Safe Haskell	None
Language	GHC2024

Proarrow.Category.Instance.PointedHask

Synopsis

data POINTED = P Type
data Pointed (a :: POINTED) (b :: POINTED) where
- Pt :: forall a1 b1. {..} -> Pointed ('P a1) ('P b1)
toHask :: ('P a ~> 'P b) -> Maybe a -> Maybe b
data These a b
- = This a
- | That b
- | These a b
memptyDefault :: Monoid a => (Unit :: POINTED) ~> 'P a
mappendDefault :: Monoid a => ('P a ** 'P a) ~> 'P a
underlyingPt :: forall k (a :: k) (b :: k). HasZeroObject k => (a ~> b) -> (Unit :: POINTED) ~> 'P (a ~> b)
enrichedPt :: forall k (a :: k) (b :: k). (Ob a, Ob b, HasZeroObject k) => ((Unit :: POINTED) ~> 'P (a ~> b)) -> a ~> b
compPt :: forall k (a :: k) (b :: k) (c :: k). (Ob a, Ob b, Ob c, HasZeroObject k) => ('P (b ~> c) ** 'P (a ~> b)) ~> 'P (a ~> c)

Documentation

data POINTED Source Comments #

Constructors

P Type

Instances

Instances details

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 #

SymMonoidal POINTED Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Methods

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

CategoryOf POINTED Source Comments #

The category of types with an added point and point-preserving morphisms.

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)

CopyDiscard POINTED Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Methods

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

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

HasBinaryCoproducts POINTED Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Associated Types

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

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

lft :: forall (a :: POINTED) (b :: POINTED). (Ob a, Ob b) => a ~> (a || b) Source Comments #

rgt :: forall (a :: POINTED) (b :: POINTED). (Ob a, Ob b) => b ~> (a || b) Source Comments #

(|||) :: forall (x :: POINTED) (a :: POINTED) (y :: POINTED). (x ~> a) -> (y ~> a) -> (x || y) ~> a Source Comments #

(+++) :: forall (a :: POINTED) (b :: POINTED) (x :: POINTED) (y :: POINTED). (a ~> x) -> (b ~> y) -> (a || b) ~> (x || y) Source Comments #

HasBinaryProducts POINTED Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Associated Types

type ('P a :: POINTED) && ('P b :: POINTED)
Instance details Defined in Proarrow.Category.Instance.PointedHask type ('P a :: POINTED) && ('P b :: POINTED) = 'P (These a b)

Methods

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

fst :: forall (a :: POINTED) (b :: POINTED). (Ob a, Ob b) => (a && b) ~> a Source Comments #

snd :: forall (a :: POINTED) (b :: POINTED). (Ob a, Ob b) => (a && b) ~> b Source Comments #

(&&&) :: forall (a :: POINTED) (x :: POINTED) (y :: POINTED). (a ~> x) -> (a ~> y) -> a ~> (x && y) Source Comments #

(***) :: forall (a :: POINTED) (b :: POINTED) (x :: POINTED) (y :: POINTED). (a ~> x) -> (b ~> y) -> (a && b) ~> (x && y) Source Comments #

Closed POINTED Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Associated Types

type ('P a :: POINTED) ~~> ('P b :: POINTED)
Instance details Defined in Proarrow.Category.Instance.PointedHask type ('P a :: POINTED) ~~> ('P b :: POINTED) = 'P (a -> Maybe b)

Methods

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

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

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

(^^^) :: forall (a :: POINTED) (b :: POINTED) (x :: POINTED) (y :: POINTED). (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) Source Comments #

HasInitialObject POINTED Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Associated Types

type InitialObject
Instance details Defined in Proarrow.Category.Instance.PointedHask type InitialObject = 'P Void

Methods

initiate :: forall (a :: POINTED). Ob a => (InitialObject :: POINTED) ~> a Source Comments #

HasTerminalObject POINTED Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Associated Types

type TerminalObject
Instance details Defined in Proarrow.Category.Instance.PointedHask type TerminalObject = 'P Void

Methods

terminate :: forall (a :: POINTED). Ob a => a ~> (TerminalObject :: POINTED) Source Comments #

Promonad Pointed Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Methods

id :: forall (a :: POINTED). Ob a => Pointed a a Source Comments #

(.) :: forall (b :: POINTED) (c :: POINTED) (a :: POINTED). Pointed b c -> Pointed a b -> Pointed a c Source Comments #

Copowered Type POINTED Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Associated Types

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

Methods

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

copower :: forall (a :: POINTED) (b :: POINTED) n. (Ob a, Ob b) => (n ~> HomObj Type a b) -> (n *. a) ~> b Source Comments #

uncopower :: forall (a :: POINTED) n (b :: POINTED). (Ob a, Ob n) => ((n *. a) ~> b) -> n ~> HomObj Type a b Source Comments #

Powered Type POINTED Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Associated Types

type ('P a :: POINTED) ^ (n :: Type)
Instance details Defined in Proarrow.Category.Instance.PointedHask type ('P a :: POINTED) ^ (n :: Type) = 'P (n -> Maybe a)

Methods

withObPower :: forall (a :: POINTED) n r. (Ob a, Ob n) => (Ob (a ^ n) => r) -> r Source Comments #

power :: forall (a :: POINTED) (b :: POINTED) n. (Ob a, Ob b) => (n ~> HomObj Type a b) -> a ~> (b ^ n) Source Comments #

unpower :: forall (b :: POINTED) n (a :: POINTED). (Ob b, Ob n) => (a ~> (b ^ n)) -> n ~> HomObj Type a b 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 #

Profunctor Pointed Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Methods

dimap :: forall (c :: POINTED) (a :: POINTED) (b :: POINTED) (d :: POINTED). (c ~> a) -> (b ~> d) -> Pointed a b -> Pointed c d Source Comments #

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

Comonoid ('P x :: POINTED) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Methods

counit :: 'P x ~> (Unit :: POINTED) Source Comments #

comult :: 'P x ~> ('P x ** 'P x) Source Comments #

Monoid ('P Void) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Methods

mempty :: (Unit :: POINTED) ~> 'P Void Source Comments #

mappend :: ('P Void ** 'P Void) ~> 'P Void Source Comments #

Monoid ('P ()) Source Comments #

Conjunction with False = Nothing, True = Just ()

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Methods

mempty :: (Unit :: POINTED) ~> 'P () Source Comments #

mappend :: ('P () ** 'P ()) ~> 'P () Source Comments #

Monoid ('P [a] :: POINTED) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

Methods

mempty :: (Unit :: POINTED) ~> 'P [a] Source Comments #

mappend :: ('P [a] ** 'P [a]) ~> 'P [a] Source Comments #

type Unit Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type Unit = 'P ()

type (~>) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type (~>) = Pointed

type InitialObject Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type InitialObject = 'P Void

type TerminalObject Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type TerminalObject = 'P Void

type Ob (a :: POINTED) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type Ob (a :: POINTED) = a ~ 'P (UN 'P a)

type UN 'P ('P a :: POINTED) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type UN 'P ('P a :: POINTED) = a

type (n :: Type) *. ('P a :: POINTED) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type (n :: Type) *. ('P a :: POINTED) = 'P (n, a)

type ('P a :: POINTED) ^ (n :: Type) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type ('P a :: POINTED) ^ (n :: Type) = 'P (n -> Maybe a)

type ('P a :: POINTED) ** ('P b :: POINTED) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type ('P a :: POINTED) ** ('P b :: POINTED) = 'P (a, b)

type ('P a :: POINTED) || ('P b :: POINTED) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type ('P a :: POINTED) || ('P b :: POINTED) = 'P (a || b)

type ('P a :: POINTED) && ('P b :: POINTED) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type ('P a :: POINTED) && ('P b :: POINTED) = 'P (These a b)

type ('P a :: POINTED) ~~> ('P b :: POINTED) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.PointedHask

type ('P a :: POINTED) ~~> ('P b :: POINTED) = 'P (a -> Maybe b)

data Pointed (a :: POINTED) (b :: POINTED) where Source Comments #

Constructors

Pt
Fields :: forall a1 b1. { unPt :: a1 -> Maybe b1 } -> Pointed ('P a1) ('P b1)

Instances

Instances details

Promonad Pointed Source Comments #
Instance details Defined in Proarrow.Category.Instance.PointedHask Methods id :: forall (a :: POINTED). Ob a => Pointed a a Source Comments # (.) :: forall (b :: POINTED) (c :: POINTED) (a :: POINTED). Pointed b c -> Pointed a b -> Pointed a c 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 #
Profunctor Pointed Source Comments #
Instance details Defined in Proarrow.Category.Instance.PointedHask Methods dimap :: forall (c :: POINTED) (a :: POINTED) (b :: POINTED) (d :: POINTED). (c ~> a) -> (b ~> d) -> Pointed a b -> Pointed c d Source Comments # (\\) :: forall (a :: POINTED) (b :: POINTED) r. ((Ob a, Ob b) => r) -> Pointed a b -> r Source Comments #

toHask :: ('P a ~> 'P b) -> Maybe a -> Maybe b Source Comments #

data These a b Source Comments #

Constructors

This a
That b
These a b

memptyDefault :: Monoid a => (Unit :: POINTED) ~> 'P a Source Comments #

Lift Hask monoids.

mappendDefault :: Monoid a => ('P a ** 'P a) ~> 'P a Source Comments #

underlyingPt :: forall k (a :: k) (b :: k). HasZeroObject k => (a ~> b) -> (Unit :: POINTED) ~> 'P (a ~> b) Source Comments #

Categories with a zero object can be seen as categories enriched in Pointed.

enrichedPt :: forall k (a :: k) (b :: k). (Ob a, Ob b, HasZeroObject k) => ((Unit :: POINTED) ~> 'P (a ~> b)) -> a ~> b Source Comments #

compPt :: forall k (a :: k) (b :: k) (c :: k). (Ob a, Ob b, Ob c, HasZeroObject k) => ('P (b ~> c) ** 'P (a ~> b)) ~> 'P (a ~> c) Source Comments #