open import Cat.Prelude

module Cat.Diagram.Pushout {o ℓ} (C : Precategory o ℓ) where

Pushouts🔗

open Precategory C
private variable
  Q X Y Z : Ob
  h i₁' i₂' : Hom X Y

A pushout $Y +_X Z$ of $f : X \to Y$ and $g : X \to Z$ is the dual construction to the pullback. Much like the pullback is a subobject of the product, the pushout is a quotient object of the coproduct. The maps $f$ and $g$ tell us which parts of the coproduct to identify.

record is-pushout {P} (f : Hom X Y) (i₁ : Hom Y P) (g : Hom X Z) (i₂ : Hom Z P)
  : Type (o ⊔ ℓ) where
    field
      square     : i₁ ∘ f ≡ i₂ ∘ g

The most important part of the pushout is a commutative square of the following shape:

This can be thought of as providing “gluing instructions”. The injection maps $i_1$ and $i_2$ embed $Y$ and $Z$ into $P$, and the maps $f$ and $g$ determine what parts of $Y$ and $Z$ we ought to identify in $P$.

The universal property ensures that we only perform the minimal number of identifications required to make the aforementioned square commute.

      universal : ∀ {Q} {i₁' : Hom Y Q} {i₂' : Hom Z Q}
                 → i₁' ∘ f ≡ i₂' ∘ g → Hom P Q
      i₁∘universal : {p : i₁' ∘ f ≡ i₂' ∘ g} → universal p ∘ i₁ ≡ i₁'
      i₂∘universal : {p : i₁' ∘ f ≡ i₂' ∘ g} → universal p ∘ i₂ ≡ i₂'

      unique : {p : i₁' ∘ f ≡ i₂' ∘ g} {colim' : Hom P Q}
             → colim' ∘ i₁ ≡ i₁'
             → colim' ∘ i₂ ≡ i₂'
             → colim' ≡ universal p

    unique₂
      : {p : i₁' ∘ f ≡ i₂' ∘ g} {colim' colim'' : Hom P Q}
      → colim' ∘ i₁ ≡ i₁' → colim' ∘ i₂ ≡ i₂'
      → colim'' ∘ i₁ ≡ i₁' → colim'' ∘ i₂ ≡ i₂'
      → colim' ≡ colim''
    unique₂ {p = o} p q r s = unique {p = o} p q ∙ sym (unique r s)

We provide a convenient packaging of the pushout and the injection maps:

record Pushout (f : Hom X Y) (g : Hom X Z) : Type (o ⊔ ℓ) where
  field
    {coapex} : Ob
    i₁       : Hom Y coapex
    i₂       : Hom Z coapex
    has-is-po  : is-pushout f i₁ g i₂

  open is-pushout has-is-po public