Safe Haskell	None
Language	GHC2024

Proarrow.Category.Instance.Cospan

Description

The category of cospans in k: objects are those of k (wrapped in CS), and a morphism a ~> b is a cospan a -> x <- b, composed by pushout. With the coproduct of k as tensor every object is a Frobenius monoid, giving a Hypergraph, compact closed, dagger category. This is the archetypal setting for undirected wiring diagrams.

Documentation

data COSPAN k Source Github #

Constructors

CS k

Instances

Instances details

(HasPushouts k, HasCoproducts k) => Monoidal (COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Associated Types

type Unit
Instance details Defined in Proarrow.Category.Instance.Cospan type Unit = 'CS (InitialObject :: k)

Methods

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

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

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

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

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

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

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

(HasPushouts k, HasCoproducts k) => SymMonoidal (COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

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

(HasPushouts k, HasCoproducts k) => Closed (COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

withObExp :: forall (a :: COSPAN k) (b :: COSPAN k) r. (Ob a, Ob b) => (Ob (a ~~> b) => r) -> r Source Github #

curry :: forall (a :: COSPAN k) (b :: COSPAN k) (c :: COSPAN k). (Ob a, Ob b) => ((a ** b) ~> c) -> a ~> (b ~~> c) Source Github #

apply :: forall (a :: COSPAN k) (b :: COSPAN k). (Ob a, Ob b) => ((a ~~> b) ** a) ~> b Source Github #

(^^^) :: forall (a :: COSPAN k) (b :: COSPAN k) (x :: COSPAN k) (y :: COSPAN k). (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) Source Github #

(HasPushouts k, HasCoproducts k) => CompactClosed (COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

distribDual :: forall (a :: COSPAN k) (b :: COSPAN k). (Ob a, Ob b) => Dual (a ** b) ~> (Dual a ** Dual b) Source Github #

dualUnit :: Dual (Unit :: COSPAN k) ~> (Unit :: COSPAN k) Source Github #

dualityUnit :: forall (a :: COSPAN k). Ob a => (Unit :: COSPAN k) ~> (a ** Dual a) Source Github #

dualityCounit :: forall (a :: COSPAN k). Ob a => (Dual a ** a) ~> (Unit :: COSPAN k) Source Github #

(HasPushouts k, HasCoproducts k) => CopyDiscard (COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

copy :: forall (a :: COSPAN k). Ob a => a ~> (a ** a) Source Github #

discard :: forall (a :: COSPAN k). Ob a => a ~> (Unit :: COSPAN k) Source Github #

(HasPushouts k, HasCoproducts k) => Hypergraph (COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

(HasPushouts k, HasCoproducts k) => StarAutonomous (COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

withObDual :: forall (a :: COSPAN k) r. Ob a => (Ob (Dual a) => r) -> r Source Github #

dual :: forall (a :: COSPAN k) (b :: COSPAN k). (a ~> b) -> Dual b ~> Dual a Source Github #

dualInv :: forall (a :: COSPAN k) (b :: COSPAN k). (Ob a, Ob b) => (Dual a ~> Dual b) -> b ~> a Source Github #

linDist :: forall (a :: COSPAN k) (b :: COSPAN k) (c :: COSPAN k). (Ob a, Ob b, Ob c) => ((a ** b) ~> Dual c) -> a ~> Dual (b ** c) Source Github #

linDistInv :: forall (a :: COSPAN k) (b :: COSPAN k) (c :: COSPAN k). (Ob a, Ob b, Ob c) => (a ~> Dual (b ** c)) -> (a ** b) ~> Dual c Source Github #

doubleNeg :: forall (a :: COSPAN k). Ob a => Dual (Dual a) ~> a Source Github #

doubleNegInv :: forall (a :: COSPAN k). Ob a => a ~> Dual (Dual a) Source Github #

HasPushouts k => CategoryOf (COSPAN k) Source Github #

The category of cospans in k: an arrow CS a ~> CS b is a pair of arrows a ~> x and b ~> x into a common object, and composition glues along a pushout.

Instance details

Defined in Proarrow.Category.Instance.Cospan

Associated Types

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

HasPushouts k => DaggerProfunctor (Cospan :: COSPAN k -> COSPAN k -> Type) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

dagger :: forall (a :: COSPAN k) (b :: COSPAN k). Cospan a b -> Cospan b a Source Github #

HasPushouts k => Promonad (Cospan :: COSPAN k -> COSPAN k -> Type) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

id :: forall (a :: COSPAN k). Ob a => Cospan a a Source Github #

(.) :: forall (b :: COSPAN k) (c :: COSPAN k) (a :: COSPAN k). Cospan b c -> Cospan a b -> Cospan a c Source Github #

(HasPushouts k, HasCoproducts k) => MonoidalProfunctor (Cospan :: COSPAN k -> COSPAN k -> Type) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

one :: Cospan (Unit :: COSPAN k) (Unit :: COSPAN k) Source Github #

(**) :: forall (x1 :: COSPAN k) (x2 :: COSPAN k) (y1 :: COSPAN k) (y2 :: COSPAN k). Cospan x1 x2 -> Cospan y1 y2 -> Cospan (x1 ** y1) (x2 ** y2) Source Github #

HasPushouts k => Profunctor (Cospan :: COSPAN k -> COSPAN k -> Type) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

dimap :: forall (c :: COSPAN k) (a :: COSPAN k) (b :: COSPAN k) (d :: COSPAN k). (c ~> a) -> (b ~> d) -> Cospan a b -> Cospan c d Source Github #

lmap :: forall (c :: COSPAN k) (a :: COSPAN k) (b :: COSPAN k). (c ~> a) -> Cospan a b -> Cospan c b Source Github #

rmap :: forall (b :: COSPAN k) (d :: COSPAN k) (a :: COSPAN k). (b ~> d) -> Cospan a b -> Cospan a d Source Github #

(\\) :: forall (a :: COSPAN k) (b :: COSPAN k) r. ((Ob a, Ob b) => r) -> Cospan a b -> r Source Github #

(HasPushouts k, HasPullbacks k) => FunctorForRep (Pullback :: COSPAN k +-> SPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

fmap :: forall (a :: COSPAN k) (b :: COSPAN k). (a ~> b) -> ((Pullback :: COSPAN k +-> SPAN k) @ a) ~> ((Pullback :: COSPAN k +-> SPAN k) @ b) Source Github #

(HasPushouts k, HasPullbacks k) => FunctorForRep (Pushout :: SPAN k +-> COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

fmap :: forall (a :: SPAN k) (b :: SPAN k). (a ~> b) -> ((Pushout :: SPAN k +-> COSPAN k) @ a) ~> ((Pushout :: SPAN k +-> COSPAN k) @ b) Source Github #

(HasPushouts k, HasCoproducts k, Ob a) => Frobenius ('CS a :: COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

(HasPushouts k, HasCoproducts k, Ob a) => CocommutativeComonoid ('CS a :: COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

(HasPushouts k, HasCoproducts k, Ob a) => CommutativeMonoid ('CS a :: COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

(HasPushouts k, HasCoproducts k, Ob a) => Comonoid ('CS a :: COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

counit :: 'CS a ~> (Unit :: COSPAN k) Source Github #

comult :: 'CS a ~> ('CS a ** 'CS a) Source Github #

(HasPushouts k, HasCoproducts k, Ob a) => Monoid ('CS a :: COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

Methods

mempty :: (Unit :: COSPAN k) ~> 'CS a Source Github #

mappend :: ('CS a ** 'CS a) ~> 'CS a Source Github #

type Unit Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

type Unit = 'CS (InitialObject :: k)

type (~>) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

type (~>) = Cospan :: COSPAN k -> COSPAN k -> Type

type Dual (a :: COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

type Dual (a :: COSPAN k) = a

type Ob (a :: COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

type Ob (a :: COSPAN k) = WrappedOb ('CS :: k -> COSPAN k) a

type (a :: COSPAN k) ~~> (b :: COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

type (a :: COSPAN k) ~~> (b :: COSPAN k) = ExpHG a b

type (Pullback :: COSPAN k +-> SPAN k) @ ('CS a :: COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

type (Pullback :: COSPAN k +-> SPAN k) @ ('CS a :: COSPAN k) = 'SP a

type (Pushout :: SPAN k +-> COSPAN k) @ ('SP a :: SPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

type (Pushout :: SPAN k +-> COSPAN k) @ ('SP a :: SPAN k) = 'CS a

type ('CS a :: COSPAN k) ** ('CS b :: COSPAN k) Source Github #

Instance details

Defined in Proarrow.Category.Instance.Cospan

type ('CS a :: COSPAN k) ** ('CS b :: COSPAN k) = 'CS (a || b)

data Cospan (a :: COSPAN k) (b :: COSPAN k) where Source Github #

Constructors

Cospan :: forall {k} (c :: k) (a1 :: k) (b1 :: k). (a1 ~> c) -> (b1 ~> c) -> Cospan ('CS a1) ('CS b1)

Instances

Instances details

HasPushouts k => DaggerProfunctor (Cospan :: COSPAN k -> COSPAN k -> Type) Source Github #
Instance details Defined in Proarrow.Category.Instance.Cospan Methods dagger :: forall (a :: COSPAN k) (b :: COSPAN k). Cospan a b -> Cospan b a Source Github #
HasPushouts k => Promonad (Cospan :: COSPAN k -> COSPAN k -> Type) Source Github #
Instance details Defined in Proarrow.Category.Instance.Cospan Methods id :: forall (a :: COSPAN k). Ob a => Cospan a a Source Github # (.) :: forall (b :: COSPAN k) (c :: COSPAN k) (a :: COSPAN k). Cospan b c -> Cospan a b -> Cospan a c Source Github #
(HasPushouts k, HasCoproducts k) => MonoidalProfunctor (Cospan :: COSPAN k -> COSPAN k -> Type) Source Github #
Instance details Defined in Proarrow.Category.Instance.Cospan Methods one :: Cospan (Unit :: COSPAN k) (Unit :: COSPAN k) Source Github # () :: forall (x1 :: COSPAN k) (x2 :: COSPAN k) (y1 :: COSPAN k) (y2 :: COSPAN k). Cospan x1 x2 -> Cospan y1 y2 -> Cospan (x1 y1) (x2 ** y2) Source Github #
HasPushouts k => Profunctor (Cospan :: COSPAN k -> COSPAN k -> Type) Source Github #
Instance details Defined in Proarrow.Category.Instance.Cospan Methods dimap :: forall (c :: COSPAN k) (a :: COSPAN k) (b :: COSPAN k) (d :: COSPAN k). (c ~> a) -> (b ~> d) -> Cospan a b -> Cospan c d Source Github # lmap :: forall (c :: COSPAN k) (a :: COSPAN k) (b :: COSPAN k). (c ~> a) -> Cospan a b -> Cospan c b Source Github # rmap :: forall (b :: COSPAN k) (d :: COSPAN k) (a :: COSPAN k). (b ~> d) -> Cospan a b -> Cospan a d Source Github # (\\) :: forall (a :: COSPAN k) (b :: COSPAN k) r. ((Ob a, Ob b) => r) -> Cospan a b -> r Source Github #

arr :: forall k (a :: k) (b :: k). CategoryOf k => (a ~> b) -> Cospan ('CS a) ('CS b) Source Github #

coarr :: forall k (a :: k) (b :: k). CategoryOf k => (a ~> b) -> Cospan ('CS b) ('CS a) Source Github #

data family Pushout :: SPAN k +-> COSPAN k Source Github #

Instances

Instances details

(HasPushouts k, HasPullbacks k) => FunctorForRep (Pushout :: SPAN k +-> COSPAN k) Source Github #
Instance details Defined in Proarrow.Category.Instance.Cospan Methods fmap :: forall (a :: SPAN k) (b :: SPAN k). (a ~> b) -> ((Pushout :: SPAN k +-> COSPAN k) @ a) ~> ((Pushout :: SPAN k +-> COSPAN k) @ b) Source Github #
type (Pushout :: SPAN k +-> COSPAN k) @ ('SP a :: SPAN k) Source Github #
Instance details Defined in Proarrow.Category.Instance.Cospan type (Pushout :: SPAN k +-> COSPAN k) @ ('SP a :: SPAN k) = 'CS a

data family Pullback :: COSPAN k +-> SPAN k Source Github #

Instances

Instances details

(HasPushouts k, HasPullbacks k) => FunctorForRep (Pullback :: COSPAN k +-> SPAN k) Source Github #
Instance details Defined in Proarrow.Category.Instance.Cospan Methods fmap :: forall (a :: COSPAN k) (b :: COSPAN k). (a ~> b) -> ((Pullback :: COSPAN k +-> SPAN k) @ a) ~> ((Pullback :: COSPAN k +-> SPAN k) @ b) Source Github #
type (Pullback :: COSPAN k +-> SPAN k) @ ('CS a :: COSPAN k) Source Github #
Instance details Defined in Proarrow.Category.Instance.Cospan type (Pullback :: COSPAN k +-> SPAN k) @ ('CS a :: COSPAN k) = 'SP a