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