open import Cat.Instances.Sets.Complete
open import Cat.Diagram.Product.Solver
open import Cat.Monoidal.Diagonals
open import Cat.Instances.Functor
open import Cat.Diagram.Terminal
open import Cat.Monoidal.Braided
open import Cat.Diagram.Product
open import Cat.Monoidal.Base
open import Cat.Prelude

import Cat.Reasoning as Cr

module Cat.Monoidal.Instances.Cartesian where

Cartesian monoidal categories🔗

Unlike with categories and bicategories, there is no handy example of monoidal category that is as canonical as how the collection of all $n\text{-categories}$ is an $(n+1)\text{-category.}$ However, we do have a certain canonical pool of examples to draw from: all the Cartesian monoidal categories, also known as finite-products categories.

module _
  {o ℓ} {C : Precategory o ℓ}
  (prods : ∀ A B → Product C A B) (term : Terminal C)
  where

  open Monoidal-category hiding (_⊗₁_)
  open Braided-monoidal
  open Symmetric-monoidal
  open Diagonals hiding (δ)
  open make-natural-iso
  open Cr C
  open Binary-products C prods
  open Terminal term

  Cartesian-monoidal : Monoidal-category C
  Cartesian-monoidal .-⊗- = ×-functor
  Cartesian-monoidal .Unit = top

There’s nothing much to say about this result: It’s pretty much just banging out the calculation. Our tensor product functor is the Cartesian product functor, and the tensor unit is the terminal object (the empty product). Associators and units are the evident maps, which are coherent by the properties of limits. Translating this intuitive explanation to a formal proof requires a lot of calculation, however:

  Cartesian-monoidal .unitor-l = to-natural-iso ni where
    ni : make-natural-iso _ _
    ni .eta x = ⟨ ! , id ⟩
    ni .inv x = π₂
    ni .eta∘inv x = Product.unique₂ (prods _ _)
      (pulll π₁∘⟨⟩ ∙ sym (!-unique _)) (cancell π₂∘⟨⟩) (!-unique₂ _ _) (idr _)
    ni .inv∘eta x = π₂∘⟨⟩
    ni .natural x y f = Product.unique₂ (prods _ _)
      (pulll π₁∘⟨⟩ ∙ pullr π₁∘⟨⟩ ∙ idl _) (pulll π₂∘⟨⟩ ∙ cancelr π₂∘⟨⟩)
      (!-unique₂ _ _) (pulll π₂∘⟨⟩ ∙ idl f)
  Cartesian-monoidal .unitor-r = to-natural-iso ni where
    ni : make-natural-iso _ _
    ni .eta x = ⟨ id , ! ⟩
    ni .inv x = π₁
    ni .eta∘inv x = Product.unique₂ (prods _ _)
      (pulll π₁∘⟨⟩ ∙ idl _) (pulll π₂∘⟨⟩ ∙ sym (!-unique _))
      (idr _) (sym (!-unique _))
    ni .inv∘eta x = π₁∘⟨⟩
    ni .natural x y f = Product.unique₂ (prods _ _)
      (pulll π₁∘⟨⟩ ·· pullr π₁∘⟨⟩ ·· idr f)
      (pulll π₂∘⟨⟩ ·· pullr π₂∘⟨⟩ ·· idl !)
      (pulll π₁∘⟨⟩ ∙ idl f)
      (!-unique₂ _ _)
  Cartesian-monoidal .associator = to-natural-iso ni where
    ni : make-natural-iso _ _
    ni .eta x = ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩
    ni .inv x = ⟨ ⟨ π₁ , π₁ ∘ π₂ ⟩ , π₂ ∘ π₂ ⟩
    ni .eta∘inv x =
      ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ ⟨ ⟨ π₁ , π₁ ∘ π₂ ⟩ , π₂ ∘ π₂ ⟩ ≡⟨ products! C prods ⟩≡
      id ∎
    ni .inv∘eta x =
      ⟨ ⟨ π₁ , π₁ ∘ π₂ ⟩ , π₂ ∘ π₂ ⟩ ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ≡⟨ products! C prods ⟩≡
      id ∎
    ni .natural x y f =
      ⟨ f .fst ∘ π₁ , ⟨ f .snd .fst ∘ π₁ , f .snd .snd ∘ π₂ ⟩ ∘ π₂ ⟩ ∘ ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩     ≡⟨ products! C prods ⟩≡
      ⟨ π₁ ∘ π₁ , ⟨ π₂ ∘ π₁ , π₂ ⟩ ⟩ ∘ ⟨ (⟨ f .fst ∘ π₁ , f .snd .fst ∘ π₂ ⟩ ∘ π₁) , (f .snd .snd ∘ π₂) ⟩ ∎
  Cartesian-monoidal .triangle = Product.unique (prods _ _) _
    (pulll π₁∘⟨⟩ ·· pullr π₁∘⟨⟩ ·· π₁∘⟨⟩ ∙ introl refl)
    (pulll π₂∘⟨⟩ ·· pullr π₂∘⟨⟩ ·· idl _)
  Cartesian-monoidal .pentagon =
    ⟨ ⟨ ⟨ π₁ , π₁ ∘ π₂ ⟩ , π₂ ∘ π₂ ⟩ ∘ π₁ , id ∘ π₂ ⟩ ∘ ⟨ ⟨ π₁ , π₁ ∘ π₂ ⟩ , π₂ ∘ π₂ ⟩ ∘ ⟨ id ∘ π₁ , ⟨ ⟨ π₁ , π₁ ∘ π₂ ⟩ , π₂ ∘ π₂ ⟩ ∘ π₂ ⟩ ≡⟨ products! C prods ⟩≡
    ⟨ ⟨ π₁ , π₁ ∘ π₂ ⟩ , π₂ ∘ π₂ ⟩ ∘ ⟨ ⟨ π₁ , π₁ ∘ π₂ ⟩ , π₂ ∘ π₂ ⟩ ∎

Cartesian monoidal categories also inherit a lot of additional structure from the categorical product. In particular, they are symmetric monoidal categories.

  Cartesian-symmetric : Symmetric-monoidal Cartesian-monoidal
  Cartesian-symmetric = to-symmetric-monoidal mk where
    swap-natural
      : ∀ {A B C D} ((f , g) : Hom A C × Hom B D)
      → (g ⊗₁ f) ∘ swap ≡ swap ∘ (f ⊗₁ g)
    swap-natural (f , g) =
      (g ⊗₁ f) ∘ swap                       ≡⟨ ⟨⟩∘ _ ⟩≡
      ⟨ (g ∘ π₁) ∘ swap , (f ∘ π₂) ∘ swap ⟩ ≡⟨ ap₂ ⟨_,_⟩ (pullr π₁∘⟨⟩) (pullr π₂∘⟨⟩) ⟩≡
      ⟨ g ∘ π₂ , f ∘ π₁ ⟩                   ≡˘⟨ ap₂ ⟨_,_⟩ π₂∘⟨⟩ π₁∘⟨⟩ ⟩≡˘
      ⟨ π₂ ∘ (f ⊗₁ g) , π₁ ∘ (f ⊗₁ g) ⟩     ≡˘⟨ ⟨⟩∘ _ ⟩≡˘
      swap ∘ (f ⊗₁ g)                       ∎

    open make-symmetric-monoidal
    mk : make-symmetric-monoidal Cartesian-monoidal
    mk .has-braiding = iso→isoⁿ
      (λ _ → invertible→iso swap swap-is-iso) swap-natural
    mk .symmetric = ⟨⟩∘ _ ∙ ap₂ ⟨_,_⟩ π₂∘⟨⟩ π₁∘⟨⟩ ∙ ⟨⟩-η
    mk .has-braiding-α→ = products! C prods

We also have a system of diagonal morphisms:

  Cartesian-diagonal : Diagonals Cartesian-monoidal
  Cartesian-diagonal .diagonals ._=>_.η A = δ
  Cartesian-diagonal .diagonals ._=>_.is-natural A B f = products! C prods
  Cartesian-diagonal .diagonal-λ→ = ap ⟨_, id ⟩ (sym (!-unique _))

Setsₓ : ∀ {ℓ} → Monoidal-category (Sets ℓ)
Setsₓ = Cartesian-monoidal Sets-products Sets-terminal