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{-# OPTIONS --without-K --safe #-}
open import Categories.Category.Core using (Category)
module Category.Diagram.Pushout {o ℓ e} (𝒞 : Category o ℓ e) where
open Category 𝒞
open import Categories.Diagram.Duality 𝒞 using (Pushout⇒coPullback)
open import Categories.Diagram.Pushout 𝒞 using (Pushout)
open import Categories.Diagram.Pushout.Properties 𝒞 using (glue; swap)
open import Categories.Morphism 𝒞 using (_≅_)
open import Categories.Morphism.Duality 𝒞 using (op-≅⇒≅)
open import Categories.Morphism.Reasoning 𝒞 using
( id-comm
; id-comm-sym
; assoc²εβ
; assoc²γδ
; assoc²γβ
; assoc²βγ
; introʳ
; elimʳ
)
private
variable
A B C D : Obj
f g h : A ⇒ B
glue-i₁ : (p : Pushout f g) → Pushout h (Pushout.i₁ p) → Pushout (h ∘ f) g
glue-i₁ p = glue p
glue-i₂ : (p₁ : Pushout f g) → Pushout (Pushout.i₂ p₁) h → Pushout f (h ∘ g)
glue-i₂ p₁ p₂ = swap (glue (swap p₁) (swap p₂))
pushout-f-id : Pushout f id
pushout-f-id {_} {_} {f} = record
{ i₁ = id
; i₂ = f
; isPushout = record
{ commute = id-comm-sym
; universal = λ {B} {h₁} {h₂} eq → h₁
; universal∘i₁≈h₁ = λ {E} {h₁} {h₂} {eq} → identityʳ
; universal∘i₂≈h₂ = λ {E} {h₁} {h₂} {eq} → eq ○ identityʳ
; unique-diagram = λ eq₁ eq₂ → Equiv.sym identityʳ ○ eq₁ ○ identityʳ
}
}
where
open HomReasoning
pushout-id-g : Pushout id g
pushout-id-g {A} {B} {g} = record
{ i₁ = g
; i₂ = id
; isPushout = record
{ commute = id-comm
; universal = λ {B} {h₁} {h₂} eq → h₂
; universal∘i₁≈h₁ = λ {E} {h₁} {h₂} {eq} → Equiv.sym eq ○ identityʳ
; universal∘i₂≈h₂ = λ {E} {h₁} {h₂} {eq} → identityʳ
; unique-diagram = λ eq₁ eq₂ → Equiv.sym identityʳ ○ eq₂ ○ identityʳ
}
}
where
open HomReasoning
extend-i₁-iso
: {f : A ⇒ B}
{g : A ⇒ C}
(p : Pushout f g)
(B≅D : B ≅ D)
→ Pushout (_≅_.from B≅D ∘ f) g
extend-i₁-iso {_} {_} {_} {_} {f} {g} p B≅D = record
{ i₁ = P.i₁ ∘ B≅D.to
; i₂ = P.i₂
; isPushout = record
{ commute = begin
(P.i₁ ∘ B≅D.to) ∘ B≅D.from ∘ f ≈⟨ assoc²γδ ⟩
P.i₁ ∘ (B≅D.to ∘ B≅D.from) ∘ f ≈⟨ refl⟩∘⟨ B≅D.isoˡ ⟩∘⟨refl ⟩
P.i₁ ∘ id ∘ f ≈⟨ refl⟩∘⟨ identityˡ ⟩
P.i₁ ∘ f ≈⟨ P.commute ⟩
P.i₂ ∘ g ∎
; universal = λ { eq → P.universal (assoc ○ eq) }
; universal∘i₁≈h₁ = λ {E} {h₁} {_} {eq} →
begin
P.universal (assoc ○ eq) ∘ P.i₁ ∘ B≅D.to ≈⟨ sym-assoc ⟩
(P.universal (assoc ○ eq) ∘ P.i₁) ∘ B≅D.to ≈⟨ P.universal∘i₁≈h₁ ⟩∘⟨refl ⟩
(h₁ ∘ B≅D.from) ∘ B≅D.to ≈⟨ assoc ⟩
h₁ ∘ B≅D.from ∘ B≅D.to ≈⟨ refl⟩∘⟨ B≅D.isoʳ ⟩
h₁ ∘ id ≈⟨ identityʳ ⟩
h₁ ∎
; universal∘i₂≈h₂ = P.universal∘i₂≈h₂
; unique-diagram = λ {_} {h} {j} ≈₁ ≈₂ →
let
≈₁′ = begin
h ∘ P.i₁ ≈⟨ introʳ B≅D.isoˡ ⟩
(h ∘ P.i₁) ∘ B≅D.to ∘ B≅D.from ≈⟨ assoc²γβ ⟩
(h ∘ P.i₁ ∘ B≅D.to) ∘ B≅D.from ≈⟨ ≈₁ ⟩∘⟨refl ⟩
(j ∘ P.i₁ ∘ B≅D.to) ∘ B≅D.from ≈⟨ assoc²βγ ⟩
(j ∘ P.i₁) ∘ B≅D.to ∘ B≅D.from ≈⟨ elimʳ B≅D.isoˡ ⟩
j ∘ P.i₁ ∎
in P.unique-diagram ≈₁′ ≈₂
}
}
where
module P = Pushout p
module B≅D = _≅_ B≅D
open HomReasoning
extend-i₂-iso
: {f : A ⇒ B}
{g : A ⇒ C}
(p : Pushout f g)
(C≅D : C ≅ D)
→ Pushout f (_≅_.from C≅D ∘ g)
extend-i₂-iso {_} {_} {_} {_} {f} {g} p C≅D = swap (extend-i₁-iso (swap p) C≅D)
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