diff options
Diffstat (limited to 'Category/Diagram/Pushout.agda')
| -rw-r--r-- | Category/Diagram/Pushout.agda | 92 |
1 files changed, 49 insertions, 43 deletions
diff --git a/Category/Diagram/Pushout.agda b/Category/Diagram/Pushout.agda index 8ca384f..1523223 100644 --- a/Category/Diagram/Pushout.agda +++ b/Category/Diagram/Pushout.agda @@ -5,8 +5,6 @@ module Category.Diagram.Pushout {o ℓ e} (𝒞 : Category o ℓ e) where open Category 𝒞 -import Categories.Diagram.Pullback op as Pb using (up-to-iso) - 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) @@ -15,8 +13,12 @@ open import Categories.Morphism.Duality 𝒞 using (op-≅⇒≅) open import Categories.Morphism.Reasoning 𝒞 using ( id-comm ; id-comm-sym - ; assoc²'' - ; assoc²' + ; assoc²εβ + ; assoc²γδ + ; assoc²γβ + ; assoc²βγ + ; introʳ + ; elimʳ ) @@ -32,31 +34,32 @@ 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)) -up-to-iso : (p p′ : Pushout f g) → Pushout.Q p ≅ Pushout.Q p′ -up-to-iso p p′ = op-≅⇒≅ (Pb.up-to-iso (Pushout⇒coPullback p) (Pushout⇒coPullback p′)) - pushout-f-id : Pushout f id pushout-f-id {_} {_} {f} = record { i₁ = id ; i₂ = f - ; commute = id-comm-sym - ; universal = λ {B} {h₁} {h₂} eq → h₁ - ; unique = λ {E} {h₁} {h₂} {eq} {j} j∘i₁≈h₁ j∘i₂≈h₂ → Equiv.sym identityʳ ○ j∘i₁≈h₁ - ; universal∘i₁≈h₁ = λ {E} {h₁} {h₂} {eq} → identityʳ - ; universal∘i₂≈h₂ = λ {E} {h₁} {h₂} {eq} → eq ○ identityʳ + ; 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 {_} {_} {g} = record +pushout-id-g {A} {B} {g} = record { i₁ = g ; i₂ = id - ; commute = id-comm - ; universal = λ {B} {h₁} {h₂} eq → h₂ - ; unique = λ {E} {h₁} {h₂} {eq} {j} j∘i₁≈h₁ j∘i₂≈h₂ → Equiv.sym identityʳ ○ j∘i₂≈h₂ - ; universal∘i₁≈h₁ = λ {E} {h₁} {h₂} {eq} → Equiv.sym eq ○ identityʳ - ; universal∘i₂≈h₂ = λ {E} {h₁} {h₂} {eq} → identityʳ + ; 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 @@ -70,31 +73,34 @@ extend-i₁-iso extend-i₁-iso {_} {_} {_} {_} {f} {g} p B≅D = record { i₁ = P.i₁ ∘ B≅D.to ; i₂ = P.i₂ - ; commute = begin - (P.i₁ ∘ B≅D.to) ∘ B≅D.from ∘ f ≈⟨ assoc²'' ⟨ - 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) } - ; unique = λ {_} {h₁} {_} {j} ≈₁ ≈₂ → - let - ≈₁′ = begin - j ∘ P.i₁ ≈⟨ refl⟩∘⟨ identityʳ ⟨ - j ∘ P.i₁ ∘ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ B≅D.isoˡ ⟨ - j ∘ P.i₁ ∘ B≅D.to ∘ B≅D.from ≈⟨ assoc²' ⟨ - (j ∘ P.i₁ ∘ B≅D.to) ∘ B≅D.from ≈⟨ ≈₁ ⟩∘⟨refl ⟩ - h₁ ∘ B≅D.from ∎ - in P.unique ≈₁′ ≈₂ - ; 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 + ; 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 |