diff options
Diffstat (limited to 'Cospan.agda')
| -rw-r--r-- | Cospan.agda | 62 |
1 files changed, 44 insertions, 18 deletions
diff --git a/Cospan.agda b/Cospan.agda index 651d883..19c422a 100644 --- a/Cospan.agda +++ b/Cospan.agda @@ -1,19 +1,22 @@ {-# OPTIONS --without-K --safe #-} open import Categories.Category using (Category) -open import Categories.Diagram.Pushout using (Pushout) -open import Categories.Diagram.Pushout.Properties using (glue; swap) -open import Categories.Object.Coproduct using (Coproduct) open import Category.Cocomplete.Bundle using (FinitelyCocompleteCategory) open import Function using (flip) open import Level using (_⊔_) -open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; sym) - module Cospan {o ℓ e} (𝒞 : FinitelyCocompleteCategory o ℓ e) where open FinitelyCocompleteCategory 𝒞 -open import Categories.Morphism U + +open import Categories.Diagram.Duality U using (Pushout⇒coPullback) +open import Categories.Diagram.Pushout U using (Pushout) +open import Categories.Diagram.Pushout.Properties U using (glue; swap) +open import Categories.Morphism U using (_≅_) +open import Categories.Morphism.Duality U using (op-≅⇒≅) + +import Categories.Diagram.Pullback op as Pb using (up-to-iso) + private @@ -55,23 +58,22 @@ record Same (P P′ : Cospan A B) : Set (ℓ ⊔ e) where from∘f₁≈f₁′ : _≅_.from iso ∘ Cospan.f₁ P ≈ Cospan.f₁ P′ from∘f₂≈f₂′ : _≅_.from iso ∘ Cospan.f₂ P ≈ Cospan.f₂ P′ -glue-i₁ : (p : Pushout U f g) → Pushout U h (Pushout.i₁ p) → Pushout U (h ∘ f) g -glue-i₁ p = glue U p +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)) -glue-i₂ : (p₁ : Pushout U f g) → Pushout U (Pushout.i₂ p₁) h → Pushout U f (h ∘ g) -glue-i₂ p₁ p₂ = swap U (glue U (swap U p₁) (swap U 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′)) compose-3-right : {c₁ : Cospan A B} {c₂ : Cospan B C} {c₃ : Cospan C D} → Same (compose c₁ (compose c₂ c₃)) (compose-3 c₁ c₂ c₃) -compose-3-right {A} {_} {_} {_} {c₁} {c₂} {c₃} = record - { iso = record - { from = P₄′.universal P₄.commute - ; to = P₄.universal P₄′.commute - ; iso = {! !} - } +compose-3-right {_} {_} {_} {_} {c₁} {c₂} {c₃} = record + { iso = up-to-iso P₄′ P₄ ; from∘f₁≈f₁′ = sym-assoc ○ P₄′.universal∘i₁≈h₁ ⟩∘⟨refl ○ assoc ; from∘f₂≈f₂′ = sym-assoc ○ P₄′.universal∘i₂≈h₂ ⟩∘⟨refl } @@ -86,13 +88,37 @@ compose-3-right {A} {_} {_} {_} {c₁} {c₂} {c₃} = record module P₂ = Pushout P₂ P₃ = pushout P₁.i₂ P₂.i₁ module P₃ = Pushout P₃ - P₄ : Pushout U C₁.f₂ (P₂.i₁ ∘ C₂.f₁) P₄ = glue-i₂ P₁ P₃ module P₄ = Pushout P₄ - P₄′ : Pushout U C₁.f₂ (P₂.i₁ ∘ C₂.f₁) P₄′ = pushout C₁.f₂ (P₂.i₁ ∘ C₂.f₁) module P₄′ = Pushout P₄′ +compose-3-left + : {c₁ : Cospan A B} + {c₂ : Cospan B C} + {c₃ : Cospan C D} + → Same (compose (compose c₁ c₂) c₃) (compose-3 c₁ c₂ c₃) +compose-3-left {_} {_} {_} {_} {c₁} {c₂} {c₃} = record + { iso = up-to-iso P₄′ P₄ + ; from∘f₁≈f₁′ = sym-assoc ○ P₄′.universal∘i₁≈h₁ ⟩∘⟨refl + ; from∘f₂≈f₂′ = sym-assoc ○ P₄′.universal∘i₂≈h₂ ⟩∘⟨refl ○ assoc + } + where + open HomReasoning + module C₁ = Cospan c₁ + module C₂ = Cospan c₂ + module C₃ = Cospan c₃ + P₁ = pushout C₁.f₂ C₂.f₁ + P₂ = pushout C₂.f₂ C₃.f₁ + module P₁ = Pushout P₁ + module P₂ = Pushout P₂ + P₃ = pushout P₁.i₂ P₂.i₁ + module P₃ = Pushout P₃ + P₄ = glue-i₁ P₂ P₃ + module P₄ = Pushout P₄ + P₄′ = pushout (P₁.i₂ ∘ C₂.f₂) C₃.f₁ + module P₄′ = Pushout P₄′ + compose-assoc : {c₁ : Cospan A B} {c₂ : Cospan B C} |