Data.Category

Category

class Category k where

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

Minimal complete definition

src, tgt, (.)

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 (->)

The category with Haskell types as objects and Haskell functions as arrows.

Methods

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

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

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

Category Cat

Cat is the category with categories as objects and funtors as arrows.

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 Unit

Unit is the category with one object.

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.

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 AdjArrow

The category with categories as objects and adjunctions as arrows.

Methods

src :: AdjArrow a b -> Obj AdjArrow a

tgt :: AdjArrow a b -> Obj AdjArrow b

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

Category Boolean

Boolean is the category with true and false as objects, and an arrow from false to true.

Methods

src :: Boolean a b -> Obj Boolean a

tgt :: Boolean a b -> Obj Boolean b

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

Category Simplex

The (augmented) simplex category is the category of finite ordinals and order preserving maps.

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

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 k => Category (Op k)

Op k is opposite category of the category k.

Methods

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

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

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

Category (f (Fix f)) => Category (Fix f)

Fix f is the fixed point category for a category combinator f.

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

Category (Kleisli m)

The category of Kleisli arrows.

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

The product category of category c1 and c2.

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)

Functor category D^C. Objects of D^C are functors from C to D. Arrows of D^C are natural transformations.

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 c -> Nat c d a b -> Nat c d a c

(Category c1, Category c2) => Category ((:>>:) c1 c2)

The directed coproduct category of categories c1 and c2.

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)

The coproduct category of categories c1 and c2.

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)

A monoid as a category with one object.

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)

The category of (F,G)-dialgebras.

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)

The comma category T \downarrow S

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

type Obj k a = k a a

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

Opposite category

data Op k a b

Constructors

Op

Fields

• unOp :: k b a

Instances

Category k => Category (Op k)

Op k is opposite category of the category k.

Methods

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

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

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

HasBinaryProducts k => HasBinaryCoproducts (Op k)

Binary products are the dual of binary coproducts.

Associated Types

type BinaryCoproduct (Op k :: * -> * -> *) x y :: *

Methods

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

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

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

(+++) :: Op k a1 b1 -> Op k a2 b2 -> Op k (BinaryCoproduct (Op k) a1 a2) (BinaryCoproduct (Op k) b1 b2)

HasBinaryCoproducts k => HasBinaryProducts (Op k)

Binary products are the dual of binary coproducts.

Associated Types

type BinaryProduct (Op k :: * -> * -> *) x y :: *

Methods

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

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

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

(***) :: Op k a1 b1 -> Op k a2 b2 -> Op k (BinaryProduct (Op k) a1 a2) (BinaryProduct (Op k) b1 b2)

HasTerminalObject k => HasInitialObject (Op k)

Terminal objects are the dual of initial objects.

Associated Types

type InitialObject (Op k :: * -> * -> *) :: *

Methods

initialObject :: Obj (Op k) (InitialObject (Op k))

initialize :: Obj (Op k) a -> Op k (InitialObject (Op k)) a

HasInitialObject k => HasTerminalObject (Op k)

Terminal objects are the dual of initial objects.

Associated Types

type TerminalObject (Op k :: * -> * -> *) :: *

Methods

terminalObject :: Obj (Op k) (TerminalObject (Op k))

terminate :: Obj (Op k) a -> Op k a (TerminalObject (Op k))

Category k => CartesianClosed (Presheaves k)

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

Associated Types

type Exponential (Presheaves k :: * -> * -> *) y z :: *

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 k)

type InitialObject (Op k) = TerminalObject k

type TerminalObject (Op k)

type TerminalObject (Op k) = InitialObject k

type BinaryCoproduct (Op k) x y

type BinaryCoproduct (Op k) x y = BinaryProduct k x y

type BinaryProduct (Op k) x y

type BinaryProduct (Op k) x y = BinaryCoproduct k x y

type Exponential (Presheaves k) y z

type Exponential (Presheaves k) y z = PShExponential k y z