open import Cat.Diagram.Product.Solver
open import Cat.Instances.Functor
open import Cat.Diagram.Terminal
open import Cat.Monoidal.Base
open import Cat.Prelude

import Cat.Diagram.Product
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 $n$-categories is an $(n+1)$-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} where
  open Cat.Diagram.Product C
  open Monoidal-category
  open make-natural-iso
  open Cr C

  Cartesian-monoidal : (∀ A B → Product A B) → Terminal C → Monoidal-category C
  Cartesian-monoidal prods term = mon where
    open Binary-products prods
    open Terminal term
    mon : Monoidal-category C
    mon .-⊗- = ×-functor
    mon .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:

    mon .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)
    mon .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 _ _)
    mon .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) ⟩ ∎
    mon .triangle = Product.unique (prods _ _) _
      (pulll π₁∘⟨⟩ ·· pullr π₁∘⟨⟩ ·· π₁∘⟨⟩ ∙ introl refl)
      (pulll π₂∘⟨⟩ ·· pullr π₂∘⟨⟩ ·· idl _)
    mon .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 ⟩ ∎