open import Cat.Instances.Product
open import Cat.Prelude

module Cat.Diagram.Product {o h} (C : Precategory o h) where

Products🔗

The product $P$ of two objects $A$ and $B$, if it exists, is the smallest object equipped with “projection” maps $P \to A$ and $P \to B$. This situation can be visualised by putting the data of a product into a commutative diagram, as the one below: To express that $P$ is the smallest object with projections to $A$ and $B$, we ask that any other object $Q$ with projections through $A$ and $B$ factors uniquely through $P$:

In the sense that (univalent) categories generalise posets, the product of $A$ and $B$ — if it exists — generalises the binary meet $A \wedge B$. Since products are unique when they exist, we may safely denote any product of $A$ and $B$ by $A \times B$.

For a diagram $A {\leftarrow}A \times B \to B$ to be a product diagram, it must be able to cough up an arrow $Q \to P$ given the data of another span $A {\leftarrow}Q \to B$, which must not only fit into the diagram above but be unique among the arrows that do so.

This factoring is called the pairing of the arrows $f : Q \to A$ and $g : Q \to B$, since in the special case where $Q$ is the terminal object (hence the two arrows are global elements of $A$ resp. $B$), the pairing $\langle f, g \rangle$ is a global element of the product $A \times B$.

record is-product {A B P} (π₁ : Hom P A) (π₂ : Hom P B) : Type (o ⊔ h) where
  field
    ⟨_,_⟩     : ∀ {Q} (p1 : Hom Q A) (p2 : Hom Q B) → Hom Q P
    π₁∘factor : ∀ {Q} {p1 : Hom Q _} {p2} → π₁ ∘ ⟨ p1 , p2 ⟩ ≡ p1
    π₂∘factor : ∀ {Q} {p1 : Hom Q _} {p2} → π₂ ∘ ⟨ p1 , p2 ⟩ ≡ p2

    unique : ∀ {Q} {p1 : Hom Q A} {p2}
           → (other : Hom Q P)
           → π₁ ∘ other ≡ p1
           → π₂ ∘ other ≡ p2
           → other ≡ ⟨ p1 , p2 ⟩

  unique₂ : ∀ {Q} {pr1 : Hom Q A} {pr2}
          → ∀ {o1} (p1 : π₁ ∘ o1 ≡ pr1) (q1 : π₂ ∘ o1 ≡ pr2)
          → ∀ {o2} (p2 : π₁ ∘ o2 ≡ pr1) (q2 : π₂ ∘ o2 ≡ pr2)
          → o1 ≡ o2
  unique₂ p1 q1 p2 q2 = unique _ p1 q1 ∙ sym (unique _ p2 q2)

  ⟨⟩∘ : ∀ {Q R} {p1 : Hom Q A} {p2 : Hom Q B} (f : Hom R Q)
      → ⟨ p1 , p2 ⟩ ∘ f ≡ ⟨ p1 ∘ f , p2 ∘ f ⟩
  ⟨⟩∘ f = unique _ (pulll π₁∘factor) (pulll π₂∘factor)

A product of $A$ and $B$ is an explicit choice of product diagram:

record Product (A B : Ob) : Type (o ⊔ h) where
  no-eta-equality
  field
    apex : Ob
    π₁ : Hom apex A
    π₂ : Hom apex B
    has-is-product : is-product π₁ π₂

  open is-product has-is-product public

open Product hiding (⟨_,_⟩ ; π₁ ; π₂ ; ⟨⟩∘)

Uniqueness🔗

Products, when they exist, are unique. It’s easiest to see this with a diagrammatic argument: If we have product diagrams $A {\leftarrow}P \to B$ and $A {\leftarrow}P' \to B$, we can fit them into a “commutative diamond” like the one below:

Since both $P$ and $P'$ are products, we know that the dashed arrows in the diagram below exist, so the overall diagram commutes: hence we have an isomorphism $P \cong P'$.

We construct the map $P \to P'$ as the pairing of the projections from $P$, and symmetrically for $P' \to P$.

×-Unique : (p1 p2 : Product A B) → apex p1 ≅ apex p2
×-Unique p1 p2 = make-iso p1→p2 p2→p1 p1→p2→p1 p2→p1→p2
  where
    module p1 = Product p1
    module p2 = Product p2

    p1→p2 : Hom (apex p1) (apex p2)
    p1→p2 = p2.⟨ p1.π₁ , p1.π₂ ⟩p2.

    p2→p1 : Hom (apex p2) (apex p1)
    p2→p1 = p1.⟨ p2.π₁ , p2.π₂ ⟩p1.

These are unique because they are maps into products which commute with the projections.

    p1→p2→p1 : p1→p2 ∘ p2→p1 ≡ id
    p1→p2→p1 =
      p2.unique₂
        (assoc _ _ _ ·· ap (_∘ _) p2.π₁∘factor ·· p1.π₁∘factor)
        (assoc _ _ _ ·· ap (_∘ _) p2.π₂∘factor ·· p1.π₂∘factor)
        (idr _) (idr _)

    p2→p1→p2 : p2→p1 ∘ p1→p2 ≡ id
    p2→p1→p2 =
      p1.unique₂
        (assoc _ _ _ ·· ap (_∘ _) p1.π₁∘factor ·· p2.π₁∘factor)
        (assoc _ _ _ ·· ap (_∘ _) p1.π₂∘factor ·· p2.π₂∘factor)
        (idr _) (idr _)

is-product-iso
  : ∀ {A A′ B B′ P} {π₁ : Hom P A} {π₂ : Hom P B}
      {f : Hom A A′} {g : Hom B B′}
  → is-invertible f
  → is-invertible g
  → is-product π₁ π₂
  → is-product (f ∘ π₁) (g ∘ π₂)
is-product-iso f-iso g-iso prod = prod′ where
  module fi = is-invertible f-iso
  module gi = is-invertible g-iso

  open is-product
  prod′ : is-product _ _
  prod′ .⟨_,_⟩ qa qb = prod .⟨_,_⟩ (fi.inv ∘ qa) (gi.inv ∘ qb)
  prod′ .π₁∘factor = pullr (prod .π₁∘factor) ∙ cancell fi.invl
  prod′ .π₂∘factor = pullr (prod .π₂∘factor) ∙ cancell gi.invl
  prod′ .unique other p q = prod .unique other
    (sym (ap (_ ∘_) (sym p) ∙ pulll (cancell fi.invr)))
    (sym (ap (_ ∘_) (sym q) ∙ pulll (cancell gi.invr)))

The product functor🔗

If $\mathcal{C}$ admits products of all pairs of objects, then the assignment $(A, B) \mapsto (A \times B)$ extends to a bifunctor $(\mathcal{C} \times \mathcal{C}) \to \mathcal{C}$.

module Cartesian (hasprods : ∀ A B → Product A B) where
  open Functor

  ×-functor : Functor (C ×ᶜ C) C
  ×-functor .F₀ (A , B) = hasprods A B .apex
  ×-functor .F₁ {a , x} {b , y} (f , g) =
    hasprods b y .has-is-product .is-product.⟨_,_⟩
      (f ∘ hasprods a x .Product.π₁) (g ∘ hasprods a x .Product.π₂)

  ×-functor .F-id {a , b} =
    unique₂ (hasprods a b)
      (hasprods a b .π₁∘factor)
      (hasprods a b .π₂∘factor)
      id-comm id-comm

  ×-functor .F-∘ {a , b} {c , d} {e , f} x y =
    unique₂ (hasprods e f)
      (hasprods e f .π₁∘factor)
      (hasprods e f .π₂∘factor)
      (  pulll (hasprods e f .π₁∘factor)
      ·· pullr (hasprods c d .π₁∘factor)
      ·· assoc _ _ _)
      (  pulll (hasprods e f .π₂∘factor)
      ·· pullr (hasprods c d .π₂∘factor)
      ·· assoc _ _ _)

We refer to a category admitting all binary products as cartesian. When working with products, a Cartesian category is the place to be, since we can work with the “canonical” product operations — rather than requiring different product data for any pair of objects we need a product for.

Here we extract the data of the “global” product-assigning operation to separate top-level definitions:

  _⊗_ : Ob → Ob → Ob
  A ⊗ B = F₀ ×-functor (A , B)

  ⟨_,_⟩ : Hom a b → Hom a c → Hom a (b ⊗ c)
  ⟨ f , g ⟩ = hasprods _ _ .has-is-product .is-product.⟨_,_⟩ f g

  π₁ : Hom (a ⊗ b) a
  π₁ = hasprods _ _ .Product.π₁

  π₂ : Hom (a ⊗ b) b
  π₂ = hasprods _ _ .Product.π₂

  π₁∘⟨⟩ : {f : Hom a b} {g : Hom a c} → π₁ ∘ ⟨ f , g ⟩ ≡ f
  π₁∘⟨⟩ = hasprods _ _ .has-is-product .is-product.π₁∘factor

  π₂∘⟨⟩ : {f : Hom a b} {g : Hom a c} → π₂ ∘ ⟨ f , g ⟩ ≡ g
  π₂∘⟨⟩ = hasprods _ _ .has-is-product .is-product.π₂∘factor

  ⟨⟩∘ : ∀ {Q R} {p1 : Hom Q a} {p2 : Hom Q b} (f : Hom R Q)
      → ⟨ p1 , p2 ⟩ ∘ f ≡ ⟨ p1 ∘ f , p2 ∘ f ⟩
  ⟨⟩∘ f = is-product.⟨⟩∘ (hasprods _ _ .has-is-product) f

  ⟨⟩-unique : ∀ {Q} {p1 : Hom Q A} {p2 : Hom Q B}
             → (other : Hom Q (A ⊗ B))
             → π₁ ∘ other ≡ p1
             → π₂ ∘ other ≡ p2
             → other ≡ ⟨ p1 , p2 ⟩
  ⟨⟩-unique = hasprods _ _ .has-is-product .is-product.unique

  ⟨⟩-η : ∀ {A B} → ⟨ π₁ , π₂ ⟩ ≡ id {A ⊗ B}
  ⟨⟩-η = sym $ ⟨⟩-unique id (idr π₁) (idr π₂)