Data.Functor.Monoidal

Documentation

class (Tensor t1 i1 cat, Tensor t0 i0 cat, Semigroupal cat t1 t0 f, Unital cat i1 i0 f) => Monoidal cat t1 i1 t0 i0 f

A Monoidal Functor is a Functor between two Monoidal Categories which preserves the monoidal structure. Eg., a homomorphism of monoidal categories.

Laws

Associativity: combine (combine fx fy) fz ⟶ combine fx (combine fy fz) ↓ ↓ f (x t1 y) t1 fz combine fx (f (y t1 z)) ↓ ↓ f ((x t1 y) t1 z) ⟶ (f x t1 (y t1 z))

Left Unitality: empty t1 f x ⟶ f empty t1 f x ↓ ↓ f x ← f (empty t0 x)

Right Unitality: f x t1 empty ⟶ f x t1 f empty ↓ ↓ f x ← f (x t0 empty)

Instances

Alternative f => Monoidal ((->) :: Type -> Type -> Type) Either Void (,) () f

Defined in Data.Functor.Monoidal

Applicative f => Monoidal ((->) :: Type -> Type -> Type) (,) () (,) () f

Defined in Data.Functor.Monoidal

class (Associative t1 cat, Associative t0 cat) => Semigroupal cat t1 t0 f where

Methods

combine :: (f x `t0` f x') `cat` f (x `t1` x')

Instances

Alternative f => Semigroupal ((->) :: Type -> Type -> Type) Either (,) f

Defined in Data.Functor.Monoidal

Methods

combine :: (f x, f x') -> f (Either x x')

Applicative f => Semigroupal ((->) :: Type -> Type -> Type) (,) (,) f

Defined in Data.Functor.Monoidal

Methods

combine :: (f x, f x') -> f (x, x')

class Unital cat i1 i0 f where

Methods

introduce :: i0 `cat` f i1

Instances

Applicative f => Unital ((->) :: Type -> Type -> Type) () () f

Defined in Data.Functor.Monoidal

Methods

introduce :: () -> f ()

Alternative f => Unital ((->) :: Type -> Type -> Type) Void () f

Defined in Data.Functor.Monoidal

Methods

introduce :: () -> f Void