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{-# OPTIONS --without-K --safe #-}
open import Categories.Category.Core using (Category)
module Coeq {o ℓ e} (C : Category o ℓ e) where
open import Categories.Diagram.Coequalizer C using (IsCoequalizer)
open import Categories.Object.Coproduct C using (Coproduct; IsCoproduct; IsCoproduct⇒Coproduct)
open import Categories.Morphism.Reasoning.Core C using (pullˡ; pushˡ; extendˡ; extendʳ; assoc²βε; assoc²εβ)
open import Function using (_$_)
open Category C
coequalizer-on-coproduct
: {A B A+B C Q₁ Q₂ : Obj}
→ {i₁ : A ⇒ A+B}
→ {i₂ : B ⇒ A+B}
→ {f g : A+B ⇒ C}
→ {q₁ : C ⇒ Q₁}
→ {q₂ : Q₁ ⇒ Q₂}
→ IsCoproduct i₁ i₂
→ IsCoequalizer (f ∘ i₁) (g ∘ i₁) q₁
→ IsCoequalizer (q₁ ∘ f ∘ i₂) (q₁ ∘ g ∘ i₂) q₂
→ IsCoequalizer f g (q₂ ∘ q₁)
coequalizer-on-coproduct {C = C} {Q₁} {Q₂} {f = f} {g} {q₁} {q₂} coprod coeq₁ coeq₂ = record
{ equality = assoc ○ equality ○ sym-assoc
; coequalize = λ eq → Cocone.u₂ eq
; universal = λ { {eq = eq} → Cocone.univ eq }
; unique = λ { {eq = eq} h≈i∘q₂∘q₁ → Cocone.uniq eq h≈i∘q₂∘q₁ }
}
where
open HomReasoning
module X₁ = IsCoequalizer coeq₁
module X₂ = IsCoequalizer coeq₂
open Coproduct (IsCoproduct⇒Coproduct coprod) using (g-η; [_,_]; unique; i₁; i₂)
on-i₁ : (q₂ ∘ q₁ ∘ f) ∘ i₁ ≈ (q₂ ∘ q₁ ∘ g) ∘ i₁
on-i₁ = assoc²βε ○ refl⟩∘⟨ X₁.equality ○ assoc²εβ
on-i₂ : (q₂ ∘ q₁ ∘ f) ∘ i₂ ≈ (q₂ ∘ q₁ ∘ g) ∘ i₂
on-i₂ = assoc²βε ○ X₂.equality ○ assoc²εβ
equality : q₂ ∘ q₁ ∘ f ≈ q₂ ∘ q₁ ∘ g
equality = begin
q₂ ∘ q₁ ∘ f ≈⟨ unique on-i₁ on-i₂ ⟨
[ (q₂ ∘ q₁ ∘ g) ∘ i₁ , (q₂ ∘ q₁ ∘ g) ∘ i₂ ] ≈⟨ g-η ⟩
q₂ ∘ q₁ ∘ g ∎
module Cocone {Q : Obj} {h : C ⇒ Q} (eq : h ∘ f ≈ h ∘ g) where
u₁ : Q₁ ⇒ Q
u₁ = X₁.coequalize (extendʳ eq)
u₂ : Q₂ ⇒ Q
u₂ = X₂.coequalize $ begin
u₁ ∘ q₁ ∘ f ∘ i₂ ≈⟨ pullˡ (Equiv.sym X₁.universal) ⟩
h ∘ f ∘ i₂ ≈⟨ extendʳ eq ⟩
h ∘ g ∘ i₂ ≈⟨ pushˡ X₁.universal ⟩
u₁ ∘ q₁ ∘ g ∘ i₂ ∎
univ : h ≈ u₂ ∘ q₂ ∘ q₁
univ = begin
h ≈⟨ X₁.universal ⟩
u₁ ∘ q₁ ≈⟨ pushˡ X₂.universal ⟩
u₂ ∘ q₂ ∘ q₁ ∎
uniq : {i : Q₂ ⇒ Q} → h ≈ i ∘ q₂ ∘ q₁ → i ≈ u₂
uniq h≈i∘q₂∘q₁ = X₂.unique (Equiv.sym (X₁.unique (h≈i∘q₂∘q₁ ○ sym-assoc)))
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