Control.Category.Constrained.Reified

GADTs that mirror the class hierarchy from Category to (at the moment) Cartesian, reifying all the available “free” composition operations.

These can be used as a “trivial base case“ for all kinds of categories: it turns out these basic operations are often not so trivial to implement, or only possible with stronger constraints than you'd like. For instance, the category of affine mappings can only be implemented directly as a category on vector spaces, because the identity mapping has zero constant offset.

By leaving the free compositions reified to runtime syntax trees, this problem can be avoided. In other applications, you may not need these cases, but can still benefit from them for optimisation (composition with id is always trivial, and so on).

Reified versions of the category classes

data ReCategory (k :: * -> * -> *) (α :: *) (β :: *) where

Constructors

ReCategory :: k α β -> ReCategory k α β

CategoryId :: Object k α => ReCategory k α α

CategoryCompo :: Object k β => ReCategory k α β -> ReCategory k β γ -> ReCategory k α γ

Instances

HasAgent k => HasAgent (ReCategory k)

Associated Types

type AgentVal (ReCategory k :: * -> * -> *) a v :: *

Methods

alg :: (Object (ReCategory k) a, Object (ReCategory k) b) => (forall q. Object (ReCategory k) q => AgentVal (ReCategory k) q a -> AgentVal (ReCategory k) q b) -> ReCategory k a b

($~) :: (Object (ReCategory k) a, Object (ReCategory k) b, Object (ReCategory k) c) => ReCategory k b c -> AgentVal (ReCategory k) a b -> AgentVal (ReCategory k) a c

Category k => Category (ReCategory k)

Associated Types

type Object (ReCategory k :: * -> * -> *) o :: Constraint

Methods

id :: Object (ReCategory k) a => ReCategory k a a

(.) :: (Object (ReCategory k) a, Object (ReCategory k) b, Object (ReCategory k) c) => ReCategory k b c -> ReCategory k a b -> ReCategory k a c

Category k => CRCategory (ReCategory k)

Associated Types

type SpecificCat (ReCategory k :: * -> * -> *) :: * -> * -> *

Methods

fromSpecific :: SpecificCat (ReCategory k) α β -> ReCategory k α β

match_concrete :: ReCategory k α β -> Maybe (SpecificCat (ReCategory k) α β)

match_id :: ReCategory k α β -> IdPattern (ReCategory k) α β

match_compose :: ReCategory k α β -> CompoPattern (ReCategory k) α β

Category k => EnhancedCat (ReCategory k) k

Methods

arr :: (Object k b, Object k c, Object (ReCategory k) b, Object (ReCategory k) c) => k b c -> ReCategory k b c

type SpecificCat (ReCategory k)

type SpecificCat (ReCategory k) = k

type Object (ReCategory k) α

type Object (ReCategory k) α = Object k α

type AgentVal (ReCategory k) α ω

type AgentVal (ReCategory k) α ω = GenericAgent (ReCategory k) α ω

data ReCartesian (k :: * -> * -> *) (α :: *) (β :: *) where

Constructors

ReCartesian :: k α β -> ReCartesian k α β

CartesianId :: Object k α => ReCartesian k α α

CartesianCompo :: Object k β => ReCartesian k α β -> ReCartesian k β γ -> ReCartesian k α γ

CartesianSwap :: (ObjectPair k α β, ObjectPair k β α) => ReCartesian k (α, β) (β, α)

CartesianAttachUnit :: (Object k α, UnitObject k ~ u, ObjectPair k α u) => ReCartesian k α (α, u)

CartesianDetachUnit :: (Object k α, UnitObject k ~ u, ObjectPair k α u) => ReCartesian k (α, u) α

CartesianRegroup :: (ObjectPair k α β, ObjectPair k β γ, ObjectPair k α (β, γ), ObjectPair k (α, β) γ) => ReCartesian k (α, (β, γ)) ((α, β), γ)

CartesianRegroup_ :: (ObjectPair k α β, ObjectPair k β γ, ObjectPair k α (β, γ), ObjectPair k (α, β) γ) => ReCartesian k ((α, β), γ) (α, (β, γ))

Instances

(HasAgent k, Cartesian k) => HasAgent (ReCartesian k)

Associated Types

type AgentVal (ReCartesian k :: * -> * -> *) a v :: *

Methods

alg :: (Object (ReCartesian k) a, Object (ReCartesian k) b) => (forall q. Object (ReCartesian k) q => AgentVal (ReCartesian k) q a -> AgentVal (ReCartesian k) q b) -> ReCartesian k a b

($~) :: (Object (ReCartesian k) a, Object (ReCartesian k) b, Object (ReCartesian k) c) => ReCartesian k b c -> AgentVal (ReCartesian k) a b -> AgentVal (ReCartesian k) a c

Cartesian k => Cartesian (ReCartesian k)

Associated Types

type PairObjects (ReCartesian k :: * -> * -> *) a b :: Constraint

type UnitObject (ReCartesian k :: * -> * -> *) :: *

Methods

swap :: (ObjectPair (ReCartesian k) a b, ObjectPair (ReCartesian k) b a) => ReCartesian k (a, b) (b, a)

attachUnit :: ((* ~ u) (UnitObject (ReCartesian k)), ObjectPair (ReCartesian k) a u) => ReCartesian k a (a, u)

detachUnit :: ((* ~ u) (UnitObject (ReCartesian k)), ObjectPair (ReCartesian k) a u) => ReCartesian k (a, u) a

regroup :: (ObjectPair (ReCartesian k) a b, ObjectPair (ReCartesian k) b c, ObjectPair (ReCartesian k) a (b, c), ObjectPair (ReCartesian k) (a, b) c) => ReCartesian k (a, (b, c)) ((a, b), c)

regroup' :: (ObjectPair (ReCartesian k) a b, ObjectPair (ReCartesian k) b c, ObjectPair (ReCartesian k) a (b, c), ObjectPair (ReCartesian k) (a, b) c) => ReCartesian k ((a, b), c) (a, (b, c))

Cartesian k => Category (ReCartesian k)

Associated Types

type Object (ReCartesian k :: * -> * -> *) o :: Constraint

Methods

id :: Object (ReCartesian k) a => ReCartesian k a a

(.) :: (Object (ReCartesian k) a, Object (ReCartesian k) b, Object (ReCartesian k) c) => ReCartesian k b c -> ReCartesian k a b -> ReCartesian k a c

Cartesian k => CRCartesian (ReCartesian k)

Methods

match_swap :: ReCartesian k α β -> SwapPattern (ReCartesian k) α β

match_attachUnit :: ReCartesian k α β -> AttachUnitPattern (ReCartesian k) α β

match_detachUnit :: ReCartesian k α β -> DetachUnitPattern (ReCartesian k) α β

match_regroup :: ReCartesian k α β -> RegroupPattern (ReCartesian k) α β

match_regroup' :: ReCartesian k α β -> Regroup'Pattern (ReCartesian k) α β

Cartesian k => CRCategory (ReCartesian k)

Associated Types

type SpecificCat (ReCartesian k :: * -> * -> *) :: * -> * -> *

Methods

fromSpecific :: SpecificCat (ReCartesian k) α β -> ReCartesian k α β

match_concrete :: ReCartesian k α β -> Maybe (SpecificCat (ReCartesian k) α β)

match_id :: ReCartesian k α β -> IdPattern (ReCartesian k) α β

match_compose :: ReCartesian k α β -> CompoPattern (ReCartesian k) α β

Cartesian k => EnhancedCat (ReCartesian k) k

Methods

arr :: (Object k b, Object k c, Object (ReCartesian k) b, Object (ReCartesian k) c) => k b c -> ReCartesian k b c

type UnitObject (ReCartesian k)

type UnitObject (ReCartesian k) = UnitObject k

type SpecificCat (ReCartesian k)

type SpecificCat (ReCartesian k) = k

type Object (ReCartesian k) a

type Object (ReCartesian k) a = Object k a

type AgentVal (ReCartesian k) α ω

type AgentVal (ReCartesian k) α ω = GenericAgent (ReCartesian k) α ω

type PairObjects (ReCartesian k) α β

type PairObjects (ReCartesian k) α β = PairObjects k α β

data ReMorphism (k :: * -> * -> *) (α :: *) (β :: *) where

Constructors

ReMorphism :: k α β -> ReMorphism k α β

MorphismId :: Object k α => ReMorphism k α α

MorphismCompo :: Object k β => ReMorphism k α β -> ReMorphism k β γ -> ReMorphism k α γ

MorphismSwap :: (ObjectPair k α β, ObjectPair k β α) => ReMorphism k (α, β) (β, α)

MorphismAttachUnit :: (Object k α, UnitObject k ~ u, ObjectPair k α u) => ReMorphism k α (α, u)

MorphismDetachUnit :: (Object k α, UnitObject k ~ u, ObjectPair k α u) => ReMorphism k (α, u) α

MorphismRegroup :: (ObjectPair k α β, ObjectPair k β γ, ObjectPair k α (β, γ), ObjectPair k (α, β) γ) => ReMorphism k (α, (β, γ)) ((α, β), γ)

MorphismRegroup_ :: (ObjectPair k α β, ObjectPair k β γ, ObjectPair k α (β, γ), ObjectPair k (α, β) γ) => ReMorphism k ((α, β), γ) (α, (β, γ))

MorphismPar :: (ObjectPair k α γ, ObjectPair k β δ) => ReMorphism k α β -> ReMorphism k γ δ -> ReMorphism k (α, γ) (β, δ)

Instances

(HasAgent k, Morphism k) => HasAgent (ReMorphism k)

Associated Types

type AgentVal (ReMorphism k :: * -> * -> *) a v :: *

Methods

alg :: (Object (ReMorphism k) a, Object (ReMorphism k) b) => (forall q. Object (ReMorphism k) q => AgentVal (ReMorphism k) q a -> AgentVal (ReMorphism k) q b) -> ReMorphism k a b

($~) :: (Object (ReMorphism k) a, Object (ReMorphism k) b, Object (ReMorphism k) c) => ReMorphism k b c -> AgentVal (ReMorphism k) a b -> AgentVal (ReMorphism k) a c

Morphism k => Cartesian (ReMorphism k)

Associated Types

type PairObjects (ReMorphism k :: * -> * -> *) a b :: Constraint

type UnitObject (ReMorphism k :: * -> * -> *) :: *

Methods

swap :: (ObjectPair (ReMorphism k) a b, ObjectPair (ReMorphism k) b a) => ReMorphism k (a, b) (b, a)

attachUnit :: ((* ~ u) (UnitObject (ReMorphism k)), ObjectPair (ReMorphism k) a u) => ReMorphism k a (a, u)

detachUnit :: ((* ~ u) (UnitObject (ReMorphism k)), ObjectPair (ReMorphism k) a u) => ReMorphism k (a, u) a

regroup :: (ObjectPair (ReMorphism k) a b, ObjectPair (ReMorphism k) b c, ObjectPair (ReMorphism k) a (b, c), ObjectPair (ReMorphism k) (a, b) c) => ReMorphism k (a, (b, c)) ((a, b), c)

regroup' :: (ObjectPair (ReMorphism k) a b, ObjectPair (ReMorphism k) b c, ObjectPair (ReMorphism k) a (b, c), ObjectPair (ReMorphism k) (a, b) c) => ReMorphism k ((a, b), c) (a, (b, c))

Morphism k => Category (ReMorphism k)

Associated Types

type Object (ReMorphism k :: * -> * -> *) o :: Constraint

Methods

id :: Object (ReMorphism k) a => ReMorphism k a a

(.) :: (Object (ReMorphism k) a, Object (ReMorphism k) b, Object (ReMorphism k) c) => ReMorphism k b c -> ReMorphism k a b -> ReMorphism k a c

Morphism k => Morphism (ReMorphism k)

Methods

first :: (ObjectPair (ReMorphism k) b d, ObjectPair (ReMorphism k) c d) => ReMorphism k b c -> ReMorphism k (b, d) (c, d)

second :: (ObjectPair (ReMorphism k) d b, ObjectPair (ReMorphism k) d c) => ReMorphism k b c -> ReMorphism k (d, b) (d, c)

(***) :: (ObjectPair (ReMorphism k) b b', ObjectPair (ReMorphism k) c c') => ReMorphism k b c -> ReMorphism k b' c' -> ReMorphism k (b, b') (c, c')

Morphism k => CRMorphism (ReMorphism k)

Methods

match_par :: ReMorphism k α β -> ParPattern (ReMorphism k) α β

Morphism k => CRCartesian (ReMorphism k)

Methods

match_swap :: ReMorphism k α β -> SwapPattern (ReMorphism k) α β

match_attachUnit :: ReMorphism k α β -> AttachUnitPattern (ReMorphism k) α β

match_detachUnit :: ReMorphism k α β -> DetachUnitPattern (ReMorphism k) α β

match_regroup :: ReMorphism k α β -> RegroupPattern (ReMorphism k) α β

match_regroup' :: ReMorphism k α β -> Regroup'Pattern (ReMorphism k) α β

Morphism k => CRCategory (ReMorphism k)

Associated Types

type SpecificCat (ReMorphism k :: * -> * -> *) :: * -> * -> *

Methods

fromSpecific :: SpecificCat (ReMorphism k) α β -> ReMorphism k α β

match_concrete :: ReMorphism k α β -> Maybe (SpecificCat (ReMorphism k) α β)

match_id :: ReMorphism k α β -> IdPattern (ReMorphism k) α β

match_compose :: ReMorphism k α β -> CompoPattern (ReMorphism k) α β

Morphism k => EnhancedCat (ReMorphism k) k

Methods

arr :: (Object k b, Object k c, Object (ReMorphism k) b, Object (ReMorphism k) c) => k b c -> ReMorphism k b c

type UnitObject (ReMorphism k)

type UnitObject (ReMorphism k) = UnitObject k

type SpecificCat (ReMorphism k)

type SpecificCat (ReMorphism k) = k

type Object (ReMorphism k) a

type Object (ReMorphism k) a = Object k a

type AgentVal (ReMorphism k) α ω

type AgentVal (ReMorphism k) α ω = GenericAgent (ReMorphism k) α ω

type PairObjects (ReMorphism k) α β

type PairObjects (ReMorphism k) α β = PairObjects k α β

data RePreArrow (k :: * -> * -> *) (α :: *) (β :: *) where

Constructors

RePreArrow :: k α β -> RePreArrow k α β

PreArrowId :: Object k α => RePreArrow k α α

PreArrowCompo :: Object k β => RePreArrow k α β -> RePreArrow k β γ -> RePreArrow k α γ

PreArrowSwap :: (ObjectPair k α β, ObjectPair k β α) => RePreArrow k (α, β) (β, α)

PreArrowAttachUnit :: (Object k α, UnitObject k ~ u, ObjectPair k α u) => RePreArrow k α (α, u)

PreArrowDetachUnit :: (Object k α, UnitObject k ~ u, ObjectPair k α u) => RePreArrow k (α, u) α

PreArrowRegroup :: (ObjectPair k α β, ObjectPair k β γ, ObjectPair k α (β, γ), ObjectPair k (α, β) γ) => RePreArrow k (α, (β, γ)) ((α, β), γ)

PreArrowRegroup_ :: (ObjectPair k α β, ObjectPair k β γ, ObjectPair k α (β, γ), ObjectPair k (α, β) γ) => RePreArrow k ((α, β), γ) (α, (β, γ))

PreArrowPar :: (ObjectPair k α γ, ObjectPair k β δ) => RePreArrow k α β -> RePreArrow k γ δ -> RePreArrow k (α, γ) (β, δ)

PreArrowFanout :: (Object k α, ObjectPair k β γ) => RePreArrow k α β -> RePreArrow k α γ -> RePreArrow k α (β, γ)

PreArrowTerminal :: Object k α => RePreArrow k α (UnitObject k)

PreArrowFst :: ObjectPair k α β => RePreArrow k (α, β) α

PreArrowSnd :: ObjectPair k α β => RePreArrow k (α, β) β

Instances

(HasAgent k, PreArrow k) => HasAgent (RePreArrow k)

Associated Types

type AgentVal (RePreArrow k :: * -> * -> *) a v :: *

Methods

alg :: (Object (RePreArrow k) a, Object (RePreArrow k) b) => (forall q. Object (RePreArrow k) q => AgentVal (RePreArrow k) q a -> AgentVal (RePreArrow k) q b) -> RePreArrow k a b

($~) :: (Object (RePreArrow k) a, Object (RePreArrow k) b, Object (RePreArrow k) c) => RePreArrow k b c -> AgentVal (RePreArrow k) a b -> AgentVal (RePreArrow k) a c

PreArrow k => Cartesian (RePreArrow k)

Associated Types

type PairObjects (RePreArrow k :: * -> * -> *) a b :: Constraint

type UnitObject (RePreArrow k :: * -> * -> *) :: *

Methods

swap :: (ObjectPair (RePreArrow k) a b, ObjectPair (RePreArrow k) b a) => RePreArrow k (a, b) (b, a)

attachUnit :: ((* ~ u) (UnitObject (RePreArrow k)), ObjectPair (RePreArrow k) a u) => RePreArrow k a (a, u)

detachUnit :: ((* ~ u) (UnitObject (RePreArrow k)), ObjectPair (RePreArrow k) a u) => RePreArrow k (a, u) a

regroup :: (ObjectPair (RePreArrow k) a b, ObjectPair (RePreArrow k) b c, ObjectPair (RePreArrow k) a (b, c), ObjectPair (RePreArrow k) (a, b) c) => RePreArrow k (a, (b, c)) ((a, b), c)

regroup' :: (ObjectPair (RePreArrow k) a b, ObjectPair (RePreArrow k) b c, ObjectPair (RePreArrow k) a (b, c), ObjectPair (RePreArrow k) (a, b) c) => RePreArrow k ((a, b), c) (a, (b, c))

PreArrow k => Category (RePreArrow k)

Associated Types

type Object (RePreArrow k :: * -> * -> *) o :: Constraint

Methods

id :: Object (RePreArrow k) a => RePreArrow k a a

(.) :: (Object (RePreArrow k) a, Object (RePreArrow k) b, Object (RePreArrow k) c) => RePreArrow k b c -> RePreArrow k a b -> RePreArrow k a c

PreArrow k => PreArrow (RePreArrow k)

Methods

(&&&) :: (Object (RePreArrow k) b, ObjectPair (RePreArrow k) c c') => RePreArrow k b c -> RePreArrow k b c' -> RePreArrow k b (c, c')

terminal :: Object (RePreArrow k) b => RePreArrow k b (UnitObject (RePreArrow k))

fst :: ObjectPair (RePreArrow k) x y => RePreArrow k (x, y) x

snd :: ObjectPair (RePreArrow k) x y => RePreArrow k (x, y) y

PreArrow k => Morphism (RePreArrow k)

Methods

first :: (ObjectPair (RePreArrow k) b d, ObjectPair (RePreArrow k) c d) => RePreArrow k b c -> RePreArrow k (b, d) (c, d)

second :: (ObjectPair (RePreArrow k) d b, ObjectPair (RePreArrow k) d c) => RePreArrow k b c -> RePreArrow k (d, b) (d, c)

(***) :: (ObjectPair (RePreArrow k) b b', ObjectPair (RePreArrow k) c c') => RePreArrow k b c -> RePreArrow k b' c' -> RePreArrow k (b, b') (c, c')

PreArrow k => CRPreArrow (RePreArrow k)

Methods

match_fan :: RePreArrow k α β -> FanPattern (RePreArrow k) α β

match_fst :: RePreArrow k α β -> FstPattern (RePreArrow k) α β

match_snd :: RePreArrow k α β -> SndPattern (RePreArrow k) α β

match_terminal :: RePreArrow k α β -> TerminalPattern (RePreArrow k) α β

PreArrow k => CRMorphism (RePreArrow k)

Methods

match_par :: RePreArrow k α β -> ParPattern (RePreArrow k) α β

PreArrow k => CRCartesian (RePreArrow k)

Methods

match_swap :: RePreArrow k α β -> SwapPattern (RePreArrow k) α β

match_attachUnit :: RePreArrow k α β -> AttachUnitPattern (RePreArrow k) α β

match_detachUnit :: RePreArrow k α β -> DetachUnitPattern (RePreArrow k) α β

match_regroup :: RePreArrow k α β -> RegroupPattern (RePreArrow k) α β

match_regroup' :: RePreArrow k α β -> Regroup'Pattern (RePreArrow k) α β

PreArrow k => CRCategory (RePreArrow k)

Associated Types

type SpecificCat (RePreArrow k :: * -> * -> *) :: * -> * -> *

Methods

fromSpecific :: SpecificCat (RePreArrow k) α β -> RePreArrow k α β

match_concrete :: RePreArrow k α β -> Maybe (SpecificCat (RePreArrow k) α β)

match_id :: RePreArrow k α β -> IdPattern (RePreArrow k) α β

match_compose :: RePreArrow k α β -> CompoPattern (RePreArrow k) α β

PreArrow k => EnhancedCat (RePreArrow k) k

Methods

arr :: (Object k b, Object k c, Object (RePreArrow k) b, Object (RePreArrow k) c) => k b c -> RePreArrow k b c

type UnitObject (RePreArrow k)

type UnitObject (RePreArrow k) = UnitObject k

type SpecificCat (RePreArrow k)

type SpecificCat (RePreArrow k) = k

type Object (RePreArrow k) a

type Object (RePreArrow k) a = Object k a

type AgentVal (RePreArrow k) α ω

type AgentVal (RePreArrow k) α ω = GenericAgent (RePreArrow k) α ω

type PairObjects (RePreArrow k) α β

type PairObjects (RePreArrow k) α β = PairObjects k α β

data ReWellPointed (k :: * -> * -> *) (α :: *) (β :: *) where

Constructors

ReWellPointed :: k α β -> ReWellPointed k α β

WellPointedId :: Object k α => ReWellPointed k α α

WellPointedCompo :: Object k β => ReWellPointed k α β -> ReWellPointed k β γ -> ReWellPointed k α γ

WellPointedSwap :: (ObjectPair k α β, ObjectPair k β α) => ReWellPointed k (α, β) (β, α)

WellPointedAttachUnit :: (Object k α, UnitObject k ~ u, ObjectPair k α u) => ReWellPointed k α (α, u)

WellPointedDetachUnit :: (Object k α, UnitObject k ~ u, ObjectPair k α u) => ReWellPointed k (α, u) α

WellPointedRegroup :: (ObjectPair k α β, ObjectPair k β γ, ObjectPair k α (β, γ), ObjectPair k (α, β) γ) => ReWellPointed k (α, (β, γ)) ((α, β), γ)

WellPointedRegroup_ :: (ObjectPair k α β, ObjectPair k β γ, ObjectPair k α (β, γ), ObjectPair k (α, β) γ) => ReWellPointed k ((α, β), γ) (α, (β, γ))

WellPointedPar :: (ObjectPair k α γ, ObjectPair k β δ) => ReWellPointed k α β -> ReWellPointed k γ δ -> ReWellPointed k (α, γ) (β, δ)

WellPointedFanout :: (Object k α, ObjectPair k β γ) => ReWellPointed k α β -> ReWellPointed k α γ -> ReWellPointed k α (β, γ)

WellPointedTerminal :: Object k α => ReWellPointed k α (UnitObject k)

WellPointedFst :: ObjectPair k α β => ReWellPointed k (α, β) α

WellPointedSnd :: ObjectPair k α β => ReWellPointed k (α, β) β

WellPointedConst :: (Object k ν, ObjectPoint k α) => α -> ReWellPointed k ν α

Instances

(HasAgent k, WellPointed k) => HasAgent (ReWellPointed k)

Associated Types

type AgentVal (ReWellPointed k :: * -> * -> *) a v :: *

Methods

alg :: (Object (ReWellPointed k) a, Object (ReWellPointed k) b) => (forall q. Object (ReWellPointed k) q => AgentVal (ReWellPointed k) q a -> AgentVal (ReWellPointed k) q b) -> ReWellPointed k a b

($~) :: (Object (ReWellPointed k) a, Object (ReWellPointed k) b, Object (ReWellPointed k) c) => ReWellPointed k b c -> AgentVal (ReWellPointed k) a b -> AgentVal (ReWellPointed k) a c

WellPointed k => Cartesian (ReWellPointed k)

Associated Types

type PairObjects (ReWellPointed k :: * -> * -> *) a b :: Constraint

type UnitObject (ReWellPointed k :: * -> * -> *) :: *

Methods

swap :: (ObjectPair (ReWellPointed k) a b, ObjectPair (ReWellPointed k) b a) => ReWellPointed k (a, b) (b, a)

attachUnit :: ((* ~ u) (UnitObject (ReWellPointed k)), ObjectPair (ReWellPointed k) a u) => ReWellPointed k a (a, u)

detachUnit :: ((* ~ u) (UnitObject (ReWellPointed k)), ObjectPair (ReWellPointed k) a u) => ReWellPointed k (a, u) a

regroup :: (ObjectPair (ReWellPointed k) a b, ObjectPair (ReWellPointed k) b c, ObjectPair (ReWellPointed k) a (b, c), ObjectPair (ReWellPointed k) (a, b) c) => ReWellPointed k (a, (b, c)) ((a, b), c)

regroup' :: (ObjectPair (ReWellPointed k) a b, ObjectPair (ReWellPointed k) b c, ObjectPair (ReWellPointed k) a (b, c), ObjectPair (ReWellPointed k) (a, b) c) => ReWellPointed k ((a, b), c) (a, (b, c))

WellPointed k => Category (ReWellPointed k)

Associated Types

type Object (ReWellPointed k :: * -> * -> *) o :: Constraint

Methods

id :: Object (ReWellPointed k) a => ReWellPointed k a a

(.) :: (Object (ReWellPointed k) a, Object (ReWellPointed k) b, Object (ReWellPointed k) c) => ReWellPointed k b c -> ReWellPointed k a b -> ReWellPointed k a c

WellPointed k => WellPointed (ReWellPointed k)

Associated Types

type PointObject (ReWellPointed k :: * -> * -> *) x :: Constraint

Methods

globalElement :: ObjectPoint (ReWellPointed k) x => x -> ReWellPointed k (UnitObject (ReWellPointed k)) x

unit :: CatTagged (ReWellPointed k) (UnitObject (ReWellPointed k))

const :: (Object (ReWellPointed k) b, ObjectPoint (ReWellPointed k) x) => x -> ReWellPointed k b x

WellPointed k => PreArrow (ReWellPointed k)

Methods

(&&&) :: (Object (ReWellPointed k) b, ObjectPair (ReWellPointed k) c c') => ReWellPointed k b c -> ReWellPointed k b c' -> ReWellPointed k b (c, c')

terminal :: Object (ReWellPointed k) b => ReWellPointed k b (UnitObject (ReWellPointed k))

fst :: ObjectPair (ReWellPointed k) x y => ReWellPointed k (x, y) x

snd :: ObjectPair (ReWellPointed k) x y => ReWellPointed k (x, y) y

WellPointed k => Morphism (ReWellPointed k)

Methods

first :: (ObjectPair (ReWellPointed k) b d, ObjectPair (ReWellPointed k) c d) => ReWellPointed k b c -> ReWellPointed k (b, d) (c, d)

second :: (ObjectPair (ReWellPointed k) d b, ObjectPair (ReWellPointed k) d c) => ReWellPointed k b c -> ReWellPointed k (d, b) (d, c)

(***) :: (ObjectPair (ReWellPointed k) b b', ObjectPair (ReWellPointed k) c c') => ReWellPointed k b c -> ReWellPointed k b' c' -> ReWellPointed k (b, b') (c, c')

WellPointed k => CRWellPointed (ReWellPointed k)

Methods

match_const :: ReWellPointed k α β -> ConstPattern (ReWellPointed k) α β

WellPointed k => CRPreArrow (ReWellPointed k)

Methods

match_fan :: ReWellPointed k α β -> FanPattern (ReWellPointed k) α β

match_fst :: ReWellPointed k α β -> FstPattern (ReWellPointed k) α β

match_snd :: ReWellPointed k α β -> SndPattern (ReWellPointed k) α β

match_terminal :: ReWellPointed k α β -> TerminalPattern (ReWellPointed k) α β

WellPointed k => CRMorphism (ReWellPointed k)

Methods

match_par :: ReWellPointed k α β -> ParPattern (ReWellPointed k) α β

WellPointed k => CRCartesian (ReWellPointed k)

Methods

match_swap :: ReWellPointed k α β -> SwapPattern (ReWellPointed k) α β

match_attachUnit :: ReWellPointed k α β -> AttachUnitPattern (ReWellPointed k) α β

match_detachUnit :: ReWellPointed k α β -> DetachUnitPattern (ReWellPointed k) α β

match_regroup :: ReWellPointed k α β -> RegroupPattern (ReWellPointed k) α β

match_regroup' :: ReWellPointed k α β -> Regroup'Pattern (ReWellPointed k) α β

WellPointed k => CRCategory (ReWellPointed k)

Associated Types

type SpecificCat (ReWellPointed k :: * -> * -> *) :: * -> * -> *

Methods

fromSpecific :: SpecificCat (ReWellPointed k) α β -> ReWellPointed k α β

match_concrete :: ReWellPointed k α β -> Maybe (SpecificCat (ReWellPointed k) α β)

match_id :: ReWellPointed k α β -> IdPattern (ReWellPointed k) α β

match_compose :: ReWellPointed k α β -> CompoPattern (ReWellPointed k) α β

WellPointed k => EnhancedCat (ReWellPointed k) k

Methods

arr :: (Object k b, Object k c, Object (ReWellPointed k) b, Object (ReWellPointed k) c) => k b c -> ReWellPointed k b c

type UnitObject (ReWellPointed k)

type UnitObject (ReWellPointed k) = UnitObject k

type SpecificCat (ReWellPointed k)

type SpecificCat (ReWellPointed k) = k

type Object (ReWellPointed k) a

type Object (ReWellPointed k) a = Object k a

type PointObject (ReWellPointed k) α

type PointObject (ReWellPointed k) α = PointObject k α

type AgentVal (ReWellPointed k) α ω

type AgentVal (ReWellPointed k) α ω = GenericAgent (ReWellPointed k) α ω

type PairObjects (ReWellPointed k) α β

type PairObjects (ReWellPointed k) α β = PairObjects k α β