Algebra.Category.Relation

Documentation

newtype Rel s a b

Constructors

Rel (a -> b -> s)

Instances

Ring s => Category (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Associated Types

type Obj (Rel s) :: k -> Constraint

Methods

(.) :: forall (a :: k) (b :: k) (c :: k). (Obj (Rel s) a, Obj (Rel s) b, Obj (Rel s) c) => Rel s b c -> Rel s a b -> Rel s a c

id :: forall (a :: k). Obj (Rel s) a => Rel s a a

Ring s => Dagger (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Methods

dagger :: forall (a :: k) (b :: k). O2 (Rel s) a b => Rel s a b -> Rel s b a

Ring s => Braided ((⊕) :: Type -> Type -> Type) (Zero :: Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Methods

swap :: forall (a :: k) (b :: k). (Obj (Rel s) a, Obj (Rel s) b) => Rel s (a ⊕ b) (b ⊕ a)

swap_ :: forall (a :: k) (b :: k). (Obj (Rel s) a, Obj (Rel s) b) => Rel s (a ⊕ b) (b ⊕ a)

Ring s => Braided ((⊗) :: Type -> Type -> Type) (One :: Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Methods

swap :: forall (a :: k) (b :: k). (Obj (Rel s) a, Obj (Rel s) b) => Rel s (a ⊗ b) (b ⊗ a)

swap_ :: forall (a :: k) (b :: k). (Obj (Rel s) a, Obj (Rel s) b) => Rel s (a ⊗ b) (b ⊗ a)

Ring s => Cartesian ((⊕) :: Type -> Type -> Type) (Zero :: Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Methods

exl :: forall (a :: k) (b :: k). O2 (Rel s) a b => Rel s (a ⊕ b) a

exr :: forall (a :: k) (b :: k). O2 (Rel s) a b => Rel s (a ⊕ b) b

dis :: forall (a :: k). Obj (Rel s) a => Rel s a Zero

dup :: forall (a :: k). Obj (Rel s) a => Rel s a (a ⊕ a)

(▵) :: forall (a :: k) (b :: k) (c :: k). (Obj (Rel s) a, Obj (Rel s) b, Obj (Rel s) c) => Rel s a b -> Rel s a c -> Rel s a (b ⊕ c)

Ring s => Cartesian ((⊗) :: Type -> Type -> Type) (One :: Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Methods

exl :: forall (a :: k) (b :: k). O2 (Rel s) a b => Rel s (a ⊗ b) a

exr :: forall (a :: k) (b :: k). O2 (Rel s) a b => Rel s (a ⊗ b) b

dis :: forall (a :: k). Obj (Rel s) a => Rel s a One

dup :: forall (a :: k). Obj (Rel s) a => Rel s a (a ⊗ a)

(▵) :: forall (a :: k) (b :: k) (c :: k). (Obj (Rel s) a, Obj (Rel s) b, Obj (Rel s) c) => Rel s a b -> Rel s a c -> Rel s a (b ⊗ c)

Ring s => CoCartesian ((⊕) :: Type -> Type -> Type) (Zero :: Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Methods

inl :: forall (a :: k) (b :: k). O2 (Rel s) a b => Rel s a (a ⊕ b)

inr :: forall (a :: k) (b :: k). O2 (Rel s) a b => Rel s b (a ⊕ b)

new :: forall (a :: k). Obj (Rel s) a => Rel s Zero a

jam :: forall (a :: k). Obj (Rel s) a => Rel s (a ⊕ a) a

(▿) :: forall (a :: k) (b :: k) (c :: k). (Obj (Rel s) a, Obj (Rel s) b, Obj (Rel s) c) => Rel s b a -> Rel s c a -> Rel s (b ⊕ c) a

Ring s => CoCartesian ((⊗) :: Type -> Type -> Type) (One :: Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Methods

inl :: forall (a :: k) (b :: k). O2 (Rel s) a b => Rel s a (a ⊗ b)

inr :: forall (a :: k) (b :: k). O2 (Rel s) a b => Rel s b (a ⊗ b)

new :: forall (a :: k). Obj (Rel s) a => Rel s One a

jam :: forall (a :: k). Obj (Rel s) a => Rel s (a ⊗ a) a

(▿) :: forall (a :: k) (b :: k) (c :: k). (Obj (Rel s) a, Obj (Rel s) b, Obj (Rel s) c) => Rel s b a -> Rel s c a -> Rel s (b ⊗ c) a

Ring s => Monoidal ((⊕) :: Type -> Type -> Type) (Zero :: Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Methods

(⊗) :: forall (a :: k) (b :: k) (c :: k) (d :: k). (Obj (Rel s) a, Obj (Rel s) b, Obj (Rel s) c, Obj (Rel s) d) => Rel s a b -> Rel s c d -> Rel s (a ⊕ c) (b ⊕ d)

assoc :: forall (a :: k) (b :: k) (c :: k). (Obj (Rel s) a, Obj (Rel s) b, Obj (Rel s) c) => Rel s ((a ⊕ b) ⊕ c) (a ⊕ (b ⊕ c))

assoc_ :: forall (a :: k) (b :: k) (c :: k). (Obj (Rel s) a, Obj (Rel s) b, Obj (Rel s) c) => Rel s (a ⊕ (b ⊕ c)) ((a ⊕ b) ⊕ c)

unitorR :: forall (a :: k). (Obj (Rel s) a, Obj (Rel s) Zero) => Rel s a (a ⊕ Zero)

unitorR_ :: forall (a :: k). (Obj (Rel s) a, Obj (Rel s) Zero) => Rel s (a ⊕ Zero) a

unitorL :: forall (a :: k). (Obj (Rel s) a, Obj (Rel s) Zero) => Rel s a (Zero ⊕ a)

unitorL_ :: forall (a :: k). (Obj (Rel s) a, Obj (Rel s) Zero) => Rel s (Zero ⊕ a) a

Ring s => Monoidal ((⊗) :: Type -> Type -> Type) (One :: Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Methods

(⊗) :: forall (a :: k) (b :: k) (c :: k) (d :: k). (Obj (Rel s) a, Obj (Rel s) b, Obj (Rel s) c, Obj (Rel s) d) => Rel s a b -> Rel s c d -> Rel s (a ⊗ c) (b ⊗ d)

assoc :: forall (a :: k) (b :: k) (c :: k). (Obj (Rel s) a, Obj (Rel s) b, Obj (Rel s) c) => Rel s ((a ⊗ b) ⊗ c) (a ⊗ (b ⊗ c))

assoc_ :: forall (a :: k) (b :: k) (c :: k). (Obj (Rel s) a, Obj (Rel s) b, Obj (Rel s) c) => Rel s (a ⊗ (b ⊗ c)) ((a ⊗ b) ⊗ c)

unitorR :: forall (a :: k). (Obj (Rel s) a, Obj (Rel s) One) => Rel s a (a ⊗ One)

unitorR_ :: forall (a :: k). (Obj (Rel s) a, Obj (Rel s) One) => Rel s (a ⊗ One) a

unitorL :: forall (a :: k). (Obj (Rel s) a, Obj (Rel s) One) => Rel s a (One ⊗ a)

unitorL_ :: forall (a :: k). (Obj (Rel s) a, Obj (Rel s) One) => Rel s (One ⊗ a) a

Ring s => Symmetric ((⊕) :: Type -> Type -> Type) (Zero :: Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Ring s => Symmetric ((⊗) :: Type -> Type -> Type) (One :: Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Ring s => Compact ((⊗) :: Type -> Type -> Type) (One :: Type) (Dual :: Type -> Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Ring s => Autonomous ((⊗) :: Type -> Type -> Type) (One :: Type) (Dual :: Type -> Type) (Dual :: Type -> Type) (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

Methods

turn :: forall (a :: k). Obj (Rel s) a => Rel s One (Dual a ⊗ a)

turn' :: forall (a :: k). Obj (Rel s) a => Rel s (a ⊗ Dual a) One

Additive s => Additive (Rel s a b)

Defined in Algebra.Category.Relation

Methods

(+) :: Rel s a b -> Rel s a b -> Rel s a b

zero :: Rel s a b

times :: Natural -> Rel s a b -> Rel s a b

type Obj (Rel s :: Type -> Type -> Type)

Defined in Algebra.Category.Relation

type Obj (Rel s :: Type -> Type -> Type) = Finite

indicate :: Ring s => Bool -> s