Safe Haskell	None
Language	Haskell2010

Proarrow.Category

Documentation

type CAT k = PRO k k Source Comments #

class Promonad ((~>) :: CAT k) => CategoryOf k Source Comments #

Associated Types

type (~>) :: CAT k infixr 0 Source Comments #

type Ob (a :: k) Source Comments #

type Ob (a :: k) = Any a

Instances

Instances details

CategoryOf BOOL Source Comments #

The category of 2 objects and one arrow between them, a.k.a. the walking arrow.

Instance details

Defined in Proarrow.Category.Instance.Bool

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Bool type (~>) = Booleans
type Ob (b :: BOOL)
Instance details Defined in Proarrow.Category.Instance.Bool type Ob (b :: BOOL) = IsBool b

CategoryOf KIND Source Comments #

The category of categories and profunctors between them.

Instance details

Defined in Proarrow.Category.Instance.Cat

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Cat type (~>) = Cat
type Ob (c :: KIND)
Instance details Defined in Proarrow.Category.Instance.Cat type Ob (c :: KIND) = (Is 'K c, CategoryOf (UN 'K c))

CategoryOf CONSTRAINT Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Constraint

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Constraint type (~>) = (:-)
type Ob (a :: CONSTRAINT)
Instance details Defined in Proarrow.Category.Instance.Constraint type Ob (a :: CONSTRAINT) = Is 'CNSTRNT a

CategoryOf LINEAR Source Comments #

Category of linear functions.

Instance details

Defined in Proarrow.Category.Instance.Linear

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Linear type (~>) = Linear
type Ob (a :: LINEAR)
Instance details Defined in Proarrow.Category.Instance.Linear type Ob (a :: LINEAR) = Is 'L a

CategoryOf POINTED Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.PointedHask type (~>) = Pointed
type Ob (a :: POINTED)
Instance details Defined in Proarrow.Category.Instance.PointedHask type Ob (a :: POINTED) = a ~ 'P (UN 'P a)

CategoryOf Nat Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Simplex

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Simplex type (~>) = Simplex
type Ob (a :: Nat)
Instance details Defined in Proarrow.Category.Instance.Simplex type Ob (a :: Nat) = IsNat a

CategoryOf VOID Source Comments #

The category with no objects, the initial category.

Instance details

Defined in Proarrow.Category.Instance.Zero

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Zero type (~>) = Zero
type Ob (a :: VOID)
Instance details Defined in Proarrow.Category.Instance.Zero type Ob (a :: VOID) = Bottom

CategoryOf () Source Comments #

The category with one object, the terminal category.

Instance details

Defined in Proarrow.Category.Instance.Unit

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Unit type (~>) = Unit
type Ob (a :: ())
Instance details Defined in Proarrow.Category.Instance.Unit type Ob (a :: ()) = a ~ '()

CategoryOf Type Source Comments #

Instance details

Defined in Proarrow.Core

Associated Types

type (~>)
Instance details Defined in Proarrow.Core type (~>) = (->)
type Ob (a :: Type)
Instance details Defined in Proarrow.Core type Ob (a :: Type) = Any a

CategoryOf (FIN n) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Fin

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Fin type (~>) = LTE :: FIN n -> FIN n -> Type

CategoryOf k => CategoryOf (LIST k) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.List

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.List type (~>) = List :: LIST k -> LIST k -> Type

Num a => CategoryOf (MatK a) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Mat

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Mat type (~>) = Mat :: MatK a -> MatK a -> Type

PreorderOf k => CategoryOf (POCATK k) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PreorderAsCategory

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.PreorderAsCategory type (~>) = PoAsCat :: POCATK k -> POCATK k -> Type

CategoryOf k => CategoryOf (REV k) Source Comments #

Instance details

Defined in Proarrow.Category.Monoidal.Rev

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Monoidal.Rev type (~>) = Rev ((~>) :: CAT k)

CategoryOf k => CategoryOf (OPPOSITE k) Source Comments #

The opposite category of the category of k.

Instance details

Defined in Proarrow.Category.Opposite

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Opposite type (~>) = Op ((~>) :: CAT k)

BiCCC k => CategoryOf (FK k) Source Comments #

Instance details

Defined in Proarrow.Helper.CCC

Associated Types

type (~>)
Instance details Defined in Proarrow.Helper.CCC type (~>) = FreeCCC :: FK k -> FK k -> Type

CategoryOf k => CategoryOf (COPROD k) Source Comments #

Instance details

Defined in Proarrow.Object.BinaryCoproduct

Associated Types

type (~>)
Instance details Defined in Proarrow.Object.BinaryCoproduct type (~>) = Coprod ((~>) :: CAT k)

CategoryOf k => CategoryOf (PROD k) Source Comments #

Instance details

Defined in Proarrow.Object.BinaryProduct

Associated Types

type (~>)
Instance details Defined in Proarrow.Object.BinaryProduct type (~>) = Prod ((~>) :: CAT k)

Monoidal k => CategoryOf [k] Source Comments #

Instance details

Defined in Proarrow.Category.Monoidal.Strictified

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Monoidal.Strictified type (~>) = Strictified :: [k] -> [k] -> Type

CategoryOf (PROFK j k) Source Comments #

Instance details

Defined in Proarrow.Category.Bicategory.Prof

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Bicategory.Prof type (~>) = Prof :: PROFK j k -> PROFK j k -> Type

Rewrite g => CategoryOf (FREE g) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Free

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Free type (~>) = Free :: FREE g -> FREE g -> Type

Promonad p => CategoryOf (KLEISLI p) Source Comments #

Every promonad makes a category.

Instance details

Defined in Proarrow.Category.Instance.Kleisli

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Kleisli type (~>) = Kleisli :: KLEISLI p -> KLEISLI p -> Type

CategoryOf (NatK j k) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Nat

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Nat type (~>) = Nat' :: NatK j k -> NatK j k -> Type

CategoryOf (Simplex j k) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Simplex

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Simplex type (~>) = BiSimplex :: Simplex j k -> Simplex j k -> Type

CategoryOf k => CategoryOf (SUBCAT ob) Source Comments #

The subcategory with objects with instances of the given constraint ob.

Instance details

Defined in Proarrow.Category.Instance.Sub

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Sub type (~>) = Sub ((~>) :: CAT k) :: SUBCAT ob -> SUBCAT ob -> Type

(j ~ '(), k ~ '()) => CategoryOf (Unit j k) Source Comments #

Instance details

Defined in Proarrow.Category.Bicategory.Terminal

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Bicategory.Terminal type (~>) = Terminal :: Unit j k -> Unit j k -> Type

CategoryOf (j +-> k) Source Comments #

The category of profunctors and natural transformations between them.

Instance details

Defined in Proarrow.Category.Instance.Prof

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Prof type (~>) = Prof :: (j +-> k) -> (j +-> k) -> Type

(CategoryOf k1, CategoryOf k2) => CategoryOf (k1, k2) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Product

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Product type (~>) = ((~>) :: CAT k1) :**: ((~>) :: CAT k2)

CategoryOf (k1 -> k2 -> k3 -> k4 -> Type) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Nat

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Nat type (~>) = Nat :: (k1 -> k2 -> k3 -> k4 -> Type) -> (k1 -> k2 -> k3 -> k4 -> Type) -> Type

CategoryOf (k1 -> k2 -> k3 -> Type) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Nat

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Nat type (~>) = Nat :: (k1 -> k2 -> k3 -> Type) -> (k1 -> k2 -> k3 -> Type) -> Type

CategoryOf (k1 -> Type) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Nat

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Nat type (~>) = Nat :: (k1 -> Type) -> (k1 -> Type) -> Type

(CategoryOf k, Ob i, Ob j) => CategoryOf (PLAINK k i j) Source Comments #

Instance details

Defined in Proarrow.Category.Bicategory.CategoryAsBi

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Bicategory.CategoryAsBi type (~>) = Category :: PLAINK k i j -> PLAINK k i j -> Type

CategoryOf k => CategoryOf (MonK k i j) Source Comments #

Instance details

Defined in Proarrow.Category.Bicategory.MonoidalAsBi

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Bicategory.MonoidalAsBi type (~>) = Mon2 :: MonK k i j -> MonK k i j -> Type

Profunctor p => CategoryOf (COLLAGE p) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Collage

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Instance.Collage type (~>) = Collage :: COLLAGE p -> COLLAGE p -> Type

(Bicategory kk, Ob0 kk k) => CategoryOf (ENDO kk k) Source Comments #

Instance details

Defined in Proarrow.Category.Monoidal.Endo

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Monoidal.Endo type (~>) = Endo :: ENDO kk k -> ENDO kk k -> Type

CategoryOf (DiscreteK ob j k) Source Comments #

Instance details

Defined in Proarrow.Category.Bicategory.Bidiscrete

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Bicategory.Bidiscrete type (~>) = Bidiscrete :: DiscreteK ob j k -> DiscreteK ob j k -> Type

CategoryOf (kk j k2) => CategoryOf (COK kk j k2) Source Comments #

Instance details

Defined in Proarrow.Category.Bicategory.Co

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Bicategory.Co type (~>) = Co :: COK kk j k2 -> COK kk j k2 -> Type

CategoryOf (kk k2 j) => CategoryOf (OPK kk j k2) Source Comments #

Instance details

Defined in Proarrow.Category.Bicategory.Op

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Bicategory.Op type (~>) = Op :: OPK kk j k2 -> OPK kk j k2 -> Type

(CategoryOf (kk j k2), Bicategory kk) => CategoryOf (Path kk j k2) Source Comments #

Instance details

Defined in Proarrow.Category.Bicategory.Strictified

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Bicategory.Strictified type (~>) = Strictified :: Path kk j k2 -> Path kk j k2 -> Type

CategoryOf (kk i j) => CategoryOf (HK kk i j) Source Comments #

Instance details

Defined in Proarrow.Category.Bicategory.Hom

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Bicategory.Hom type (~>) = HomW :: HK kk i j -> HK kk i j -> Type

CategoryOf (kk i j) => CategoryOf (SUBCAT tag kk i j) Source Comments #

The subcategory with objects with instances of the given constraint `IsOb tag`.

Instance details

Defined in Proarrow.Category.Bicategory.Sub

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Bicategory.Sub type (~>) = Sub :: SUBCAT tag kk i j -> SUBCAT tag kk i j -> Type

CategoryOf (kk i j) => CategoryOf (WKK kk i j) Source Comments #

Instance details

Defined in Proarrow.Category.Equipment.BiAsEquipment

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Equipment.BiAsEquipment type (~>) = W :: WKK kk i j -> WKK kk i j -> Type

CategoryOf (kk i j) => CategoryOf (QKK kk i j) Source Comments #

Instance details

Defined in Proarrow.Category.Equipment.Quintet

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Equipment.Quintet type (~>) = Q2 :: QKK kk i j -> QKK kk i j -> Type

IsOptic w c d => CategoryOf (OPTIC w c d) Source Comments #

Instance details

Defined in Proarrow.Category.Monoidal.Optic

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Monoidal.Optic type (~>) = OpticCat :: OPTIC w c d -> OPTIC w c d -> Type

(CategoryOf (jj (Fst ik) (Fst jl)), CategoryOf (kk (Snd ik) (Snd jl))) => CategoryOf (PRODK jj kk ik jl) Source Comments #

Instance details

Defined in Proarrow.Category.Bicategory.Product

Associated Types

type (~>)
Instance details Defined in Proarrow.Category.Bicategory.Product type (~>) = Prod :: PRODK jj kk ik jl -> PRODK jj kk ik jl -> Type

dimapDefault :: forall {k} p (c :: k) (a :: k) (b :: k) (d :: k). Promonad p => p c a -> p b d -> p a b -> p c d Source Comments #