Data.Category

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

Category k2 => Category (Op k2 :: k1 -> k1 -> Type)

Op k is opposite category of the category k.

Defined in Data.Category

Methods

src :: Op k2 a b -> Obj (Op k2) a

tgt :: Op k2 a b -> Obj (Op k2) b

(.) :: Op k2 b c -> Op k2 a b -> Op k2 a c

Category Unit

Unit is the category with one object.

Defined in Data.Category.Unit

Methods

src :: Unit a b -> Obj Unit a

tgt :: Unit a b -> Obj Unit b

(.) :: Unit b c -> Unit a b -> Unit a c

Category Void

Void is the category with no objects.

Defined in Data.Category.Void

Methods

src :: Void a b -> Obj Void a

tgt :: Void a b -> Obj Void b

(.) :: 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.

Defined in Data.Category.Simplex

Methods

src :: Simplex a b -> Obj Simplex a

tgt :: Simplex a b -> Obj Simplex b

(.) :: Simplex b c -> Simplex a b -> Simplex a c

Category Cube

Defined in Data.Category.Cube

Methods

src :: Cube a b -> Obj Cube a

tgt :: Cube a b -> Obj Cube b

(.) :: 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.

Defined in Data.Category.Boolean

Methods

src :: Boolean a b -> Obj Boolean a

tgt :: Boolean a b -> Obj Boolean b

(.) :: 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.

Defined in Data.Category.Coproduct

Methods

src :: Cograph f a b -> Obj (Cograph f) a

tgt :: Cograph f a b -> Obj (Cograph f) b

(.) :: Cograph f b c -> Cograph f a b -> Cograph f a c

Category (Kleisli m :: Type -> Type -> Type)

The category of Kleisli arrows.

Defined in Data.Category.Kleisli

Methods

src :: Kleisli m a b -> Obj (Kleisli m) a

tgt :: Kleisli m a b -> Obj (Kleisli m) b

(.) :: 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.

Defined in Data.Category.Fix

Methods

src :: Fix f a b -> Obj (Fix f) a

tgt :: Fix f a b -> Obj (Fix f) b

(.) :: 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

Defined in Data.Category.Enriched

Methods

src :: Underlying k a b -> Obj (Underlying k) a

tgt :: Underlying k a b -> Obj (Underlying k) b

(.) :: 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.

Defined in Data.Category

Methods

src :: (a -> b) -> Obj (->) a

tgt :: (a -> b) -> Obj (->) b

(.) :: (b -> c) -> (a -> b) -> a -> c

(Category c1, Category c2) => Category (c1 :**: c2 :: Type -> Type -> Type)

The product category of categories c1 and c2.

Defined in Data.Category.Product

Methods

src :: (c1 :**: c2) a b -> Obj (c1 :**: c2) a

tgt :: (c1 :**: c2) a b -> Obj (c1 :**: c2) b

(.) :: (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.

Defined in Data.Category.NaturalTransformation

Methods

src :: Nat c d a b -> Obj (Nat c d) a

tgt :: Nat c d a b -> Obj (Nat c d) b

(.) :: 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)

Defined in Data.Category.Coproduct

Methods

src :: (c1 :>>: c2) a b -> Obj (c1 :>>: c2) a

tgt :: (c1 :>>: c2) a b -> Obj (c1 :>>: c2) b

(.) :: (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.

Defined in Data.Category.Coproduct

Methods

src :: (c1 :++: c2) a b -> Obj (c1 :++: c2) a

tgt :: (c1 :++: c2) a b -> Obj (c1 :++: c2) b

(.) :: (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.

Defined in Data.Category.Monoidal

Methods

src :: MonoidAsCategory f m a b -> Obj (MonoidAsCategory f m) a

tgt :: MonoidAsCategory f m a b -> Obj (MonoidAsCategory f m) b

(.) :: 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.

Defined in Data.Category.Dialg

Methods

src :: Dialg f g a b -> Obj (Dialg f g) a

tgt :: Dialg f g a b -> Obj (Dialg f g) b

(.) :: 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

Defined in Data.Category.Comma

Methods

src :: (t :/\: s) a b -> Obj (t :/\: s) a

tgt :: (t :/\: s) a b -> Obj (t :/\: s) b

(.) :: (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.

Defined in Data.Category.Functor

Methods

src :: Cat a b -> Obj Cat a

tgt :: Cat a b -> Obj Cat b

(.) :: Cat b c -> Cat a b -> Cat a c

Category AdjArrow

The category with categories as objects and adjunctions as arrows.

Defined in Data.Category.Adjunction

Methods

src :: AdjArrow a b -> Obj AdjArrow a

tgt :: AdjArrow a b -> Obj AdjArrow b

(.) :: 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 family Kind (k :: o -> o -> *) :: * where ...

Kind k is the kind of the objects of the category k.

Equations

Kind (k :: o -> o -> *) = o

Opposite category

newtype Op k a b

Constructors

Op

Fields

• unOp :: k b a

Instances

Category k2 => Category (Op k2 :: k1 -> k1 -> Type)

Op k is opposite category of the category k.

Defined in Data.Category

Methods

src :: Op k2 a b -> Obj (Op k2) a

tgt :: Op k2 a b -> Obj (Op k2) b

(.) :: 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.

Defined in Data.Category.Limit

Associated Types

type BinaryCoproduct (Op k2) x y :: Kind k

Methods

inj1 :: Obj (Op k2) x -> Obj (Op k2) y -> Op k2 x (BinaryCoproduct (Op k2) x y)

inj2 :: Obj (Op k2) x -> Obj (Op k2) y -> Op k2 y (BinaryCoproduct (Op k2) x y)

(|||) :: Op k2 x a -> Op k2 y a -> Op k2 (BinaryCoproduct (Op k2) x y) a

(+++) :: 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.

Defined in Data.Category.Limit

Associated Types

type BinaryProduct (Op k2) x y :: Kind k

Methods

proj1 :: Obj (Op k2) x -> Obj (Op k2) y -> Op k2 (BinaryProduct (Op k2) x y) x

proj2 :: Obj (Op k2) x -> Obj (Op k2) y -> Op k2 (BinaryProduct (Op k2) x y) y

(&&&) :: Op k2 a x -> Op k2 a y -> Op k2 a (BinaryProduct (Op k2) x y)

(***) :: 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.

Defined in Data.Category.Limit

Associated Types

type InitialObject (Op k2) :: Kind k

Methods

initialObject :: Obj (Op k2) (InitialObject (Op k2))

initialize :: 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.

Defined in Data.Category.Limit

Associated Types

type TerminalObject (Op k2) :: Kind k

Methods

terminalObject :: Obj (Op k2) (TerminalObject (Op k2))

terminate :: 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.

Defined in Data.Category.CartesianClosed

Associated Types

type Exponential (Presheaves k) y z :: Kind k

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)

type InitialObject (Op k2 :: k1 -> k1 -> Type)

Defined in Data.Category.Limit

type InitialObject (Op k2 :: k1 -> k1 -> Type) = TerminalObject k2

type TerminalObject (Op k2 :: k1 -> k1 -> Type)

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))

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))

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 (Nat (Op k) ((->) :: Type -> Type -> Type))) (z :: Kind (Nat (Op k) ((->) :: Type -> Type -> Type)))

Defined in Data.Category.CartesianClosed

type Exponential (Presheaves k :: Type -> Type -> Type) (y :: Kind (Nat (Op k) ((->) :: Type -> Type -> Type))) (z :: Kind (Nat (Op k) ((->) :: Type -> Type -> Type))) = PShExponential k y z