Safe Haskell	None
Language	Haskell2010

Proarrow.Category.Monoidal

Documentation

class (Monoidal j, Monoidal k, Profunctor p) => MonoidalProfunctor (p :: j +-> k) where Source Comments #

Methods

par0 :: p (Unit :: k) (Unit :: j) Source Comments #

par :: forall (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). p x1 x2 -> p y1 y2 -> p (x1 ** y1) (x2 ** y2) Source Comments #

Instances

Instances details

MonoidalProfunctor Booleans Source Comments #
Instance details Defined in Proarrow.Category.Instance.Bool Methods par0 :: Booleans (Unit :: BOOL) (Unit :: BOOL) Source Comments # par :: forall (x1 :: BOOL) (x2 :: BOOL) (y1 :: BOOL) (y2 :: BOOL). Booleans x1 x2 -> Booleans y1 y2 -> Booleans (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor Cat Source Comments #
Instance details Defined in Proarrow.Category.Instance.Cat Methods par0 :: Cat (Unit :: KIND) (Unit :: KIND) Source Comments # par :: forall (x1 :: KIND) (x2 :: KIND) (y1 :: KIND) (y2 :: KIND). Cat x1 x2 -> Cat y1 y2 -> Cat (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor (:-) Source Comments #
Instance details Defined in Proarrow.Category.Instance.Constraint Methods par0 :: (Unit :: CONSTRAINT) :- (Unit :: CONSTRAINT) Source Comments # par :: forall (x1 :: CONSTRAINT) (x2 :: CONSTRAINT) (y1 :: CONSTRAINT) (y2 :: CONSTRAINT). (x1 :- x2) -> (y1 :- y2) -> (x1 y1) :- (x2 y2) Source Comments #
MonoidalProfunctor Linear Source Comments #
Instance details Defined in Proarrow.Category.Instance.Linear Methods par0 :: Linear (Unit :: LINEAR) (Unit :: LINEAR) Source Comments # par :: forall (x1 :: LINEAR) (x2 :: LINEAR) (y1 :: LINEAR) (y2 :: LINEAR). Linear x1 x2 -> Linear y1 y2 -> Linear (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor Forget Source Comments #	Forget is a lax monoidal functor
Instance details Defined in Proarrow.Category.Instance.Linear Methods par0 :: Forget (Unit :: Type) (Unit :: LINEAR) Source Comments # par :: forall x1 (x2 :: LINEAR) y1 (y2 :: LINEAR). Forget x1 x2 -> Forget y1 y2 -> Forget (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor Pointed Source Comments #
Instance details Defined in Proarrow.Category.Instance.PointedHask Methods par0 :: Pointed (Unit :: POINTED) (Unit :: POINTED) Source Comments # par :: forall (x1 :: POINTED) (x2 :: POINTED) (y1 :: POINTED) (y2 :: POINTED). Pointed x1 x2 -> Pointed y1 y2 -> Pointed (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor Simplex Source Comments #
Instance details Defined in Proarrow.Category.Instance.Simplex Methods par0 :: Simplex (Unit :: Nat) (Unit :: Nat) Source Comments # par :: forall (x1 :: Nat) (x2 :: Nat) (y1 :: Nat) (y2 :: Nat). Simplex x1 x2 -> Simplex y1 y2 -> Simplex (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor Unit Source Comments #
Instance details Defined in Proarrow.Object.BinaryProduct Methods par0 :: Unit (Unit :: ()) (Unit :: ()) Source Comments # par :: forall (x1 :: ()) (x2 :: ()) (y1 :: ()) (y2 :: ()). Unit x1 x2 -> Unit y1 y2 -> Unit (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor Free Source Comments #	Free is a lax monoidal functor
Instance details Defined in Proarrow.Category.Instance.Linear Methods par0 :: Free (Unit :: LINEAR) (Unit :: Type) Source Comments # par :: forall (x1 :: LINEAR) x2 (y1 :: LINEAR) y2. Free x1 x2 -> Free y1 y2 -> Free (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor (Cont r :: Type -> Type -> Type) Source Comments #	Only premonoidal not monoidal?
Instance details Defined in Proarrow.Promonad.Cont Methods par0 :: Cont r (Unit :: Type) (Unit :: Type) Source Comments # par :: Cont r x1 x2 -> Cont r y1 y2 -> Cont r (x1 y1) (x2 y2) Source Comments #
SymMonoidal k => MonoidalProfunctor (Fold :: k -> k -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Fold Methods par0 :: Fold (Unit :: k) (Unit :: k) Source Comments # par :: forall (x1 :: k) (x2 :: k) (y1 :: k) (y2 :: k). Fold x1 x2 -> Fold y1 y2 -> Fold (x1 y1) (x2 y2) Source Comments #
Monoidal k => MonoidalProfunctor (Id :: k -> k -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Identity Methods par0 :: Id (Unit :: k) (Unit :: k) Source Comments # par :: forall (x1 :: k) (x2 :: k) (y1 :: k) (y2 :: k). Id x1 x2 -> Id y1 y2 -> Id (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor (->) Source Comments #
Instance details Defined in Proarrow.Object.BinaryProduct Methods par0 :: (Unit :: Type) -> (Unit :: Type) Source Comments # par :: (x1 -> x2) -> (y1 -> y2) -> (x1 y1) -> (x2 y2) Source Comments #
MonoidalProfunctor p => MonoidalProfunctor (Fix p :: j -> j -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Fix Methods par0 :: Fix p (Unit :: j) (Unit :: j) Source Comments # par :: forall (x1 :: j) (x2 :: j) (y1 :: j) (y2 :: j). Fix p x1 x2 -> Fix p y1 y2 -> Fix p (x1 y1) (x2 y2) Source Comments #
(Monoidal j, Monoidal k) => MonoidalProfunctor (TerminalProfunctor :: k -> j -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Terminal Methods par0 :: TerminalProfunctor (Unit :: k) (Unit :: j) Source Comments # par :: forall (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). TerminalProfunctor x1 x2 -> TerminalProfunctor y1 y2 -> TerminalProfunctor (x1 y1) (x2 y2) Source Comments #
(SymMonoidal k, Ob s) => MonoidalProfunctor (State s :: k -> k -> Type) Source Comments #	Note: This is only premonoidal, not monoidal.
Instance details Defined in Proarrow.Promonad.State Methods par0 :: State s (Unit :: k) (Unit :: k) Source Comments # par :: forall (x1 :: k) (x2 :: k) (y1 :: k) (y2 :: k). State s x1 x2 -> State s y1 y2 -> State s (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor (RepCostar Free) Source Comments #	Free is also a colax monoidal functor
Instance details Defined in Proarrow.Category.Instance.Linear Methods par0 :: RepCostar Free (Unit :: Type) (Unit :: LINEAR) Source Comments # par :: forall x1 (x2 :: LINEAR) y1 (y2 :: LINEAR). RepCostar Free x1 x2 -> RepCostar Free y1 y2 -> RepCostar Free (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor (RepCostar Forget) Source Comments #	Forget is also a colax monoidal functor
Instance details Defined in Proarrow.Category.Instance.Linear Methods par0 :: RepCostar Forget (Unit :: LINEAR) (Unit :: Type) Source Comments # par :: forall (x1 :: LINEAR) x2 (y1 :: LINEAR) y2. RepCostar Forget x1 x2 -> RepCostar Forget y1 y2 -> RepCostar Forget (x1 y1) (x2 y2) Source Comments #
(Monoid c, Monoidal j, Monoidal k) => MonoidalProfunctor (Constant c :: k -> j -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Constant Methods par0 :: Constant c (Unit :: k) (Unit :: j) Source Comments # par :: forall (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). Constant c x1 x2 -> Constant c y1 y2 -> Constant c (x1 y1) (x2 y2) Source Comments #
(MonoidalProfunctor r, Adjunction l r, Representable l, Representable r, Monoidal j, Monoidal k) => MonoidalProfunctor (RepCostar l :: k -> j -> Type) Source Comments #
Instance details Defined in Proarrow.Adjunction Methods par0 :: RepCostar l (Unit :: k) (Unit :: j) Source Comments # par :: forall (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). RepCostar l x1 x2 -> RepCostar l y1 y2 -> RepCostar l (x1 y1) (x2 y2) Source Comments #
(Applicative f, Monoidal j, Monoidal k) => MonoidalProfunctor (Star f :: k -> j -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Star Methods par0 :: Star f (Unit :: k) (Unit :: j) Source Comments # par :: forall (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). Star f x1 x2 -> Star f y1 y2 -> Star f (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor p => MonoidalProfunctor (Wrapped p :: k -> j -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Wrapped Methods par0 :: Wrapped p (Unit :: k) (Unit :: j) Source Comments # par :: forall (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). Wrapped p x1 x2 -> Wrapped p y1 y2 -> Wrapped p (x1 y1) (x2 y2) Source Comments #
(Cartesian j, Cartesian k, Functor f) => MonoidalProfunctor (Costar f :: j -> k -> Type) Source Comments #	Every functor between cartesian categories is a colax monoidal functor.
Instance details Defined in Proarrow.Profunctor.Costar Methods par0 :: Costar f (Unit :: j) (Unit :: k) Source Comments # par :: forall (x1 :: j) (x2 :: k) (y1 :: j) (y2 :: k). Costar f x1 x2 -> Costar f y1 y2 -> Costar f (x1 y1) (x2 y2) Source Comments #
(Comonoid r, SelfAction k, SymMonoidal k) => MonoidalProfunctor (Reader ('OP r) :: k -> k -> Type) Source Comments #	Note: This is only premonoidal, not monoidal, unless the comonoid is cocommutative.
Instance details Defined in Proarrow.Promonad.Reader Methods par0 :: Reader ('OP r) (Unit :: k) (Unit :: k) Source Comments # par :: forall (x1 :: k) (x2 :: k) (y1 :: k) (y2 :: k). Reader ('OP r) x1 x2 -> Reader ('OP r) y1 y2 -> Reader ('OP r) (x1 y1) (x2 y2) Source Comments #
(Monoid w, SelfAction k) => MonoidalProfunctor (Writer w :: k -> k -> Type) Source Comments #	Note: This is only premonoidal, not monoidal, unless the monoid is commutative.
Instance details Defined in Proarrow.Promonad.Writer Methods par0 :: Writer w (Unit :: k) (Unit :: k) Source Comments # par :: forall (x1 :: k) (x2 :: k) (y1 :: k) (y2 :: k). Writer w x1 x2 -> Writer w y1 y2 -> Writer w (x1 y1) (x2 y2) Source Comments #
(MonoidalProfunctor p, MonoidalProfunctor q) => MonoidalProfunctor (p :*: q :: k -> j -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Product Methods par0 :: (p :: q) (Unit :: k) (Unit :: j) Source Comments # par :: forall (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). (p :: q) x1 x2 -> (p :: q) y1 y2 -> (p :: q) (x1 y1) (x2 y2) Source Comments #
(SymMonoidal j, SymMonoidal k, MonoidalProfunctor p, MonoidalProfunctor q) => MonoidalProfunctor (Day p q :: j -> k -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Day Methods par0 :: Day p q (Unit :: j) (Unit :: k) Source Comments # par :: forall (x1 :: j) (x2 :: k) (y1 :: j) (y2 :: k). Day p q x1 x2 -> Day p q y1 y2 -> Day p q (x1 y1) (x2 y2) Source Comments #
(MonoidalProfunctor p, MonoidalProfunctor q) => MonoidalProfunctor (p :.: q :: k -> j2 -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Composition Methods par0 :: (p :.: q) (Unit :: k) (Unit :: j2) Source Comments # par :: forall (x1 :: k) (x2 :: j2) (y1 :: k) (y2 :: j2). (p :.: q) x1 x2 -> (p :.: q) y1 y2 -> (p :.: q) (x1 y1) (x2 y2) Source Comments #
(Alternative f, Monoidal k, Distributive j) => MonoidalProfunctor (CoprodDom (Star f) :: k -> COPROD j -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Star Methods par0 :: CoprodDom (Star f) (Unit :: k) (Unit :: COPROD j) Source Comments # par :: forall (x1 :: k) (x2 :: COPROD j) (y1 :: k) (y2 :: COPROD j). CoprodDom (Star f) x1 x2 -> CoprodDom (Star f) y1 y2 -> CoprodDom (Star f) (x1 y1) (x2 y2) Source Comments #
(TracedMonoidal k, Ob (Unit :: k)) => MonoidalProfunctor (IntConstruction :: INT k -> INT k -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Instance.IntConstruction Methods par0 :: IntConstruction (Unit :: INT k) (Unit :: INT k) Source Comments # par :: forall (x1 :: INT k) (x2 :: INT k) (y1 :: INT k) (y2 :: INT k). IntConstruction x1 x2 -> IntConstruction y1 y2 -> IntConstruction (x1 y1) (x2 y2) Source Comments #
CategoryOf k => MonoidalProfunctor (List :: LIST k -> LIST k -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Instance.List Methods par0 :: List (Unit :: LIST k) (Unit :: LIST k) Source Comments # par :: forall (x1 :: LIST k) (x2 :: LIST k) (y1 :: LIST k) (y2 :: LIST k). List x1 x2 -> List y1 y2 -> List (x1 y1) (x2 y2) Source Comments #
Num a => MonoidalProfunctor (Mat :: MatK a -> MatK a -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Instance.Mat Methods par0 :: Mat (Unit :: MatK a) (Unit :: MatK a) Source Comments # par :: forall (x1 :: MatK a) (x2 :: MatK a) (y1 :: MatK a) (y2 :: MatK a). Mat x1 x2 -> Mat y1 y2 -> Mat (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor (PoAsCat :: POCATK Constraint -> POCATK Constraint -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Instance.PreorderAsCategory Methods par0 :: PoAsCat (Unit :: POCATK Constraint) (Unit :: POCATK Constraint) Source Comments # par :: forall (x1 :: POCATK Constraint) (x2 :: POCATK Constraint) (y1 :: POCATK Constraint) (y2 :: POCATK Constraint). PoAsCat x1 x2 -> PoAsCat y1 y2 -> PoAsCat (x1 y1) (x2 y2) Source Comments #
BiCCC k => MonoidalProfunctor (FreeCCC :: FK k -> FK k -> Type) Source Comments #
Instance details Defined in Proarrow.Helper.CCC Methods par0 :: FreeCCC (Unit :: FK k) (Unit :: FK k) Source Comments # par :: forall (x1 :: FK k) (x2 :: FK k) (y1 :: FK k) (y2 :: FK k). FreeCCC x1 x2 -> FreeCCC y1 y2 -> FreeCCC (x1 y1) (x2 y2) Source Comments #
Monoidal k => MonoidalProfunctor (Strictified :: [k] -> [k] -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Monoidal.Strictified Methods par0 :: Strictified (Unit :: [k]) (Unit :: [k]) Source Comments # par :: forall (x1 :: [k]) (x2 :: [k]) (y1 :: [k]) (y2 :: [k]). Strictified x1 x2 -> Strictified y1 y2 -> Strictified (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor p => MonoidalProfunctor (Rev p :: REV k -> REV j -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Monoidal.Rev Methods par0 :: Rev p (Unit :: REV k) (Unit :: REV j) Source Comments # par :: forall (x1 :: REV k) (x2 :: REV j) (y1 :: REV k) (y2 :: REV j). Rev p x1 x2 -> Rev p y1 y2 -> Rev p (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor p => MonoidalProfunctor (Op p :: OPPOSITE j -> OPPOSITE k -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Opposite Methods par0 :: Op p (Unit :: OPPOSITE j) (Unit :: OPPOSITE k) Source Comments # par :: forall (x1 :: OPPOSITE j) (x2 :: OPPOSITE k) (y1 :: OPPOSITE j) (y2 :: OPPOSITE k). Op p x1 x2 -> Op p y1 y2 -> Op p (x1 y1) (x2 y2) Source Comments #
(Functor f, Distributive j, Distributive k) => MonoidalProfunctor (Coprod (Star f) :: COPROD k -> COPROD j -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Star Methods par0 :: Coprod (Star f) (Unit :: COPROD k) (Unit :: COPROD j) Source Comments # par :: forall (x1 :: COPROD k) (x2 :: COPROD j) (y1 :: COPROD k) (y2 :: COPROD j). Coprod (Star f) x1 x2 -> Coprod (Star f) y1 y2 -> Coprod (Star f) (x1 y1) (x2 y2) Source Comments #
(HasCoproducts j, HasCoproducts k) => MonoidalProfunctor (Coprod (TerminalProfunctor :: k -> j -> Type) :: COPROD k -> COPROD j -> Type) Source Comments #
Instance details Defined in Proarrow.Object.BinaryCoproduct Methods par0 :: Coprod (TerminalProfunctor :: k -> j -> Type) (Unit :: COPROD k) (Unit :: COPROD j) Source Comments # par :: forall (x1 :: COPROD k) (x2 :: COPROD j) (y1 :: COPROD k) (y2 :: COPROD j). Coprod (TerminalProfunctor :: k -> j -> Type) x1 x2 -> Coprod (TerminalProfunctor :: k -> j -> Type) y1 y2 -> Coprod (TerminalProfunctor :: k -> j -> Type) (x1 y1) (x2 y2) Source Comments #
(Profunctor f, Profunctor g, MonoidalProfunctor (Coprod f), MonoidalProfunctor (Coprod g)) => MonoidalProfunctor (Coprod (f :.: g) :: COPROD k -> COPROD j2 -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Composition Methods par0 :: Coprod (f :.: g) (Unit :: COPROD k) (Unit :: COPROD j2) Source Comments # par :: forall (x1 :: COPROD k) (x2 :: COPROD j2) (y1 :: COPROD k) (y2 :: COPROD j2). Coprod (f :.: g) x1 x2 -> Coprod (f :.: g) y1 y2 -> Coprod (f :.: g) (x1 y1) (x2 y2) Source Comments #
HasCoproducts k => MonoidalProfunctor (Coprod (Id :: k -> k -> Type) :: COPROD k -> COPROD k -> Type) Source Comments #
Instance details Defined in Proarrow.Object.BinaryCoproduct Methods par0 :: Coprod (Id :: k -> k -> Type) (Unit :: COPROD k) (Unit :: COPROD k) Source Comments # par :: forall (x1 :: COPROD k) (x2 :: COPROD k) (y1 :: COPROD k) (y2 :: COPROD k). Coprod (Id :: k -> k -> Type) x1 x2 -> Coprod (Id :: k -> k -> Type) y1 y2 -> Coprod (Id :: k -> k -> Type) (x1 y1) (x2 y2) Source Comments #
(HasProducts k, Category cat) => MonoidalProfunctor (Prod cat :: PROD k -> PROD k -> Type) Source Comments #
Instance details Defined in Proarrow.Object.BinaryProduct Methods par0 :: Prod cat (Unit :: PROD k) (Unit :: PROD k) Source Comments # par :: forall (x1 :: PROD k) (x2 :: PROD k) (y1 :: PROD k) (y2 :: PROD k). Prod cat x1 x2 -> Prod cat y1 y2 -> Prod cat (x1 y1) (x2 y2) Source Comments #
(Promonad p, MonoidalProfunctor p) => MonoidalProfunctor (Kleisli :: KLEISLI p -> KLEISLI p -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Instance.Kleisli Methods par0 :: Kleisli (Unit :: KLEISLI p) (Unit :: KLEISLI p) Source Comments # par :: forall (x1 :: KLEISLI p) (x2 :: KLEISLI p) (y1 :: KLEISLI p) (y2 :: KLEISLI p). Kleisli x1 x2 -> Kleisli y1 y2 -> Kleisli (x1 y1) (x2 y2) Source Comments #
(Monoidal j, Monoidal k) => MonoidalProfunctor (Prof :: (j +-> k) -> (j +-> k) -> Type) Source Comments #
Instance details Defined in Proarrow.Profunctor.Day Methods par0 :: Prof (Unit :: j +-> k) (Unit :: j +-> k) Source Comments # par :: forall (x1 :: j +-> k) (x2 :: j +-> k) (y1 :: j +-> k) (y2 :: j +-> k). Prof x1 x2 -> Prof y1 y2 -> Prof (x1 y1) (x2 y2) Source Comments #
MonoidalProfunctor (Nat :: (Type -> Type) -> (Type -> Type) -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Instance.Nat Methods par0 :: Nat (Unit :: Type -> Type) (Unit :: Type -> Type) Source Comments # par :: forall (x1 :: Type -> Type) (x2 :: Type -> Type) (y1 :: Type -> Type) (y2 :: Type -> Type). Nat x1 x2 -> Nat y1 y2 -> Nat (x1 y1) (x2 y2) Source Comments #
(MonoidalProfunctor p, ob (Unit :: k), forall (a :: k) (b :: k). (ob a, ob b) => IsObMult ob a b) => MonoidalProfunctor (Sub p :: SUBCAT ob -> SUBCAT ob -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Instance.Sub Methods par0 :: Sub p (Unit :: SUBCAT ob) (Unit :: SUBCAT ob) Source Comments # par :: forall (x1 :: SUBCAT ob) (x2 :: SUBCAT ob) (y1 :: SUBCAT ob) (y2 :: SUBCAT ob). Sub p x1 x2 -> Sub p y1 y2 -> Sub p (x1 y1) (x2 y2) Source Comments #
(MonoidalProfunctor p, MonoidalProfunctor q) => MonoidalProfunctor (p :**: q :: (k1, k2) -> (j1, j2) -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Instance.Product Methods par0 :: (p :: q) (Unit :: (k1, k2)) (Unit :: (j1, j2)) Source Comments # par :: forall (x1 :: (k1, k2)) (x2 :: (j1, j2)) (y1 :: (k1, k2)) (y2 :: (j1, j2)). (p :: q) x1 x2 -> (p :: q) y1 y2 -> (p :: q) (x1 y1) (x2 y2) Source Comments #
(Bicategory kk, Ob0 kk k, Ob (I :: kk k k)) => MonoidalProfunctor (Endo :: ENDO kk k -> ENDO kk k -> Type) Source Comments #
Instance details Defined in Proarrow.Category.Monoidal.Endo Methods par0 :: Endo (Unit :: ENDO kk k) (Unit :: ENDO kk k) Source Comments # par :: forall (x1 :: ENDO kk k) (x2 :: ENDO kk k) (y1 :: ENDO kk k) (y2 :: ENDO kk k). Endo x1 x2 -> Endo y1 y2 -> Endo (x1 y1) (x2 y2) Source Comments #

class (CategoryOf k, MonoidalProfunctor ((~>) :: CAT k), Ob (Unit :: k)) => Monoidal k where Source Comments #

Associated Types

type Unit :: k Source Comments #

type (a :: k) ** (b :: k) :: k Source Comments #

Methods

withOb2 :: forall (a :: k) (b :: k) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: k). Ob a => ((Unit :: k) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: k). Ob a => a ~> ((Unit :: k) ** a) Source Comments #

rightUnitor :: forall (a :: k). Ob a => (a ** (Unit :: k)) ~> a Source Comments #

rightUnitorInv :: forall (a :: k). Ob a => a ~> (a ** (Unit :: k)) Source Comments #

associator :: forall (a :: k) (b :: k) (c :: k). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: k) (b :: k) (c :: k). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

Instances

Instances details

Monoidal BOOL Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Bool

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.Bool type Unit = TerminalObject :: BOOL
type (a :: BOOL) ** (b :: BOOL)
Instance details Defined in Proarrow.Category.Instance.Bool type (a :: BOOL) ** (b :: BOOL) = a && b

Methods

withOb2 :: forall (a :: BOOL) (b :: BOOL) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: BOOL). Ob a => ((Unit :: BOOL) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: BOOL). Ob a => a ~> ((Unit :: BOOL) ** a) Source Comments #

rightUnitor :: forall (a :: BOOL). Ob a => (a ** (Unit :: BOOL)) ~> a Source Comments #

rightUnitorInv :: forall (a :: BOOL). Ob a => a ~> (a ** (Unit :: BOOL)) Source Comments #

associator :: forall (a :: BOOL) (b :: BOOL) (c :: BOOL). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: BOOL) (b :: BOOL) (c :: BOOL). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

Monoidal KIND Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Cat

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.Cat type Unit = 'K ()
type (l :: KIND) ** (r :: KIND)
Instance details Defined in Proarrow.Category.Instance.Cat type (l :: KIND) ** (r :: KIND) = 'K (UN 'K l, UN 'K r)

Methods

withOb2 :: forall (a :: KIND) (b :: KIND) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: KIND). Ob a => ((Unit :: KIND) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: KIND). Ob a => a ~> ((Unit :: KIND) ** a) Source Comments #

rightUnitor :: forall (a :: KIND). Ob a => (a ** (Unit :: KIND)) ~> a Source Comments #

rightUnitorInv :: forall (a :: KIND). Ob a => a ~> (a ** (Unit :: KIND)) Source Comments #

associator :: forall (a :: KIND) (b :: KIND) (c :: KIND). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: KIND) (b :: KIND) (c :: KIND). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

Monoidal CONSTRAINT Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Constraint

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.Constraint type Unit = TerminalObject :: CONSTRAINT
type (a :: CONSTRAINT) ** (b :: CONSTRAINT)
Instance details Defined in Proarrow.Category.Instance.Constraint type (a :: CONSTRAINT) ** (b :: CONSTRAINT) = a && b

Methods

withOb2 :: forall (a :: CONSTRAINT) (b :: CONSTRAINT) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: CONSTRAINT). Ob a => ((Unit :: CONSTRAINT) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: CONSTRAINT). Ob a => a ~> ((Unit :: CONSTRAINT) ** a) Source Comments #

rightUnitor :: forall (a :: CONSTRAINT). Ob a => (a ** (Unit :: CONSTRAINT)) ~> a Source Comments #

rightUnitorInv :: forall (a :: CONSTRAINT). Ob a => a ~> (a ** (Unit :: CONSTRAINT)) Source Comments #

associator :: forall (a :: CONSTRAINT) (b :: CONSTRAINT) (c :: CONSTRAINT). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: CONSTRAINT) (b :: CONSTRAINT) (c :: CONSTRAINT). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

Monoidal LINEAR Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Linear

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.Linear type Unit = 'L ()
type ('L a :: LINEAR) ** ('L b :: LINEAR)
Instance details Defined in Proarrow.Category.Instance.Linear type ('L a :: LINEAR) ** ('L b :: LINEAR) = 'L (a, b)

Methods

withOb2 :: forall (a :: LINEAR) (b :: LINEAR) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: LINEAR). Ob a => ((Unit :: LINEAR) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: LINEAR). Ob a => a ~> ((Unit :: LINEAR) ** a) Source Comments #

rightUnitor :: forall (a :: LINEAR). Ob a => (a ** (Unit :: LINEAR)) ~> a Source Comments #

rightUnitorInv :: forall (a :: LINEAR). Ob a => a ~> (a ** (Unit :: LINEAR)) Source Comments #

associator :: forall (a :: LINEAR) (b :: LINEAR) (c :: LINEAR). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: LINEAR) (b :: LINEAR) (c :: LINEAR). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

Monoidal POINTED Source Comments #

The smash product of pointed sets. Monoids relative to the smash product are absorption monoids.

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.PointedHask type Unit = 'P ()
type ('P a :: POINTED) ** ('P b :: POINTED)
Instance details Defined in Proarrow.Category.Instance.PointedHask type ('P a :: POINTED) ** ('P b :: POINTED) = 'P (a, b)

Methods

withOb2 :: forall (a :: POINTED) (b :: POINTED) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: POINTED). Ob a => ((Unit :: POINTED) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: POINTED). Ob a => a ~> ((Unit :: POINTED) ** a) Source Comments #

rightUnitor :: forall (a :: POINTED). Ob a => (a ** (Unit :: POINTED)) ~> a Source Comments #

rightUnitorInv :: forall (a :: POINTED). Ob a => a ~> (a ** (Unit :: POINTED)) Source Comments #

associator :: forall (a :: POINTED) (b :: POINTED) (c :: POINTED). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: POINTED) (b :: POINTED) (c :: POINTED). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

Monoidal Nat Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Simplex

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.Simplex type Unit = 'Z
type (a :: Nat) ** (b :: Nat)
Instance details Defined in Proarrow.Category.Instance.Simplex type (a :: Nat) ** (b :: Nat) = a + b

Methods

withOb2 :: forall (a :: Nat) (b :: Nat) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: Nat). Ob a => ((Unit :: Nat) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: Nat). Ob a => a ~> ((Unit :: Nat) ** a) Source Comments #

rightUnitor :: forall (a :: Nat). Ob a => (a ** (Unit :: Nat)) ~> a Source Comments #

rightUnitorInv :: forall (a :: Nat). Ob a => a ~> (a ** (Unit :: Nat)) Source Comments #

associator :: forall (a :: Nat) (b :: Nat) (c :: Nat). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: Nat) (b :: Nat) (c :: Nat). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

Monoidal () Source Comments #

Instance details

Defined in Proarrow.Object.BinaryProduct

Associated Types

type Unit
Instance details Defined in Proarrow.Object.BinaryProduct type Unit = TerminalObject :: ()
type (a :: ()) ** (b :: ())
Instance details Defined in Proarrow.Object.BinaryProduct type (a :: ()) ** (b :: ()) = a && b

Methods

withOb2 :: forall (a :: ()) (b :: ()) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: ()). Ob a => ((Unit :: ()) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: ()). Ob a => a ~> ((Unit :: ()) ** a) Source Comments #

rightUnitor :: forall (a :: ()). Ob a => (a ** (Unit :: ())) ~> a Source Comments #

rightUnitorInv :: forall (a :: ()). Ob a => a ~> (a ** (Unit :: ())) Source Comments #

associator :: forall (a :: ()) (b :: ()) (c :: ()). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: ()) (b :: ()) (c :: ()). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

Monoidal Type Source Comments #

Instance details

Defined in Proarrow.Object.BinaryProduct

Associated Types

type Unit
Instance details Defined in Proarrow.Object.BinaryProduct type Unit = TerminalObject :: Type
type (a :: Type) ** (b :: Type)
Instance details Defined in Proarrow.Object.BinaryProduct type (a :: Type) ** (b :: Type) = a && b

Methods

withOb2 :: (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: Ob a => ((Unit :: Type) ** a) ~> a Source Comments #

leftUnitorInv :: Ob a => a ~> ((Unit :: Type) ** a) Source Comments #

rightUnitor :: Ob a => (a ** (Unit :: Type)) ~> a Source Comments #

rightUnitorInv :: Ob a => a ~> (a ** (Unit :: Type)) Source Comments #

associator :: (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

(TracedMonoidal k, Ob (Unit :: k)) => Monoidal (INT k) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.IntConstruction

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.IntConstruction type Unit = 'I (Unit :: k) (Unit :: k)

Methods

withOb2 :: forall (a :: INT k) (b :: INT k) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: INT k). Ob a => ((Unit :: INT k) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: INT k). Ob a => a ~> ((Unit :: INT k) ** a) Source Comments #

rightUnitor :: forall (a :: INT k). Ob a => (a ** (Unit :: INT k)) ~> a Source Comments #

rightUnitorInv :: forall (a :: INT k). Ob a => a ~> (a ** (Unit :: INT k)) Source Comments #

associator :: forall (a :: INT k) (b :: INT k) (c :: INT k). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: INT k) (b :: INT k) (c :: INT k). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

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

Instance details

Defined in Proarrow.Category.Instance.List

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.List type Unit = 'L ('[] :: [k])

Methods

withOb2 :: forall (a :: LIST k) (b :: LIST k) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: LIST k). Ob a => ((Unit :: LIST k) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: LIST k). Ob a => a ~> ((Unit :: LIST k) ** a) Source Comments #

rightUnitor :: forall (a :: LIST k). Ob a => (a ** (Unit :: LIST k)) ~> a Source Comments #

rightUnitorInv :: forall (a :: LIST k). Ob a => a ~> (a ** (Unit :: LIST k)) Source Comments #

associator :: forall (a :: LIST k) (b :: LIST k) (c :: LIST k). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: LIST k) (b :: LIST k) (c :: LIST k). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

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

Instance details

Defined in Proarrow.Category.Instance.Mat

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.Mat type Unit = 'M ('S 'Z) :: MatK a

Methods

withOb2 :: forall (a0 :: MatK a) (b :: MatK a) r. (Ob a0, Ob b) => (Ob (a0 ** b) => r) -> r Source Comments #

leftUnitor :: forall (a0 :: MatK a). Ob a0 => ((Unit :: MatK a) ** a0) ~> a0 Source Comments #

leftUnitorInv :: forall (a0 :: MatK a). Ob a0 => a0 ~> ((Unit :: MatK a) ** a0) Source Comments #

rightUnitor :: forall (a0 :: MatK a). Ob a0 => (a0 ** (Unit :: MatK a)) ~> a0 Source Comments #

rightUnitorInv :: forall (a0 :: MatK a). Ob a0 => a0 ~> (a0 ** (Unit :: MatK a)) Source Comments #

associator :: forall (a0 :: MatK a) (b :: MatK a) (c :: MatK a). (Ob a0, Ob b, Ob c) => ((a0 ** b) ** c) ~> (a0 ** (b ** c)) Source Comments #

associatorInv :: forall (a0 :: MatK a) (b :: MatK a) (c :: MatK a). (Ob a0, Ob b, Ob c) => (a0 ** (b ** c)) ~> ((a0 ** b) ** c) Source Comments #

Monoidal (POCATK Constraint) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PreorderAsCategory

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.PreorderAsCategory type Unit = TerminalObject :: POCATK Constraint
type (a :: POCATK Constraint) ** (b :: POCATK Constraint)
Instance details Defined in Proarrow.Category.Instance.PreorderAsCategory type (a :: POCATK Constraint) ** (b :: POCATK Constraint) = a && b

Methods

withOb2 :: forall (a :: POCATK Constraint) (b :: POCATK Constraint) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: POCATK Constraint). Ob a => ((Unit :: POCATK Constraint) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: POCATK Constraint). Ob a => a ~> ((Unit :: POCATK Constraint) ** a) Source Comments #

rightUnitor :: forall (a :: POCATK Constraint). Ob a => (a ** (Unit :: POCATK Constraint)) ~> a Source Comments #

rightUnitorInv :: forall (a :: POCATK Constraint). Ob a => a ~> (a ** (Unit :: POCATK Constraint)) Source Comments #

associator :: forall (a :: POCATK Constraint) (b :: POCATK Constraint) (c :: POCATK Constraint). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: POCATK Constraint) (b :: POCATK Constraint) (c :: POCATK Constraint). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

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

Instance details

Defined in Proarrow.Category.Monoidal.Rev

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Monoidal.Rev type Unit = 'R (Unit :: k)

Methods

withOb2 :: forall (a :: REV k) (b :: REV k) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: REV k). Ob a => ((Unit :: REV k) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: REV k). Ob a => a ~> ((Unit :: REV k) ** a) Source Comments #

rightUnitor :: forall (a :: REV k). Ob a => (a ** (Unit :: REV k)) ~> a Source Comments #

rightUnitorInv :: forall (a :: REV k). Ob a => a ~> (a ** (Unit :: REV k)) Source Comments #

associator :: forall (a :: REV k) (b :: REV k) (c :: REV k). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: REV k) (b :: REV k) (c :: REV k). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

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

Instance details

Defined in Proarrow.Category.Opposite

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Opposite type Unit = 'OP (Unit :: k)

Methods

withOb2 :: forall (a :: OPPOSITE k) (b :: OPPOSITE k) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: OPPOSITE k). Ob a => ((Unit :: OPPOSITE k) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: OPPOSITE k). Ob a => a ~> ((Unit :: OPPOSITE k) ** a) Source Comments #

rightUnitor :: forall (a :: OPPOSITE k). Ob a => (a ** (Unit :: OPPOSITE k)) ~> a Source Comments #

rightUnitorInv :: forall (a :: OPPOSITE k). Ob a => a ~> (a ** (Unit :: OPPOSITE k)) Source Comments #

associator :: forall (a :: OPPOSITE k) (b :: OPPOSITE k) (c :: OPPOSITE k). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: OPPOSITE k) (b :: OPPOSITE k) (c :: OPPOSITE k). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

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

Instance details

Defined in Proarrow.Helper.CCC

Associated Types

type Unit
Instance details Defined in Proarrow.Helper.CCC type Unit = TerminalObject :: FK k

Methods

withOb2 :: forall (a :: FK k) (b :: FK k) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: FK k). Ob a => ((Unit :: FK k) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: FK k). Ob a => a ~> ((Unit :: FK k) ** a) Source Comments #

rightUnitor :: forall (a :: FK k). Ob a => (a ** (Unit :: FK k)) ~> a Source Comments #

rightUnitorInv :: forall (a :: FK k). Ob a => a ~> (a ** (Unit :: FK k)) Source Comments #

associator :: forall (a :: FK k) (b :: FK k) (c :: FK k). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: FK k) (b :: FK k) (c :: FK k). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

HasCoproducts k => Monoidal (COPROD k) Source Comments #

Instance details

Defined in Proarrow.Object.BinaryCoproduct

Associated Types

type Unit
Instance details Defined in Proarrow.Object.BinaryCoproduct type Unit = 'COPR (InitialObject :: k)

Methods

withOb2 :: forall (a :: COPROD k) (b :: COPROD k) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: COPROD k). Ob a => ((Unit :: COPROD k) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: COPROD k). Ob a => a ~> ((Unit :: COPROD k) ** a) Source Comments #

rightUnitor :: forall (a :: COPROD k). Ob a => (a ** (Unit :: COPROD k)) ~> a Source Comments #

rightUnitorInv :: forall (a :: COPROD k). Ob a => a ~> (a ** (Unit :: COPROD k)) Source Comments #

associator :: forall (a :: COPROD k) (b :: COPROD k) (c :: COPROD k). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: COPROD k) (b :: COPROD k) (c :: COPROD k). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

HasProducts k => Monoidal (PROD k) Source Comments #

Instance details

Defined in Proarrow.Object.BinaryProduct

Associated Types

type Unit
Instance details Defined in Proarrow.Object.BinaryProduct type Unit = TerminalObject :: PROD k

Methods

withOb2 :: forall (a :: PROD k) (b :: PROD k) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: PROD k). Ob a => ((Unit :: PROD k) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: PROD k). Ob a => a ~> ((Unit :: PROD k) ** a) Source Comments #

rightUnitor :: forall (a :: PROD k). Ob a => (a ** (Unit :: PROD k)) ~> a Source Comments #

rightUnitorInv :: forall (a :: PROD k). Ob a => a ~> (a ** (Unit :: PROD k)) Source Comments #

associator :: forall (a :: PROD k) (b :: PROD k) (c :: PROD k). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: PROD k) (b :: PROD k) (c :: PROD k). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

Monoidal k => Monoidal [k] Source Comments #

Instance details

Defined in Proarrow.Category.Monoidal.Strictified

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Monoidal.Strictified type Unit = '[] :: [k]

Methods

withOb2 :: forall (a :: [k]) (b :: [k]) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: [k]). Ob a => ((Unit :: [k]) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: [k]). Ob a => a ~> ((Unit :: [k]) ** a) Source Comments #

rightUnitor :: forall (a :: [k]). Ob a => (a ** (Unit :: [k])) ~> a Source Comments #

rightUnitorInv :: forall (a :: [k]). Ob a => a ~> (a ** (Unit :: [k])) Source Comments #

associator :: forall (a :: [k]) (b :: [k]) (c :: [k]). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: [k]) (b :: [k]) (c :: [k]). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

(Promonad p, MonoidalProfunctor p) => Monoidal (KLEISLI p) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Kleisli

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.Kleisli type Unit = 'KL (Unit :: k) :: KLEISLI p

Methods

withOb2 :: forall (a :: KLEISLI p) (b :: KLEISLI p) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: KLEISLI p). Ob a => ((Unit :: KLEISLI p) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: KLEISLI p). Ob a => a ~> ((Unit :: KLEISLI p) ** a) Source Comments #

rightUnitor :: forall (a :: KLEISLI p). Ob a => (a ** (Unit :: KLEISLI p)) ~> a Source Comments #

rightUnitorInv :: forall (a :: KLEISLI p). Ob a => a ~> (a ** (Unit :: KLEISLI p)) Source Comments #

associator :: forall (a :: KLEISLI p) (b :: KLEISLI p) (c :: KLEISLI p). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: KLEISLI p) (b :: KLEISLI p) (c :: KLEISLI p). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

(Monoidal k, ob (Unit :: k), forall (a :: k) (b :: k). (ob a, ob b) => IsObMult ob a b) => Monoidal (SUBCAT ob) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Sub

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.Sub type Unit = 'SUB (Unit :: k) :: SUBCAT ob

Methods

withOb2 :: forall (a :: SUBCAT ob) (b :: SUBCAT ob) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: SUBCAT ob). Ob a => ((Unit :: SUBCAT ob) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: SUBCAT ob). Ob a => a ~> ((Unit :: SUBCAT ob) ** a) Source Comments #

rightUnitor :: forall (a :: SUBCAT ob). Ob a => (a ** (Unit :: SUBCAT ob)) ~> a Source Comments #

rightUnitorInv :: forall (a :: SUBCAT ob). Ob a => a ~> (a ** (Unit :: SUBCAT ob)) Source Comments #

associator :: forall (a :: SUBCAT ob) (b :: SUBCAT ob) (c :: SUBCAT ob). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: SUBCAT ob) (b :: SUBCAT ob) (c :: SUBCAT ob). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

(Monoidal j, Monoidal k) => Monoidal (PRO j k) Source Comments #

Instance details

Defined in Proarrow.Profunctor.Day

Associated Types

type Unit
Instance details Defined in Proarrow.Profunctor.Day type Unit = DayUnit :: j -> k -> Type

Methods

withOb2 :: forall (a :: PRO j k) (b :: PRO j k) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: PRO j k). Ob a => ((Unit :: PRO j k) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: PRO j k). Ob a => a ~> ((Unit :: PRO j k) ** a) Source Comments #

rightUnitor :: forall (a :: PRO j k). Ob a => (a ** (Unit :: PRO j k)) ~> a Source Comments #

rightUnitorInv :: forall (a :: PRO j k). Ob a => a ~> (a ** (Unit :: PRO j k)) Source Comments #

associator :: forall (a :: PRO j k) (b :: PRO j k) (c :: PRO j k). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: PRO j k) (b :: PRO j k) (c :: PRO j k). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

(Monoidal j, Monoidal k) => Monoidal (j, k) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Product

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.Product type Unit = '(Unit :: j, Unit :: k)

Methods

withOb2 :: forall (a :: (j, k)) (b :: (j, k)) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: (j, k)). Ob a => ((Unit :: (j, k)) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: (j, k)). Ob a => a ~> ((Unit :: (j, k)) ** a) Source Comments #

rightUnitor :: forall (a :: (j, k)). Ob a => (a ** (Unit :: (j, k))) ~> a Source Comments #

rightUnitorInv :: forall (a :: (j, k)). Ob a => a ~> (a ** (Unit :: (j, k))) Source Comments #

associator :: forall (a :: (j, k)) (b :: (j, k)) (c :: (j, k)). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: (j, k)) (b :: (j, k)) (c :: (j, k)). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

Monoidal (Type -> Type) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Nat

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.Nat type Unit = Identity
type (f :: Type -> Type) ** (g :: Type -> Type)
Instance details Defined in Proarrow.Category.Instance.Nat type (f :: Type -> Type) ** (g :: Type -> Type) = Compose f g

Methods

withOb2 :: forall (a :: Type -> Type) (b :: Type -> Type) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: Type -> Type). Ob a => ((Unit :: Type -> Type) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: Type -> Type). Ob a => a ~> ((Unit :: Type -> Type) ** a) Source Comments #

rightUnitor :: forall (a :: Type -> Type). Ob a => (a ** (Unit :: Type -> Type)) ~> a Source Comments #

rightUnitorInv :: forall (a :: Type -> Type). Ob a => a ~> (a ** (Unit :: Type -> Type)) Source Comments #

associator :: forall (a :: Type -> Type) (b :: Type -> Type) (c :: Type -> Type). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: Type -> Type) (b :: Type -> Type) (c :: Type -> Type). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

(Bicategory kk, Ob0 kk k, Ob (I :: kk k k)) => Monoidal (ENDO kk k) Source Comments #

The monoidal subcategory of a bicategory for a single object.

Instance details

Defined in Proarrow.Category.Monoidal.Endo

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Monoidal.Endo type Unit = 'E (I :: kk k k)

Methods

withOb2 :: forall (a :: ENDO kk k) (b :: ENDO kk k) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r Source Comments #

leftUnitor :: forall (a :: ENDO kk k). Ob a => ((Unit :: ENDO kk k) ** a) ~> a Source Comments #

leftUnitorInv :: forall (a :: ENDO kk k). Ob a => a ~> ((Unit :: ENDO kk k) ** a) Source Comments #

rightUnitor :: forall (a :: ENDO kk k). Ob a => (a ** (Unit :: ENDO kk k)) ~> a Source Comments #

rightUnitorInv :: forall (a :: ENDO kk k). Ob a => a ~> (a ** (Unit :: ENDO kk k)) Source Comments #

associator :: forall (a :: ENDO kk k) (b :: ENDO kk k) (c :: ENDO kk k). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv :: forall (a :: ENDO kk k) (b :: ENDO kk k) (c :: ENDO kk k). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

obj2 :: forall {k} (a :: k) (b :: k). (Monoidal k, Ob a, Ob b) => Obj (a ** b) Source Comments #

leftUnitor' :: forall k (a :: k) (b :: k). Monoidal k => (a ~> b) -> ((Unit :: k) ** a) ~> b Source Comments #

leftUnitorInv' :: forall k (a :: k) (b :: k). Monoidal k => (a ~> b) -> a ~> ((Unit :: k) ** b) Source Comments #

rightUnitor' :: forall k (a :: k) (b :: k). Monoidal k => (a ~> b) -> (a ** (Unit :: k)) ~> b Source Comments #

rightUnitorInv' :: forall k (a :: k) (b :: k). Monoidal k => (a ~> b) -> a ~> (b ** (Unit :: k)) Source Comments #

associator' :: forall {k} (a :: k) (b :: k) (c :: k). Monoidal k => Obj a -> Obj b -> Obj c -> ((a ** b) ** c) ~> (a ** (b ** c)) Source Comments #

associatorInv' :: forall {k} (a :: k) (b :: k) (c :: k). Monoidal k => Obj a -> Obj b -> Obj c -> (a ** (b ** c)) ~> ((a ** b) ** c) Source Comments #

leftUnitorWith :: forall {k} (a :: k) (b :: k). (Monoidal k, Ob a) => (b ~> (Unit :: k)) -> (b ** a) ~> a Source Comments #

leftUnitorInvWith :: forall {k} (a :: k) (b :: k). (Monoidal k, Ob a) => ((Unit :: k) ~> b) -> a ~> (b ** a) Source Comments #

rightUnitorWith :: forall {k} (a :: k) (b :: k). (Monoidal k, Ob a) => (b ~> (Unit :: k)) -> (a ** b) ~> a Source Comments #

rightUnitorInvWith :: forall {k} (a :: k) (b :: k). (Monoidal k, Ob a) => ((Unit :: k) ~> b) -> a ~> (a ** b) Source Comments #

unitObj :: Monoidal k => Obj (Unit :: k) Source Comments #

class Monoidal k => SymMonoidal k where Source Comments #

Methods

swap :: forall (a :: k) (b :: k). (Ob a, Ob b) => (a ** b) ~> (b ** a) Source Comments #

Instances

Instances details

SymMonoidal BOOL Source Comments #
Instance details Defined in Proarrow.Category.Instance.Bool Methods swap :: forall (a :: BOOL) (b :: BOOL). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
SymMonoidal KIND Source Comments #
Instance details Defined in Proarrow.Category.Instance.Cat Methods swap :: forall (a :: KIND) (b :: KIND). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
SymMonoidal CONSTRAINT Source Comments #
Instance details Defined in Proarrow.Category.Instance.Constraint Methods swap :: forall (a :: CONSTRAINT) (b :: CONSTRAINT). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
SymMonoidal LINEAR Source Comments #
Instance details Defined in Proarrow.Category.Instance.Linear Methods swap :: forall (a :: LINEAR) (b :: LINEAR). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
SymMonoidal () Source Comments #
Instance details Defined in Proarrow.Object.BinaryProduct Methods swap :: forall (a :: ()) (b :: ()). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
SymMonoidal Type Source Comments #
Instance details Defined in Proarrow.Object.BinaryProduct Methods swap :: (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
(TracedMonoidal k, Ob (Unit :: k)) => SymMonoidal (INT k) Source Comments #
Instance details Defined in Proarrow.Category.Instance.IntConstruction Methods swap :: forall (a :: INT k) (b :: INT k). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
Num a => SymMonoidal (MatK a) Source Comments #
Instance details Defined in Proarrow.Category.Instance.Mat Methods swap :: forall (a0 :: MatK a) (b :: MatK a). (Ob a0, Ob b) => (a0 b) ~> (b a0) Source Comments #
SymMonoidal k => SymMonoidal (REV k) Source Comments #
Instance details Defined in Proarrow.Category.Monoidal.Rev Methods swap :: forall (a :: REV k) (b :: REV k). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
SymMonoidal k => SymMonoidal (OPPOSITE k) Source Comments #
Instance details Defined in Proarrow.Category.Opposite Methods swap :: forall (a :: OPPOSITE k) (b :: OPPOSITE k). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
BiCCC k => SymMonoidal (FK k) Source Comments #
Instance details Defined in Proarrow.Helper.CCC Methods swap :: forall (a :: FK k) (b :: FK k). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
HasCoproducts k => SymMonoidal (COPROD k) Source Comments #
Instance details Defined in Proarrow.Object.BinaryCoproduct Methods swap :: forall (a :: COPROD k) (b :: COPROD k). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
HasProducts k => SymMonoidal (PROD k) Source Comments #
Instance details Defined in Proarrow.Object.BinaryProduct Methods swap :: forall (a :: PROD k) (b :: PROD k). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
SymMonoidal k => SymMonoidal [k] Source Comments #
Instance details Defined in Proarrow.Category.Monoidal.Strictified Methods swap :: forall (a :: [k]) (b :: [k]). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
(Promonad p, MonoidalProfunctor p, SymMonoidal k) => SymMonoidal (KLEISLI p) Source Comments #
Instance details Defined in Proarrow.Category.Instance.Kleisli Methods swap :: forall (a :: KLEISLI p) (b :: KLEISLI p). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
(SymMonoidal k, ob (Unit :: k), forall (a :: k) (b :: k). (ob a, ob b) => IsObMult ob a b) => SymMonoidal (SUBCAT ob) Source Comments #
Instance details Defined in Proarrow.Category.Instance.Sub Methods swap :: forall (a :: SUBCAT ob) (b :: SUBCAT ob). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
(SymMonoidal j, SymMonoidal k) => SymMonoidal (PRO j k) Source Comments #
Instance details Defined in Proarrow.Profunctor.Day Methods swap :: forall (a :: PRO j k) (b :: PRO j k). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #
(SymMonoidal j, SymMonoidal k) => SymMonoidal (j, k) Source Comments #
Instance details Defined in Proarrow.Category.Instance.Product Methods swap :: forall (a :: (j, k)) (b :: (j, k)). (Ob a, Ob b) => (a b) ~> (b a) Source Comments #

swap' :: forall {k} (a :: k) (a' :: k) (b :: k) (b' :: k). SymMonoidal k => (a ~> a') -> (b ~> b') -> (a ** b) ~> (b' ** a') Source Comments #

first :: forall {k} (c :: k) (a :: k) (b :: k). (Monoidal k, Ob c) => (a ~> b) -> (a ** c) ~> (b ** c) Source Comments #

second :: forall {k} (c :: k) (a :: k) (b :: k). (Monoidal k, Ob c) => (a ~> b) -> (c ** a) ~> (c ** b) Source Comments #

swapInner' :: forall k (a :: k) (a' :: k) (b :: k) (b' :: k) (c :: k) (c' :: k) (d :: k) (d' :: k). SymMonoidal k => (a ~> a') -> (b ~> b') -> (c ~> c') -> (d ~> d') -> ((a ** b) ** (c ** d)) ~> ((a' ** c') ** (b' ** d')) Source Comments #

swapInner :: forall {k} (a :: k) (b :: k) (c :: k) (d :: k). (SymMonoidal k, Ob a, Ob b, Ob c, Ob d) => ((a ** b) ** (c ** d)) ~> ((a ** c) ** (b ** d)) Source Comments #

swapFst :: forall {k} (a :: k) (b :: k) (c :: k) (d :: k). (SymMonoidal k, Ob a, Ob b, Ob c, Ob d) => ((a ** b) ** (c ** d)) ~> ((c ** b) ** (a ** d)) Source Comments #

swapSnd :: forall {k} (a :: k) (b :: k) (c :: k) (d :: k). (SymMonoidal k, Ob a, Ob b, Ob c, Ob d) => ((a ** b) ** (c ** d)) ~> ((a ** d) ** (c ** b)) Source Comments #

swapOuter :: forall {k} (a :: k) (b :: k) (c :: k) (d :: k). (SymMonoidal k, Ob a, Ob b, Ob c, Ob d) => ((a ** b) ** (c ** d)) ~> ((d ** b) ** (c ** a)) Source Comments #