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

Coproducts🔗

The coproduct $P$ of two objects $A$ and $B$ (if it exists), is the smallest object equipped with “injection” maps $A \to P,$ $B \to P .$ It is dual to the product.

We witness this notion of “smallest object” with a universal property; Given any other $Q$ that also admits injection maps from $A$ and $B,$ we must have a unique map $P \to Q$ that factors the injections into $Q .$ This is best explained by a commutative diagram:

record is-coproduct {A B P} (in₀ : Hom A P) (in₁ : Hom B P) : Type (o ⊔ h) where
  field
    [_,_]      : ∀ {Q} (inj0 : Hom A Q) (inj1 : Hom B Q) → Hom P Q
    in₀∘factor : ∀ {Q} {inj0 : Hom A Q} {inj1} → [ inj0 , inj1 ] ∘ in₀ ≡ inj0
    in₁∘factor : ∀ {Q} {inj0 : Hom A Q} {inj1} → [ inj0 , inj1 ] ∘ in₁ ≡ inj1

    unique : ∀ {Q} {inj0 : Hom A Q} {inj1}
           → (other : Hom P Q)
           → other ∘ in₀ ≡ inj0
           → other ∘ in₁ ≡ inj1
           → other ≡ [ inj0 , inj1 ]

  unique₂ : ∀ {Q} {inj0 : Hom A Q} {inj1}
          → ∀ o1 (p1 : o1 ∘ in₀  ≡ inj0) (q1 : o1 ∘ in₁ ≡ inj1)
          → ∀ o2 (p2 : o2 ∘ in₀  ≡ inj0) (q2 : o2 ∘ in₁ ≡ inj1)
          → o1 ≡ o2
  unique₂ o1 p1 q1 o2 p2 q2 = unique o1 p1 q1 ∙ sym (unique o2 p2 q2)

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

record Coproduct (A B : Ob) : Type (o ⊔ h) where
  field
    coapex : Ob
    in₀ : Hom A coapex
    in₁ : Hom B coapex
    has-is-coproduct : is-coproduct in₀ in₁

  open is-coproduct has-is-coproduct public

Uniqueness🔗

The uniqueness argument presented here is dual to the argument for the product.

  +-Unique : (c1 c2 : Coproduct A B) → coapex c1 ≅ coapex c2
  +-Unique c1 c2 = make-iso c1→c2 c2→c1 c1→c2→c1 c2→c1→c2
    where
      module c1 = Coproduct c1
      module c2 = Coproduct c2

      c1→c2 : Hom (coapex c1) (coapex c2)
      c1→c2 = c1.[ c2.in₀ , c2.in₁ ]

      c2→c1 : Hom (coapex c2) (coapex c1)
      c2→c1 = c2.[ c1.in₀ , c1.in₁ ]

      c1→c2→c1 : c1→c2 ∘ c2→c1 ≡ id
      c1→c2→c1 =
        c2.unique₂ _
          (pullr c2.in₀∘factor ∙ c1.in₀∘factor)
          (pullr c2.in₁∘factor ∙ c1.in₁∘factor)
          id (idl _) (idl _)

      c2→c1→c2 : c2→c1 ∘ c1→c2 ≡ id
      c2→c1→c2 =
        c1.unique₂ _
          (pullr c1.in₀∘factor ∙ c2.in₀∘factor)
          (pullr c1.in₁∘factor ∙ c2.in₁∘factor)
          id (idl _) (idl _)

Categories with all binary coproducts🔗

Categories with all binary coproducts are quite common, so we define a module for working with them.

has-coproducts : Type _
has-coproducts = ∀ a b → Coproduct a b

module Binary-coproducts (all-coproducts : has-coproducts) where

  module coproduct {a} {b} = Coproduct (all-coproducts a b)

  open coproduct renaming
    (unique to []-unique; in₀∘factor to in₀∘[]; in₁∘factor to in₁∘[]) public
  open Functor

  infixr 7 _⊕₀_
  infix 50 _⊕₁_

  _⊕₀_ : Ob → Ob → Ob
  a ⊕₀ b = coproduct.coapex {a} {b}

  _⊕₁_ : ∀ {a b x y} → Hom a x → Hom b y → Hom (a ⊕₀ b) (x ⊕₀ y)
  f ⊕₁ g = [ in₀ ∘ f , in₁ ∘ g ]