (b :: k) (c :: k) (a :: k). Void b c -> Void a b -> Void a c #
Category Simplex #	The (augmented) simplex category is the category of finite ordinals and order preserving maps.
Instance details Defined in Data.Category.Simplex Methods src :: forall (a :: k) (b :: k). Simplex a b -> Obj Simplex a # tgt :: forall (a :: k) (b :: k). Simplex a b -> Obj Simplex b # (.) :: forall (b :: k) (c :: k) (a :: k). Simplex b c -> Simplex a b -> Simplex a c #
Category Cube #
Instance details Defined in Data.Category.Cube Methods src :: forall (a :: k) (b :: k). Cube a b -> Obj Cube a # tgt :: forall (a :: k) (b :: k). Cube a b -> Obj Cube b # (.) :: forall (b :: k) (c :: k) (a :: k). Cube b c -> Cube a b -> Cube a c #
Category Boolean #	`Boolean` is the category with true and false as objects, and an arrow from false to true.
Instance details Defined in Data.Category.Boolean Methods src :: forall (a :: k) (b :: k). Boolean a b -> Obj Boolean a # tgt :: forall (a :: k) (b :: k). Boolean a b -> Obj Boolean b # (.) :: forall (b :: k) (c :: k) (a :: k). Boolean b c -> Boolean a b -> Boolean a c #
(Functor f, Dom f ~ (Op c :**: d), Cod f ~ ((->) :: Type -> Type -> Type), Category c, Category d) => Category (Cograph f :: Type -> Type -> Type) #	The cograph of the profunctor `f`.
Instance details Defined in Data.Category.Coproduct Methods src :: forall (a :: k) (b :: k). Cograph f a b -> Obj (Cograph f) a # tgt :: forall (a :: k) (b :: k). Cograph f a b -> Obj (Cograph f) b # (.) :: forall (b :: k) (c :: k) (a :: k). Cograph f b c -> Cograph f a b -> Cograph f a c #
Category (Kleisli m :: Type -> Type -> Type) #	The category of Kleisli arrows.
Instance details Defined in Data.Category.Kleisli Methods src :: forall (a :: k) (b :: k). Kleisli m a b -> Obj (Kleisli m) a # tgt :: forall (a :: k) (b :: k). Kleisli m a b -> Obj (Kleisli m) b # (.) :: forall (b :: k) (c :: k) (a :: k). Kleisli m b c -> Kleisli m a b -> Kleisli m a c #
Category (f (Fix f)) => Category (Fix f :: Type -> Type -> Type) #	`Fix f` is the fixed point category for a category combinator `f`.
Instance details Defined in Data.Category.Fix Methods src :: forall (a :: k) (b :: k). Fix f a b -> Obj (Fix f) a # tgt :: forall (a :: k) (b :: k). Fix f a b -> Obj (Fix f) b # (.) :: forall (b :: k) (c :: k) (a :: k). Fix f b c -> Fix f a b -> Fix f a c #
ECategory k => Category (Underlying k :: Type -> Type -> Type) #	The underlying category of an enriched category
Instance details Defined in Data.Category.Enriched Methods src :: forall (a :: k0) (b :: k0). Underlying k a b -> Obj (Underlying k) a # tgt :: forall (a :: k0) (b :: k0). Underlying k a b -> Obj (Underlying k) b # (.) :: forall (b :: k0) (c :: k0) (a :: k0). Underlying k b c -> Underlying k a b -> Underlying k a c #
Category ((->) :: Type -> Type -> Type) #	The category with Haskell types as objects and Haskell functions as arrows.
Instance details Defined in Data.Category Methods src :: forall (a :: k) (b :: k). (a -> b) -> Obj (->) a # tgt :: forall (a :: k) (b :: k). (a -> b) -> Obj (->) b # (.) :: forall (b :: k) (c :: k) (a :: k). (b -> c) -> (a -> b) -> a -> c #
(Category c1, Category c2) => Category (c1 :**: c2 :: Type -> Type -> Type) #	The product category of categories `c1` and `c2`.
Instance details Defined in Data.Category.Product Methods src :: forall (a :: k) (b :: k). (c1 :: c2) a b -> Obj (c1 :: c2) a # tgt :: forall (a :: k) (b :: k). (c1 :: c2) a b -> Obj (c1 :: c2) b # (.) :: forall (b :: k) (c :: k) (a :: k). (c1 :: c2) b c -> (c1 :: c2) a b -> (c1 :**: c2) a c #
Category d => Category (Nat c d :: Type -> Type -> Type) #	Functor category D^C. Objects of D^C are functors from C to D. Arrows of D^C are natural transformations.
Instance details Defined in Data.Category.NaturalTransformation Methods src :: forall (a :: k) (b :: k). Nat c d a b -> Obj (Nat c d) a # tgt :: forall (a :: k) (b :: k). Nat c d a b -> Obj (Nat c d) b # (.) :: forall (b :: k) (c0 :: k) (a :: k). Nat c d b c0 -> Nat c d a b -> Nat c d a c0 #
(Category c1, Category c2) => Category (c1 :>>: c2 :: Type -> Type -> Type) #
Instance details Defined in Data.Category.Coproduct Methods src :: forall (a :: k) (b :: k). (c1 :>>: c2) a b -> Obj (c1 :>>: c2) a # tgt :: forall (a :: k) (b :: k). (c1 :>>: c2) a b -> Obj (c1 :>>: c2) b # (.) :: forall (b :: k) (c :: k) (a :: k). (c1 :>>: c2) b c -> (c1 :>>: c2) a b -> (c1 :>>: c2) a c #
(Category c1, Category c2) => Category (c1 :++: c2 :: Type -> Type -> Type) #	The coproduct category of categories `c1` and `c2`.
Instance details Defined in Data.Category.Coproduct Methods src :: forall (a :: k) (b :: k). (c1 :++: c2) a b -> Obj (c1 :++: c2) a # tgt :: forall (a :: k) (b :: k). (c1 :++: c2) a b -> Obj (c1 :++: c2) b # (.) :: forall (b :: k) (c :: k) (a :: k). (c1 :++: c2) b c -> (c1 :++: c2) a b -> (c1 :++: c2) a c #
Category (MonoidAsCategory f m :: Type -> Type -> Type) #	A monoid as a category with one object.
Instance details Defined in Data.Category.Monoidal Methods src :: forall (a :: k) (b :: k). MonoidAsCategory f m a b -> Obj (MonoidAsCategory f m) a # tgt :: forall (a :: k) (b :: k). MonoidAsCategory f m a b -> Obj (MonoidAsCategory f m) b # (.) :: forall (b :: k) (c :: k) (a :: k). MonoidAsCategory f m b c -> MonoidAsCategory f m a b -> MonoidAsCategory f m a c #
Category (Dialg f g :: Type -> Type -> Type) #	The category of (F,G)-dialgebras.
Instance details Defined in Data.Category.Dialg Methods src :: forall (a :: k) (b :: k). Dialg f g a b -> Obj (Dialg f g) a # tgt :: forall (a :: k) (b :: k). Dialg f g a b -> Obj (Dialg f g) b # (.) :: forall (b :: k) (c :: k) (a :: k). Dialg f g b c -> Dialg f g a b -> Dialg f g a c #
(Category (Dom t), Category (Dom s)) => Category (t :/\: s :: Type -> Type -> Type) #	The comma category T \downarrow S
Instance details Defined in Data.Category.Comma Methods src :: forall (a :: k) (b :: k). (t :/\: s) a b -> Obj (t :/\: s) a # tgt :: forall (a :: k) (b :: k). (t :/\: s) a b -> Obj (t :/\: s) b # (.) :: forall (b :: k) (c :: k) (a :: k). (t :/\: s) b c -> (t :/\: s) a b -> (t :/\: s) a c #
Category Cat #	`Cat` is the category with categories as objects and funtors as arrows.
Instance details Defined in Data.Category.Functor Methods src :: forall (a :: k) (b :: k). Cat a b -> Obj Cat a # tgt :: forall (a :: k) (b :: k). Cat a b -> Obj Cat b # (.) :: forall (b :: k) (c :: k) (a :: k). Cat b c -> Cat a b -> Cat a c #
Category AdjArrow #	The category with categories as objects and adjunctions as arrows.
Instance details Defined in Data.Category.Adjunction Methods src :: forall (a :: k) (b :: k). AdjArrow a b -> Obj AdjArrow a # tgt :: forall (a :: k) (b :: k). AdjArrow a b -> Obj AdjArrow b # (.) :: forall (b :: k) (c :: k) (a :: k). AdjArrow b c -> AdjArrow a b -> AdjArrow a c #

type Obj k a = k a a #

Whenever objects are required at value level, they are represented by their identity arrows.

type Kind (cat :: k -> k -> *) = k #

Opposite category

newtype Op k a b #

Constructors

Op
Fields unOp :: k b a

Instances

Instances details

Category k2 => Category (Op k2 :: k1 -> k1 -> Type) #	`Op k` is opposite category of the category `k`.
Instance details Defined in Data.Category Methods src :: forall (a :: k) (b :: k). Op k2 a b -> Obj (Op k2) a # tgt :: forall (a :: k) (b :: k). Op k2 a b -> Obj (Op k2) b # (.) :: forall (b :: k) (c :: k) (a :: k). Op k2 b c -> Op k2 a b -> Op k2 a c #
HasBinaryProducts k2 => HasBinaryCoproducts (Op k2 :: k1 -> k1 -> Type) #	Binary products are the dual of binary coproducts.
Instance details Defined in Data.Category.Limit Associated Types type BinaryCoproduct (Op k2) x y :: Kind k # Methods inj1 :: forall (x :: k) (y :: k). Obj (Op k2) x -> Obj (Op k2) y -> Op k2 x (BinaryCoproduct (Op k2) x y) # inj2 :: forall (x :: k) (y :: k). Obj (Op k2) x -> Obj (Op k2) y -> Op k2 y (BinaryCoproduct (Op k2) x y) # (\|\|\|) :: forall (x :: k) (a :: k) (y :: k). Op k2 x a -> Op k2 y a -> Op k2 (BinaryCoproduct (Op k2) x y) a # (+++) :: forall (a1 :: k) (b1 :: k) (a2 :: k) (b2 :: k). Op k2 a1 b1 -> Op k2 a2 b2 -> Op k2 (BinaryCoproduct (Op k2) a1 a2) (BinaryCoproduct (Op k2) b1 b2) #
HasBinaryCoproducts k2 => HasBinaryProducts (Op k2 :: k1 -> k1 -> Type) #	Binary products are the dual of binary coproducts.
Instance details Defined in Data.Category.Limit Associated Types type BinaryProduct (Op k2) x y :: Kind k # Methods proj1 :: forall (x :: k) (y :: k). Obj (Op k2) x -> Obj (Op k2) y -> Op k2 (BinaryProduct (Op k2) x y) x # proj2 :: forall (x :: k) (y :: k). Obj (Op k2) x -> Obj (Op k2) y -> Op k2 (BinaryProduct (Op k2) x y) y # (&&&) :: forall (a :: k) (x :: k) (y :: k). Op k2 a x -> Op k2 a y -> Op k2 a (BinaryProduct (Op k2) x y) # (***) :: forall (a1 :: k) (b1 :: k) (a2 :: k) (b2 :: k). Op k2 a1 b1 -> Op k2 a2 b2 -> Op k2 (BinaryProduct (Op k2) a1 a2) (BinaryProduct (Op k2) b1 b2) #
HasTerminalObject k2 => HasInitialObject (Op k2 :: k1 -> k1 -> Type) #	Terminal objects are the dual of initial objects.
Instance details Defined in Data.Category.Limit Associated Types type InitialObject (Op k2) :: Kind k # Methods initialObject :: Obj (Op k2) (InitialObject (Op k2)) # initialize :: forall (a :: k). Obj (Op k2) a -> Op k2 (InitialObject (Op k2)) a #
HasInitialObject k2 => HasTerminalObject (Op k2 :: k1 -> k1 -> Type) #	Terminal objects are the dual of initial objects.
Instance details Defined in Data.Category.Limit Associated Types type TerminalObject (Op k2) :: Kind k # Methods terminalObject :: Obj (Op k2) (TerminalObject (Op k2)) # terminate :: forall (a :: k). Obj (Op k2) a -> Op k2 a (TerminalObject (Op k2)) #
Category k => CartesianClosed (Presheaves k :: Type -> Type -> Type) #	The category of presheaves on a category `C` is cartesian closed for any `C`.
Instance details Defined in Data.Category.CartesianClosed Associated Types type Exponential (Presheaves k) y z :: Kind k # Methods apply :: forall (y :: k0) (z :: k0). Obj (Presheaves k) y -> Obj (Presheaves k) z -> Presheaves k (BinaryProduct (Presheaves k) (Exponential (Presheaves k) y z) y) z # tuple :: forall (y :: k0) (z :: k0). Obj (Presheaves k) y -> Obj (Presheaves k) z -> Presheaves k z (Exponential (Presheaves k) y (BinaryProduct (Presheaves k) z y)) # (^^^) :: forall (z1 :: k0) (z2 :: k0) (y2 :: k0) (y1 :: k0). Presheaves k z1 z2 -> Presheaves k y2 y1 -> Presheaves k (Exponential (Presheaves k) y1 z1) (Exponential (Presheaves k) y2 z2) #
type InitialObject (Op k2 :: k1 -> k1 -> Type) #
Instance details Defined in Data.Category.Limit type InitialObject (Op k2 :: k1 -> k1 -> Type) = TerminalObject k2
type TerminalObject (Op k2 :: k1 -> k1 -> Type) #
Instance details Defined in Data.Category.Limit type TerminalObject (Op k2 :: k1 -> k1 -> Type) = InitialObject k2
type BinaryCoproduct (Op k2 :: k1 -> k1 -> Type) (x :: Kind (Op k2)) (y :: Kind (Op k2)) #
Instance details Defined in Data.Category.Limit type BinaryCoproduct (Op k2 :: k1 -> k1 -> Type) (x :: Kind (Op k2)) (y :: Kind (Op k2)) = BinaryProduct k2 x y
type BinaryProduct (Op k2 :: k1 -> k1 -> Type) (x :: Kind (Op k2)) (y :: Kind (Op k2)) #
Instance details Defined in Data.Category.Limit type BinaryProduct (Op k2 :: k1 -> k1 -> Type) (x :: Kind (Op k2)) (y :: Kind (Op k2)) = BinaryCoproduct k2 x y
type Exponential (Presheaves k :: Type -> Type -> Type) (y :: Kind (Presheaves k)) (z :: Kind (Presheaves k)) #
Instance details Defined in Data.Category.CartesianClosed type Exponential (Presheaves k :: Type -> Type -> Type) (y :: Kind (Presheaves k)) (z :: Kind (Presheaves k)) = PShExponential k y z