{-# OPTIONS --without-K --safe #-}
 
module Categories.Functor.Monoidal where
 
open import Level
open import Data.Product using (Σ; _,_)
 
open import Categories.Category
open import Categories.Category.Product
open import Categories.Category.Monoidal
 
open import Categories.Functor hiding (id)
open import Categories.NaturalTransformation hiding (id)
open import Categories.NaturalTransformation.NaturalIsomorphism
 
import Categories.Morphism as Mor
 
private
  variable
    o o′ ℓ ℓ′ e e′ : Level
 
module _  (C : MonoidalCategory o ℓ e) (D : MonoidalCategory o′ ℓ′ e′) where
  private
    module C = MonoidalCategory C
    module D = MonoidalCategory D
    open Mor D.U
 
  -- lax monoidal functor
  record IsMonoidalFunctor (F : Functor C.U D.U) : Set (o ⊔ ℓ ⊔ ℓ′ ⊔ e′) where
    open Functor F
 
    field
      ε      : D.U [ D.unit , F₀ C.unit ]
      ⊗-homo : NaturalTransformation (D.⊗ ∘F (F ⁂ F)) (F ∘F C.⊗)
 
    module ⊗-homo = NaturalTransformation ⊗-homo
 
    -- coherence condition
    open D
    open Commutation D.U
 
    field
      associativity : ∀ {X Y Z} →
                      [ (F₀ X ⊗₀ F₀ Y) ⊗₀ F₀ Z ⇒ F₀ (X C.⊗₀ Y C.⊗₀ Z) ]⟨
                        ⊗-homo.η (X , Y) ⊗₁ id                        ⇒⟨ F₀ (X C.⊗₀ Y) ⊗₀ F₀ Z ⟩
                        ⊗-homo.η (X C.⊗₀ Y , Z)                       ⇒⟨ F₀ ((X C.⊗₀ Y) C.⊗₀ Z) ⟩
                        F₁ C.associator.from
                      ≈ associator.from                               ⇒⟨ F₀ X ⊗₀ F₀ Y ⊗₀ F₀ Z ⟩
                        id ⊗₁ ⊗-homo.η (Y , Z)                        ⇒⟨ F₀ X ⊗₀ F₀ (Y C.⊗₀ Z) ⟩
                        ⊗-homo.η (X , Y C.⊗₀ Z)
                      ⟩
      unitaryˡ      : ∀ {X} →
                      [ unit ⊗₀ F₀ X ⇒ F₀ X   ]⟨
                        ε ⊗₁ id               ⇒⟨ F₀ C.unit ⊗₀ F₀ X ⟩
                        ⊗-homo.η (C.unit , X) ⇒⟨ F₀ (C.unit C.⊗₀ X) ⟩
                        F₁ C.unitorˡ.from
                      ≈ unitorˡ.from
                      ⟩
      unitaryʳ      : ∀ {X} →
                      [ F₀ X ⊗₀ unit ⇒ F₀ X   ]⟨
                        id ⊗₁ ε               ⇒⟨ F₀ X ⊗₀ F₀ C.unit ⟩
                        ⊗-homo.η (X , C.unit) ⇒⟨ F₀ (X C.⊗₀ C.unit) ⟩
                        F₁ C.unitorʳ.from
                      ≈ unitorʳ.from
                      ⟩
 
  record MonoidalFunctor : Set (o ⊔ ℓ ⊔ e ⊔ o′ ⊔ ℓ′ ⊔ e′) where
    field
      F          : Functor C.U D.U
      isMonoidal : IsMonoidalFunctor F
 
    open Functor F public
    open IsMonoidalFunctor isMonoidal public
 
 
  -- strong monoidal functor
  record IsStrongMonoidalFunctor (F : Functor C.U D.U) : Set (o ⊔ ℓ ⊔ ℓ′ ⊔ e′) where
    open Functor F
 
    field
      ε      : D.unit ≅ F₀ C.unit
      ⊗-homo : D.⊗ ∘F (F ⁂ F) ≃ F ∘F C.⊗
 
    module ε      = _≅_ ε
    module ⊗-homo = NaturalIsomorphism ⊗-homo
 
    -- coherence condition
    open D
    open Commutation D.U
 
    field
      associativity : ∀ {X Y Z} →
                      [ (F₀ X ⊗₀ F₀ Y) ⊗₀ F₀ Z ⇒ F₀ (X C.⊗₀ Y C.⊗₀ Z) ]⟨
                        ⊗-homo.⇒.η (X , Y) ⊗₁ id                        ⇒⟨ F₀ (X C.⊗₀ Y) ⊗₀ F₀ Z ⟩
                        ⊗-homo.⇒.η (X C.⊗₀ Y , Z)                      ⇒⟨ F₀ ((X C.⊗₀ Y) C.⊗₀ Z) ⟩
                        F₁ C.associator.from
                      ≈ associator.from                                 ⇒⟨ F₀ X ⊗₀ F₀ Y ⊗₀ F₀ Z ⟩
                        id ⊗₁ ⊗-homo.⇒.η (Y , Z)                        ⇒⟨ F₀ X ⊗₀ F₀ (Y C.⊗₀ Z) ⟩
                        ⊗-homo.⇒.η (X , Y C.⊗₀ Z)
                      ⟩
      unitaryˡ      : ∀ {X} →
                      [ unit ⊗₀ F₀ X ⇒ F₀ X      ]⟨
                        ε.from ⊗₁ id             ⇒⟨ F₀ C.unit ⊗₀ F₀ X ⟩
                        ⊗-homo.⇒.η (C.unit , X) ⇒⟨ F₀ (C.unit C.⊗₀ X) ⟩
                        F₁ C.unitorˡ.from
                      ≈ unitorˡ.from
                      ⟩
      unitaryʳ      : ∀ {X} →
                      [ F₀ X ⊗₀ unit ⇒ F₀ X      ]⟨
                        id ⊗₁ ε.from             ⇒⟨ F₀ X ⊗₀ F₀ C.unit ⟩
                        ⊗-homo.⇒.η (X , C.unit) ⇒⟨ F₀ (X C.⊗₀ C.unit) ⟩
                        F₁ C.unitorʳ.from
                      ≈ unitorʳ.from
                      ⟩
 
    isMonoidal : IsMonoidalFunctor F
    isMonoidal = record
      { ε             = ε.from
      ; ⊗-homo        = ⊗-homo.F⇒G
      ; associativity = associativity
      ; unitaryˡ      = unitaryˡ
      ; unitaryʳ      = unitaryʳ
      }
 
  record StrongMonoidalFunctor : Set (o ⊔ ℓ ⊔ e ⊔ o′ ⊔ ℓ′ ⊔ e′) where
    field
      F : Functor C.U D.U
      isStrongMonoidal : IsStrongMonoidalFunctor F
 
    open Functor F public
    open IsStrongMonoidalFunctor isStrongMonoidal public
 
    monoidalFunctor : MonoidalFunctor
    monoidalFunctor = record { F = F ; isMonoidal = isMonoidal }