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{-# OPTIONS --without-K --safe #-}
module Coeq where
open import Categories.Category.Core using (Category)
open import Categories.Category.Instance.Nat using (Nat)
open import Categories.Diagram.Coequalizer Nat using (IsCoequalizer)
open import Categories.Object.Coproduct Nat using (Coproduct; IsCoproduct; IsCoproduct⇒Coproduct)
open Category Nat
-- TODO: any category
≈-i₁-i₂
: {A B A+B C : Obj}
→ {i₁ : A ⇒ A+B}
→ {i₂ : B ⇒ A+B}
→ (f g : A+B ⇒ C)
→ IsCoproduct i₁ i₂
→ f ∘ i₁ ≈ g ∘ i₁
→ f ∘ i₂ ≈ g ∘ i₂
→ f ≈ g
≈-i₁-i₂ f g coprod ≈₁ ≈₂ = begin
f ≈⟨ Equiv.sym g-η ⟩
[ f ∘ i₁ , f ∘ i₂ ] ≈⟨ []-cong₂ ≈₁ ≈₂ ⟩
[ g ∘ i₁ , g ∘ i₂ ] ≈⟨ g-η ⟩
g ∎
where
open Coproduct (IsCoproduct⇒Coproduct coprod)
open HomReasoning
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} i₁ i₂ f g q₁ q₂ coprod coeq₁ coeq₂ = record
{ equality = ≈-i₁-i₂
(q₂ ∘ q₁ ∘ f)
(q₂ ∘ q₁ ∘ g)
coprod
(∘-resp-≈ʳ {g = q₂} X₁.equality)
X₂.equality
; coequalize = λ {Q} {q} q∘f≈q∘g → let module X = Cocone {Q} {q} q∘f≈q∘g in X.u₂
; universal = λ {Q} {q} {q∘f≈q∘g} → let module X = Cocone {Q} {q} q∘f≈q∘g in X.q≈u₂∘q₂∘q₁
; unique = λ {Q} {q} {i} {q∘f≈q∘g} q≈i∘q₂∘q₁ →
let module X = Cocone {Q} {q} q∘f≈q∘g in X.q≈i∘q₂∘q₁⇒i≈u₂ q≈i∘q₂∘q₁
}
where
module X₁ = IsCoequalizer coeq₁
module X₂ = IsCoequalizer coeq₂
module Cocone {Q : Obj} {q : C ⇒ Q} (q∘f≈q∘g : q ∘ f ≈ q ∘ g) where
open HomReasoning
u₁ : Q₁ ⇒ Q
u₁ = X₁.coequalize (∘-resp-≈ˡ q∘f≈q∘g)
q≈u₁∘q₁ : q ≈ u₁ ∘ q₁
q≈u₁∘q₁ = X₁.universal
u₁∘q₁∘f∘i₂≈u₁∘q₁∘g∘i₂ : u₁ ∘ q₁ ∘ f ∘ i₂ ≈ u₁ ∘ q₁ ∘ g ∘ i₂
u₁∘q₁∘f∘i₂≈u₁∘q₁∘g∘i₂ = begin
u₁ ∘ q₁ ∘ f ∘ i₂ ≈⟨ ∘-resp-≈ˡ (Equiv.sym q≈u₁∘q₁) ⟩
q ∘ f ∘ i₂ ≈⟨ ∘-resp-≈ˡ q∘f≈q∘g ⟩
q ∘ g ∘ i₂ ≈⟨ ∘-resp-≈ˡ q≈u₁∘q₁ ⟩
u₁ ∘ q₁ ∘ g ∘ i₂ ∎
u₂ : Q₂ ⇒ Q
u₂ = X₂.coequalize u₁∘q₁∘f∘i₂≈u₁∘q₁∘g∘i₂
u₁≈u₂∘q₂ : u₁ ≈ u₂ ∘ q₂
u₁≈u₂∘q₂ = X₂.universal
q≈u₂∘q₂∘q₁ : q ≈ u₂ ∘ q₂ ∘ q₁
q≈u₂∘q₂∘q₁ = begin
q ≈⟨ q≈u₁∘q₁ ⟩
u₁ ∘ q₁ ≈⟨ ∘-resp-≈ˡ u₁≈u₂∘q₂ ⟩
u₂ ∘ q₂ ∘ q₁ ∎
q≈i∘q₂∘q₁⇒i≈u₂ : {i : Q₂ ⇒ Q} → q ≈ i ∘ q₂ ∘ q₁ → i ≈ u₂
q≈i∘q₂∘q₁⇒i≈u₂ q≈i∘q₂∘q₁ = X₂.unique (Equiv.sym (X₁.unique q≈i∘q₂∘q₁))
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