License	BSD-style (see the file LICENSE)
Maintainer	sjoerd@w3future.com
Stability	experimental
Portability	non-portable
Safe Haskell	None
Language	Haskell2010

Data.Category.Monoidal

Description

Synopsis

class (Functor f, Dom f ~ (Cod f :**: Cod f)) => TensorProduct f where
- type Unit f :: *
- unitObject :: f -> Obj (Cod f) (Unit f)
- leftUnitor :: Cod f ~ k => f -> Obj k a -> k (f :% (Unit f, a)) a
- leftUnitorInv :: Cod f ~ k => f -> Obj k a -> k a (f :% (Unit f, a))
- rightUnitor :: Cod f ~ k => f -> Obj k a -> k (f :% (a, Unit f)) a
- rightUnitorInv :: Cod f ~ k => f -> Obj k a -> k a (f :% (a, Unit f))
- associator :: Cod f ~ k => f -> Obj k a -> Obj k b -> Obj k c -> k (f :% (f :% (a, b), c)) (f :% (a, f :% (b, c)))
- associatorInv :: Cod f ~ k => f -> Obj k a -> Obj k b -> Obj k c -> k (f :% (a, f :% (b, c))) (f :% (f :% (a, b), c))
class TensorProduct f => SymmetricTensorProduct f where
- swap :: Cod f ~ k => f -> Obj k a -> Obj k b -> k (f :% (a, b)) (f :% (b, a))
data MonoidObject f a = MonoidObject {
- unit :: Cod f (Unit f) a
- multiply :: Cod f (f :% (a, a)) a
}
trivialMonoid :: TensorProduct f => f -> MonoidObject f (Unit f)
coproductMonoid :: (HasInitialObject k, HasBinaryCoproducts k) => Obj k a -> MonoidObject (CoproductFunctor k) a
data ComonoidObject f a = ComonoidObject {
- counit :: Cod f a (Unit f)
- comultiply :: Cod f a (f :% (a, a))
}
trivialComonoid :: TensorProduct f => f -> ComonoidObject f (Unit f)
productComonoid :: (HasTerminalObject k, HasBinaryProducts k) => Obj k a -> ComonoidObject (ProductFunctor k) a
data MonoidAsCategory f m a b where
- MonoidValue :: (TensorProduct f, Dom f ~ (k :**: k), Cod f ~ k) => f -> MonoidObject f m -> k (Unit f) m -> MonoidAsCategory f m m m
type Monad f = MonoidObject (EndoFunctorCompose (Dom f)) f
mkMonad :: (Functor f, Dom f ~ k, Cod f ~ k) => f -> (forall a. Obj k a -> Component (Id k) f a) -> (forall a. Obj k a -> Component (f :.: f) f a) -> Monad f
monadFunctor :: Monad f -> f
idMonad :: Category k => Monad (Id k)
type Comonad f = ComonoidObject (EndoFunctorCompose (Dom f)) f
mkComonad :: (Functor f, Dom f ~ k, Cod f ~ k) => f -> (forall a. Obj k a -> Component f (Id k) a) -> (forall a. Obj k a -> Component f (f :.: f) a) -> Comonad f
idComonad :: Category k => Comonad (Id k)
adjunctionMonad :: Adjunction c d f g -> Monad (g :.: f)
adjunctionMonadT :: Dom m ~ c => Adjunction c d f g -> Monad m -> Monad ((g :.: m) :.: f)
adjunctionComonad :: Adjunction c d f g -> Comonad (f :.: g)
adjunctionComonadT :: Dom w ~ d => Adjunction c d f g -> Comonad w -> Comonad ((f :.: w) :.: g)

Documentation

class (Functor f, Dom f ~ (Cod f :**: Cod f)) => TensorProduct f where Source #

A monoidal category is a category with some kind of tensor product. A tensor product is a bifunctor, with a unit object.

Associated Types

type Unit f :: * Source #

Methods

unitObject :: f -> Obj (Cod f) (Unit f) Source #

leftUnitor :: Cod f ~ k => f -> Obj k a -> k (f :% (Unit f, a)) a Source #

leftUnitorInv :: Cod f ~ k => f -> Obj k a -> k a (f :% (Unit f, a)) Source #

rightUnitor :: Cod f ~ k => f -> Obj k a -> k (f :% (a, Unit f)) a Source #

rightUnitorInv :: Cod f ~ k => f -> Obj k a -> k a (f :% (a, Unit f)) Source #

associator :: Cod f ~ k => f -> Obj k a -> Obj k b -> Obj k c -> k (f :% (f :% (a, b), c)) (f :% (a, f :% (b, c))) Source #

associatorInv :: Cod f ~ k => f -> Obj k a -> Obj k b -> Obj k c -> k (f :% (a, f :% (b, c))) (f :% (f :% (a, b), c)) Source #

Instances

TensorProduct Add Source #	Ordinal addition makes the simplex category a monoidal category, with `0` as unit.
Instance details Defined in Data.Category.Simplex Associated Types type Unit Add :: Type Source # Methods unitObject :: Add -> Obj (Cod Add) (Unit Add) Source # leftUnitor :: Cod Add ~ k => Add -> Obj k a -> k (Add :% (Unit Add, a)) a Source # leftUnitorInv :: Cod Add ~ k => Add -> Obj k a -> k a (Add :% (Unit Add, a)) Source # rightUnitor :: Cod Add ~ k => Add -> Obj k a -> k (Add :% (a, Unit Add)) a Source # rightUnitorInv :: Cod Add ~ k => Add -> Obj k a -> k a (Add :% (a, Unit Add)) Source # associator :: Cod Add ~ k => Add -> Obj k a -> Obj k b -> Obj k c -> k (Add :% (Add :% (a, b), c)) (Add :% (a, Add :% (b, c))) Source # associatorInv :: Cod Add ~ k => Add -> Obj k a -> Obj k b -> Obj k c -> k (Add :% (a, Add :% (b, c))) (Add :% (Add :% (a, b), c)) Source #
TensorProduct Add Source #
Instance details Defined in Data.Category.Cube Associated Types type Unit Add :: Type Source # Methods unitObject :: Add -> Obj (Cod Add) (Unit Add) Source # leftUnitor :: Cod Add ~ k => Add -> Obj k a -> k (Add :% (Unit Add, a)) a Source # leftUnitorInv :: Cod Add ~ k => Add -> Obj k a -> k a (Add :% (Unit Add, a)) Source # rightUnitor :: Cod Add ~ k => Add -> Obj k a -> k (Add :% (a, Unit Add)) a Source # rightUnitorInv :: Cod Add ~ k => Add -> Obj k a -> k a (Add :% (a, Unit Add)) Source # associator :: Cod Add ~ k => Add -> Obj k a -> Obj k b -> Obj k c -> k (Add :% (Add :% (a, b), c)) (Add :% (a, Add :% (b, c))) Source # associatorInv :: Cod Add ~ k => Add -> Obj k a -> Obj k b -> Obj k c -> k (Add :% (a, Add :% (b, c))) (Add :% (Add :% (a, b), c)) Source #
Category k => TensorProduct (EndoFunctorCompose k) Source #	Functor composition makes the category of endofunctors monoidal, with the identity functor as unit.
Instance details Defined in Data.Category.Monoidal Associated Types type Unit (EndoFunctorCompose k) :: Type Source # Methods unitObject :: EndoFunctorCompose k -> Obj (Cod (EndoFunctorCompose k)) (Unit (EndoFunctorCompose k)) Source # leftUnitor :: Cod (EndoFunctorCompose k) ~ k0 => EndoFunctorCompose k -> Obj k0 a -> k0 (EndoFunctorCompose k :% (Unit (EndoFunctorCompose k), a)) a Source # leftUnitorInv :: Cod (EndoFunctorCompose k) ~ k0 => EndoFunctorCompose k -> Obj k0 a -> k0 a (EndoFunctorCompose k :% (Unit (EndoFunctorCompose k), a)) Source # rightUnitor :: Cod (EndoFunctorCompose k) ~ k0 => EndoFunctorCompose k -> Obj k0 a -> k0 (EndoFunctorCompose k :% (a, Unit (EndoFunctorCompose k))) a Source # rightUnitorInv :: Cod (EndoFunctorCompose k) ~ k0 => EndoFunctorCompose k -> Obj k0 a -> k0 a (EndoFunctorCompose k :% (a, Unit (EndoFunctorCompose k))) Source # associator :: Cod (EndoFunctorCompose k) ~ k0 => EndoFunctorCompose k -> Obj k0 a -> Obj k0 b -> Obj k0 c -> k0 (EndoFunctorCompose k :% (EndoFunctorCompose k :% (a, b), c)) (EndoFunctorCompose k :% (a, EndoFunctorCompose k :% (b, c))) Source # associatorInv :: Cod (EndoFunctorCompose k) ~ k0 => EndoFunctorCompose k -> Obj k0 a -> Obj k0 b -> Obj k0 c -> k0 (EndoFunctorCompose k :% (a, EndoFunctorCompose k :% (b, c))) (EndoFunctorCompose k :% (EndoFunctorCompose k :% (a, b), c)) Source #
(HasInitialObject k, HasBinaryCoproducts k) => TensorProduct (CoproductFunctor k) Source #	If a category has all coproducts, then the coproduct functor makes it a monoidal category, with the initial object as unit.
Instance details Defined in Data.Category.Monoidal Associated Types type Unit (CoproductFunctor k) :: Type Source # Methods unitObject :: CoproductFunctor k -> Obj (Cod (CoproductFunctor k)) (Unit (CoproductFunctor k)) Source # leftUnitor :: Cod (CoproductFunctor k) ~ k0 => CoproductFunctor k -> Obj k0 a -> k0 (CoproductFunctor k :% (Unit (CoproductFunctor k), a)) a Source # leftUnitorInv :: Cod (CoproductFunctor k) ~ k0 => CoproductFunctor k -> Obj k0 a -> k0 a (CoproductFunctor k :% (Unit (CoproductFunctor k), a)) Source # rightUnitor :: Cod (CoproductFunctor k) ~ k0 => CoproductFunctor k -> Obj k0 a -> k0 (CoproductFunctor k :% (a, Unit (CoproductFunctor k))) a Source # rightUnitorInv :: Cod (CoproductFunctor k) ~ k0 => CoproductFunctor k -> Obj k0 a -> k0 a (CoproductFunctor k :% (a, Unit (CoproductFunctor k))) Source # associator :: Cod (CoproductFunctor k) ~ k0 => CoproductFunctor k -> Obj k0 a -> Obj k0 b -> Obj k0 c -> k0 (CoproductFunctor k :% (CoproductFunctor k :% (a, b), c)) (CoproductFunctor k :% (a, CoproductFunctor k :% (b, c))) Source # associatorInv :: Cod (CoproductFunctor k) ~ k0 => CoproductFunctor k -> Obj k0 a -> Obj k0 b -> Obj k0 c -> k0 (CoproductFunctor k :% (a, CoproductFunctor k :% (b, c))) (CoproductFunctor k :% (CoproductFunctor k :% (a, b), c)) Source #
(HasTerminalObject k, HasBinaryProducts k) => TensorProduct (ProductFunctor k) Source #	If a category has all products, then the product functor makes it a monoidal category, with the terminal object as unit.
Instance details Defined in Data.Category.Monoidal Associated Types type Unit (ProductFunctor k) :: Type Source # Methods unitObject :: ProductFunctor k -> Obj (Cod (ProductFunctor k)) (Unit (ProductFunctor k)) Source # leftUnitor :: Cod (ProductFunctor k) ~ k0 => ProductFunctor k -> Obj k0 a -> k0 (ProductFunctor k :% (Unit (ProductFunctor k), a)) a Source # leftUnitorInv :: Cod (ProductFunctor k) ~ k0 => ProductFunctor k -> Obj k0 a -> k0 a (ProductFunctor k :% (Unit (ProductFunctor k), a)) Source # rightUnitor :: Cod (ProductFunctor k) ~ k0 => ProductFunctor k -> Obj k0 a -> k0 (ProductFunctor k :% (a, Unit (ProductFunctor k))) a Source # rightUnitorInv :: Cod (ProductFunctor k) ~ k0 => ProductFunctor k -> Obj k0 a -> k0 a (ProductFunctor k :% (a, Unit (ProductFunctor k))) Source # associator :: Cod (ProductFunctor k) ~ k0 => ProductFunctor k -> Obj k0 a -> Obj k0 b -> Obj k0 c -> k0 (ProductFunctor k :% (ProductFunctor k :% (a, b), c)) (ProductFunctor k :% (a, ProductFunctor k :% (b, c))) Source # associatorInv :: Cod (ProductFunctor k) ~ k0 => ProductFunctor k -> Obj k0 a -> Obj k0 b -> Obj k0 c -> k0 (ProductFunctor k :% (a, ProductFunctor k :% (b, c))) (ProductFunctor k :% (ProductFunctor k :% (a, b), c)) Source #
(TensorProduct t, Cod t ~ f (Fix f)) => TensorProduct (WrapTensor (Fix f) t) Source #	`Fix f` inherits tensor products from `f (Fix f)`.
Instance details Defined in Data.Category.Fix Associated Types type Unit (WrapTensor (Fix f) t) :: Type Source # Methods unitObject :: WrapTensor (Fix f) t -> Obj (Cod (WrapTensor (Fix f) t)) (Unit (WrapTensor (Fix f) t)) Source # leftUnitor :: Cod (WrapTensor (Fix f) t) ~ k => WrapTensor (Fix f) t -> Obj k a -> k (WrapTensor (Fix f) t :% (Unit (WrapTensor (Fix f) t), a)) a Source # leftUnitorInv :: Cod (WrapTensor (Fix f) t) ~ k => WrapTensor (Fix f) t -> Obj k a -> k a (WrapTensor (Fix f) t :% (Unit (WrapTensor (Fix f) t), a)) Source # rightUnitor :: Cod (WrapTensor (Fix f) t) ~ k => WrapTensor (Fix f) t -> Obj k a -> k (WrapTensor (Fix f) t :% (a, Unit (WrapTensor (Fix f) t))) a Source # rightUnitorInv :: Cod (WrapTensor (Fix f) t) ~ k => WrapTensor (Fix f) t -> Obj k a -> k a (WrapTensor (Fix f) t :% (a, Unit (WrapTensor (Fix f) t))) Source # associator :: Cod (WrapTensor (Fix f) t) ~ k => WrapTensor (Fix f) t -> Obj k a -> Obj k b -> Obj k c -> k (WrapTensor (Fix f) t :% (WrapTensor (Fix f) t :% (a, b), c)) (WrapTensor (Fix f) t :% (a, WrapTensor (Fix f) t :% (b, c))) Source # associatorInv :: Cod (WrapTensor (Fix f) t) ~ k => WrapTensor (Fix f) t -> Obj k a -> Obj k b -> Obj k c -> k (WrapTensor (Fix f) t :% (a, WrapTensor (Fix f) t :% (b, c))) (WrapTensor (Fix f) t :% (WrapTensor (Fix f) t :% (a, b), c)) Source #

class TensorProduct f => SymmetricTensorProduct f where Source #

Methods

swap :: Cod f ~ k => f -> Obj k a -> Obj k b -> k (f :% (a, b)) (f :% (b, a)) Source #

Instances

(HasInitialObject k, HasBinaryCoproducts k) => SymmetricTensorProduct (CoproductFunctor k) Source #
Instance details Defined in Data.Category.Monoidal Methods swap :: Cod (CoproductFunctor k) ~ k0 => CoproductFunctor k -> Obj k0 a -> Obj k0 b -> k0 (CoproductFunctor k :% (a, b)) (CoproductFunctor k :% (b, a)) Source #
(HasTerminalObject k, HasBinaryProducts k) => SymmetricTensorProduct (ProductFunctor k) Source #
Instance details Defined in Data.Category.Monoidal Methods swap :: Cod (ProductFunctor k) ~ k0 => ProductFunctor k -> Obj k0 a -> Obj k0 b -> k0 (ProductFunctor k :% (a, b)) (ProductFunctor k :% (b, a)) Source #

data MonoidObject f a Source #

MonoidObject f a defines a monoid a in a monoidal category with tensor product f.

Constructors

MonoidObject
Fields unit :: Cod f (Unit f) a multiply :: Cod f (f :% (a, a)) a

trivialMonoid :: TensorProduct f => f -> MonoidObject f (Unit f) Source #

coproductMonoid :: (HasInitialObject k, HasBinaryCoproducts k) => Obj k a -> MonoidObject (CoproductFunctor k) a Source #

data ComonoidObject f a Source #

ComonoidObject f a defines a comonoid a in a comonoidal category with tensor product f.

Constructors

ComonoidObject
Fields counit :: Cod f a (Unit f) comultiply :: Cod f a (f :% (a, a))

trivialComonoid :: TensorProduct f => f -> ComonoidObject f (Unit f) Source #

productComonoid :: (HasTerminalObject k, HasBinaryProducts k) => Obj k a -> ComonoidObject (ProductFunctor k) a Source #

data MonoidAsCategory f m a b where Source #

Constructors

MonoidValue :: (TensorProduct f, Dom f ~ (k :**: k), Cod f ~ k) => f -> MonoidObject f m -> k (Unit f) m -> MonoidAsCategory f m m m

Instances

Category (MonoidAsCategory f m :: Type -> Type -> Type) Source #	A monoid as a category with one object.
Instance details Defined in Data.Category.Monoidal Methods src :: MonoidAsCategory f m a b -> Obj (MonoidAsCategory f m) a Source # tgt :: MonoidAsCategory f m a b -> Obj (MonoidAsCategory f m) b Source # (.) :: MonoidAsCategory f m b c -> MonoidAsCategory f m a b -> MonoidAsCategory f m a c Source #

type Monad f = MonoidObject (EndoFunctorCompose (Dom f)) f Source #

A monad is a monoid in the category of endofunctors.

mkMonad :: (Functor f, Dom f ~ k, Cod f ~ k) => f -> (forall a. Obj k a -> Component (Id k) f a) -> (forall a. Obj k a -> Component (f :.: f) f a) -> Monad f Source #

monadFunctor :: Monad f -> f Source #

idMonad :: Category k => Monad (Id k) Source #

type Comonad f = ComonoidObject (EndoFunctorCompose (Dom f)) f Source #

A comonad is a comonoid in the category of endofunctors.

mkComonad :: (Functor f, Dom f ~ k, Cod f ~ k) => f -> (forall a. Obj k a -> Component f (Id k) a) -> (forall a. Obj k a -> Component f (f :.: f) a) -> Comonad f Source #

idComonad :: Category k => Comonad (Id k) Source #

adjunctionMonad :: Adjunction c d f g -> Monad (g :.: f) Source #

Every adjunction gives rise to an associated monad.

adjunctionMonadT :: Dom m ~ c => Adjunction c d f g -> Monad m -> Monad ((g :.: m) :.: f) Source #

Every adjunction gives rise to an associated monad transformer.

adjunctionComonad :: Adjunction c d f g -> Comonad (f :.: g) Source #

Every adjunction gives rise to an associated comonad.

adjunctionComonadT :: Dom w ~ d => Adjunction c d f g -> Comonad w -> Comonad ((f :.: w) :.: g) Source #

Every adjunction gives rise to an associated comonad transformer.