proarrow
Safe HaskellNone
LanguageGHC2024

Proarrow.Category.Instance.PointedHask

Synopsis

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 #

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)

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

Constructors

Pt 

Fields

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 

Instances

Instances details
Generic (These a b) Source Comments # 
Instance details

Defined in Proarrow.Category.Instance.PointedHask

Associated Types

type Rep (These a b) 
Instance details

Defined in Proarrow.Category.Instance.PointedHask

type Rep (These a b) = D1 ('MetaData "These" "Proarrow.Category.Instance.PointedHask" "proarrow-0-inplace" 'False) (C1 ('MetaCons "This" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedStrict) (Rec0 a)) :+: (C1 ('MetaCons "That" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedStrict) (Rec0 b)) :+: C1 ('MetaCons "These" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedStrict) (Rec0 a) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedStrict) (Rec0 b))))

Methods

from :: These a b -> Rep (These a b) x Comments #

to :: Rep (These a b) x -> These a b Comments #

(Show a, Show b) => Show (These a b) Source Comments # 
Instance details

Defined in Proarrow.Category.Instance.PointedHask

Methods

showsPrec :: Int -> These a b -> ShowS Comments #

show :: These a b -> String Comments #

showList :: [These a b] -> ShowS Comments #

(Eq a, Eq b) => Eq (These a b) Source Comments # 
Instance details

Defined in Proarrow.Category.Instance.PointedHask

Methods

(==) :: These a b -> These a b -> Bool Comments #

(/=) :: These a b -> These a b -> Bool Comments #

type Rep (These a b) Source Comments # 
Instance details

Defined in Proarrow.Category.Instance.PointedHask

type Rep (These a b) = D1 ('MetaData "These" "Proarrow.Category.Instance.PointedHask" "proarrow-0-inplace" 'False) (C1 ('MetaCons "This" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedStrict) (Rec0 a)) :+: (C1 ('MetaCons "That" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedStrict) (Rec0 b)) :+: C1 ('MetaCons "These" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedStrict) (Rec0 a) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedStrict) (Rec0 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 #