Data.Category.Monoidal

Documentation

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

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 :: *

Methods

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))

Instances

TensorProduct Add

Ordinal addition makes the simplex category a monoidal category, with 0 as unit.

TensorProduct Add

Category k => TensorProduct (EndoFunctorCompose k)

Functor composition makes the category of endofunctors monoidal, with the identity functor as unit.

(HasInitialObject k, HasBinaryCoproducts k) => TensorProduct (CoproductFunctor k)

If a category has all coproducts, then the coproduct functor makes it a monoidal category, with the initial object as unit.

(HasTerminalObject k, HasBinaryProducts k) => TensorProduct (ProductFunctor k)

If a category has all products, then the product functor makes it a monoidal category, with the terminal object as unit.

data MonoidObject f a

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

Constructors

MonoidObject

Fields

unit :: Cod f ~ k => k (Unit f) a

multiply :: Cod f ~ k => k (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 f a defines a comonoid a in a comonoidal category with tensor product f.

Constructors

ComonoidObject

Fields

counit :: Cod f ~ k => k a (Unit f)

comultiply :: Cod f ~ k => k 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

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)

A monoid as a category with one object.

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

A monad is a monoid in the category of endofunctors.

mkMonad :: (Functor f, Dom f ~ k, Cod f ~ k, Category 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

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

A comonad is a comonoid in the category of endofunctors.

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

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

Every adjunction gives rise to an associated monad.

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

Every adjunction gives rise to an associated comonad.