open import Cat.Prelude

import Cat.Reasoning

module Cat.Diagram.Pushout where

Pushouts🔗

module _ {o ℓ} (C : Precategory o ℓ) where
  open Cat.Reasoning 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
        universal∘i₁ : {p : i₁' ∘ f ≡ i₂' ∘ g} → universal p ∘ i₁ ≡ i₁'
        universal∘i₂ : {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