Data.Functor.Monoidal

Semigroupal

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

Given monoidal categories \((\mathcal{C}, \otimes, I_{\mathcal{C}})\) and \((\mathcal{D}, \bullet, I_{\mathcal{D}})\). A functor \(F : \mathcal{C} \to \mathcal{D}\) is Semigroupal if it supports a natural transformation \(\phi_{A,B} : F\ A \bullet F\ B \to F\ (A \otimes B)\), which we call combine.

Laws

Associativity:

\[ \require{AMScd} \begin{CD} (F A \bullet F B) \bullet F C @>>{\alpha_{\mathcal{D}}}> F A \bullet (F B \bullet F C) \\ @VV{\phi_{A,B} \bullet 1}V @VV{1 \bullet \phi_{B,C}}V \\ F (A \otimes B) \bullet F C @. F A \bullet (F (B \otimes C)) \\ @VV{\phi_{A \otimes B,C}}V @VV{\phi_{A,B \otimes C}}V \\ F ((A \otimes B) \otimes C) @>>{F \alpha_{\mathcal{C}}}> F (A \otimes (B \otimes C)) \\ \end{CD} \]

combine . grmap combine . bwd assoc ≡ fmap (bwd assoc) . combine . glmap combine

Methods

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

combine is a natural transformation from functors \(\mathcal{C} × \mathcal{C}\) to \(\mathcal{D}\).

Examples

>>> combine @(->) @(,) @(,) @Maybe (Just True, Just "hello")
Just (True,"hello")

>>> combine @(->) @(,) @(,) @Maybe (Just True, Nothing)
Nothing

>>> combine @(->) @Either @(,) @Maybe (Just True, Nothing)
Just (Left True)

>>> combine @(->) @Either @(,) @Maybe (Nothing, Just "hello")
Just (Right "hello")

Instances

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

Defined in Data.Functor.Monoidal

Methods

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

Semialign f => Semigroupal (->) These (,) f

Defined in Data.Functor.Monoidal

Methods

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

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

Defined in Data.Functor.Monoidal

Methods

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

Unital

class Unital cat i1 i0 f where

Given monoidal categories \((\mathcal{C}, \otimes, I_{\mathcal{C}})\) and \((\mathcal{D}, \bullet, I_{\mathcal{D}})\). A functor \(F : \mathcal{C} \to \mathcal{D}\) is Unital if it supports a morphism \(\phi : I_{\mathcal{D}} \to F\ I_{\mathcal{C}}\), which we call introduce.

Methods

introduce :: cat i0 (f i1)

introduce maps from the identity in \(\mathcal{C}\) to the identity in \(\mathcal{D}\).

Examples

>>> introduce @(->) @() @() @Maybe ()
Just ()

>>> :t introduce @(->) @Void @() @Maybe
introduce @(->) @Void @() @Maybe :: () -> Maybe Void

>>> introduce @(->) @Void @() @Maybe ()
Nothing

Instances

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

Defined in Data.Functor.Monoidal

Methods

introduce :: () -> f Void

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

Defined in Data.Functor.Monoidal

Methods

introduce :: () -> f ()

Monoidal

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

Given monoidal categories \((\mathcal{C}, \otimes, I_{\mathcal{C}})\) and \((\mathcal{D}, \bullet, I_{\mathcal{D}})\). A functor \(F : \mathcal{C} \to \mathcal{D}\) is Monoidal if it maps between \(\mathcal{C}\) and \(\mathcal{D}\) while preserving their monoidal structure. Eg., a homomorphism of monoidal categories.

See NCatlab for more details.

Laws

Right Unitality:

\[ \require{AMScd} \begin{CD} F A \bullet I_{\mathcal{D}} @>{1 \bullet \phi}>> F A \bullet F I_{\mathcal{C}};\\ @VV{\rho_{\mathcal{D}}}V @VV{\phi A,I_{\mathcal{C}}}V \\ F A @<<{F \rho_{\mathcal{C}}}< F (A \otimes I_{\mathcal{C}}); \end{CD} \]

  combine . grmap introduce ≡ bwd unitr . fwd unitr

Left Unitality:

\[ \begin{CD} I_{\mathcal{D}} \bullet F B @>{\phi \bullet 1}>> F I_{\mathcal{C}} \bullet F B;\\ @VV{\lambda_{\mathcal{D}}}V @VV{\phi I_{\mathcal{C}},B}V \\ F A @<<{F \lambda_{\mathcal{C}}}< F (A \otimes I_{\mathcal{C}} \otimes B); \end{CD} \]

  combine . glmap introduce ≡ fmap (bwd unitl) . fwd unitl

Instances

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

Defined in Data.Functor.Monoidal

(Alternative f, Semialign f) => Monoidal (->) These Void (,) () f

Defined in Data.Functor.Monoidal

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

Defined in Data.Functor.Monoidal