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{-# OPTIONS --without-K --safe #-}
open import Categories.Category using (Category)
open import Category.Cocomplete.Finitely.Bundle using (FinitelyCocompleteCategory)
open import Function using (flip)
open import Level using (_⊔_)
module Category.Instance.Cospans {o ℓ e} (𝒞 : FinitelyCocompleteCategory o ℓ e) where
module 𝒞 = FinitelyCocompleteCategory 𝒞
open 𝒞 using (U; pushout)
open Category U hiding (_≈_)
open import Categories.Diagram.Pushout U using (Pushout)
open import Categories.Diagram.Pushout.Properties U using (pushout-resp-≈; up-to-iso)
open import Categories.Morphism U using (_≅_; module ≅)
open import Categories.Morphism.Reasoning U
using
( switch-fromtoˡ
; pullˡ ; pullʳ
; cancelˡ
)
open import Category.Diagram.Pushout U
using
( glue-i₁ ; glue-i₂
; pushout-f-id ; pushout-id-g
; extend-i₁-iso ; extend-i₂-iso
)
open import Category.Diagram.Cospan 𝒞 using (Cospan; compose; compose-3; identity; _≈_; cospan; ≈-trans; ≈-sym; ≈-equiv)
private
variable
A B C D : Obj
private
variable
f g h : Cospan A B
compose-idˡ : compose f identity ≈ f
compose-idˡ {f = cospan {N} f₁ f₂} = record
{ ≅N = ≅P
; from∘f₁≈f₁ = from∘f₁≈f₁
; from∘f₂≈f₂ = from∘f₂≈f₂
}
where
open HomReasoning
P P′ : Pushout f₂ id
P = pushout f₂ id
P′ = pushout-f-id {f = f₂}
module P = Pushout P
≅P : P.Q ≅ N
≅P = up-to-iso P P′
module ≅P = _≅_ ≅P
abstract
from∘f₁≈f₁ : ≅P.from ∘ P.i₁ ∘ f₁ 𝒞.≈ f₁
from∘f₁≈f₁ = begin
≅P.from ∘ P.i₁ ∘ f₁ ≈⟨ cancelˡ P.universal∘i₁≈h₁ ⟩
f₁ ∎
from∘f₂≈f₂ : ≅P.from ∘ P.i₂ ∘ id 𝒞.≈ f₂
from∘f₂≈f₂ = begin
≅P.from ∘ P.i₂ ∘ id ≈⟨ refl⟩∘⟨ identityʳ ⟩
≅P.from ∘ P.i₂ ≈⟨ P.universal∘i₂≈h₂ ⟩
f₂ ∎
compose-idʳ : compose identity f ≈ f
compose-idʳ {f = cospan {N} f₁ f₂} = record
{ ≅N = ≅P
; from∘f₁≈f₁ = from∘f₁≈f₁
; from∘f₂≈f₂ = from∘f₂≈f₂
}
where
open HomReasoning
P P′ : Pushout id f₁
P = pushout id f₁
module P = Pushout P
P′ = pushout-id-g {g = f₁}
≅P : P.Q ≅ N
≅P = up-to-iso P P′
module ≅P = _≅_ ≅P
abstract
from∘f₁≈f₁ : ≅P.from ∘ P.i₁ ∘ id 𝒞.≈ f₁
from∘f₁≈f₁ = begin
≅P.from ∘ P.i₁ ∘ id ≈⟨ refl⟩∘⟨ identityʳ ⟩
≅P.from ∘ P.i₁ ≈⟨ P.universal∘i₁≈h₁ ⟩
f₁ ∎
from∘f₂≈f₂ : ≅P.from ∘ P.i₂ ∘ f₂ 𝒞.≈ f₂
from∘f₂≈f₂ = begin
≅P.from ∘ P.i₂ ∘ f₂ ≈⟨ cancelˡ P.universal∘i₂≈h₂ ⟩
f₂ ∎
compose-id² : compose identity identity ≈ identity {A}
compose-id² = compose-idˡ
compose-3-right : compose f (compose g h) ≈ compose-3 f g h
compose-3-right {f = f} {g = g} {h = h} = record
{ ≅N = ≅N
; from∘f₁≈f₁ = from∘f₁≈f₁
; from∘f₂≈f₂ = from∘f₂≈f₂
}
where
open HomReasoning
module C₁ = Cospan f
module C₂ = Cospan g
module C₃ = Cospan h
P₁ : Pushout C₁.f₂ C₂.f₁
P₁ = pushout C₁.f₂ C₂.f₁
P₂ : Pushout C₂.f₂ C₃.f₁
P₂ = pushout C₂.f₂ C₃.f₁
module P₁ = Pushout P₁
module P₂ = Pushout P₂
P₃ : Pushout P₁.i₂ P₂.i₁
P₃ = pushout P₁.i₂ P₂.i₁
module P₃ = Pushout P₃
P₄ P₄′ : Pushout C₁.f₂ (P₂.i₁ ∘ C₂.f₁)
P₄ = glue-i₂ P₁ P₃
P₄′ = pushout C₁.f₂ (P₂.i₁ ∘ C₂.f₁)
module P₄ = Pushout P₄
module P₄′ = Pushout P₄′
≅N : P₄′.Q ≅ P₄.Q
≅N = up-to-iso P₄′ P₄
module ≅N = _≅_ ≅N
abstract
from∘f₁≈f₁ : ≅N.from ∘ P₄′.i₁ ∘ C₁.f₁ 𝒞.≈ P₃.i₁ ∘ P₁.i₁ ∘ C₁.f₁
from∘f₁≈f₁ = sym-assoc ○ P₄′.universal∘i₁≈h₁ ⟩∘⟨refl ○ assoc
from∘f₂≈f₂ : ≅N.from ∘ P₄′.i₂ ∘ P₂.i₂ ∘ C₃.f₂ 𝒞.≈ P₄.i₂ ∘ P₂.i₂ ∘ C₃.f₂
from∘f₂≈f₂ = sym-assoc ○ P₄′.universal∘i₂≈h₂ ⟩∘⟨refl
compose-3-left : compose (compose f g) h ≈ compose-3 f g h
compose-3-left {f = f} {g = g} {h = h} = record
{ ≅N = up-to-iso P₄′ P₄
; from∘f₁≈f₁ = from∘f₁≈f₁
; from∘f₂≈f₂ = from∘f₂≈f₂
}
where
open HomReasoning
module C₁ = Cospan f
module C₂ = Cospan g
module C₃ = Cospan h
P₁ : Pushout C₁.f₂ C₂.f₁
P₁ = pushout C₁.f₂ C₂.f₁
P₂ : Pushout C₂.f₂ C₃.f₁
P₂ = pushout C₂.f₂ C₃.f₁
module P₁ = Pushout P₁
module P₂ = Pushout P₂
P₃ : Pushout P₁.i₂ P₂.i₁
P₃ = pushout P₁.i₂ P₂.i₁
module P₃ = Pushout P₃
P₄ P₄′ : Pushout (P₁.i₂ ∘ C₂.f₂) C₃.f₁
P₄ = glue-i₁ P₂ P₃
P₄′ = pushout (P₁.i₂ ∘ C₂.f₂) C₃.f₁
module P₄ = Pushout P₄
module P₄′ = Pushout P₄′
≅N : P₄′.Q ≅ P₄.Q
≅N = up-to-iso P₄′ P₄
module ≅N = _≅_ ≅N
abstract
from∘f₁≈f₁ : ≅N.from ∘ P₄′.i₁ ∘ P₁.i₁ ∘ C₁.f₁ 𝒞.≈ P₄.i₁ ∘ P₁.i₁ ∘ C₁.f₁
from∘f₁≈f₁ = sym-assoc ○ P₄′.universal∘i₁≈h₁ ⟩∘⟨refl
from∘f₂≈f₂ : ≅N.from ∘ P₄′.i₂ ∘ C₃.f₂ 𝒞.≈ P₃.i₂ ∘ P₂.i₂ ∘ C₃.f₂
from∘f₂≈f₂ = sym-assoc ○ P₄′.universal∘i₂≈h₂ ⟩∘⟨refl ○ assoc
compose-assoc
: {c₁ : Cospan A B}
{c₂ : Cospan B C}
{c₃ : Cospan C D}
→ compose c₁ (compose c₂ c₃) ≈ (compose (compose c₁ c₂) c₃)
compose-assoc = ≈-trans compose-3-right (≈-sym compose-3-left)
compose-sym-assoc
: {c₁ : Cospan A B}
{c₂ : Cospan B C}
{c₃ : Cospan C D}
→ compose (compose c₁ c₂) c₃ ≈ compose c₁ (compose c₂ c₃)
compose-sym-assoc = ≈-trans compose-3-left (≈-sym compose-3-right)
compose-equiv
: {c₂ c₂′ : Cospan B C}
{c₁ c₁′ : Cospan A B}
→ c₂ ≈ c₂′
→ c₁ ≈ c₁′
→ compose c₁ c₂ ≈ compose c₁′ c₂′
compose-equiv {_} {_} {_} {c₂} {c₂′} {c₁} {c₁′} ≈C₂ ≈C₁ = record
{ ≅N = ≅P
; from∘f₁≈f₁ = from∘f₁≈f₁
; from∘f₂≈f₂ = from∘f₂≈f₂
}
where
module C₁ = Cospan c₁
module C₁′ = Cospan c₁′
module C₂ = Cospan c₂
module C₂′ = Cospan c₂′
P : Pushout C₁.f₂ C₂.f₁
P = pushout C₁.f₂ C₂.f₁
P′ : Pushout C₁′.f₂ C₂′.f₁
P′ = pushout C₁′.f₂ C₂′.f₁
module ≈C₁ = _≈_ ≈C₁
module ≈C₂ = _≈_ ≈C₂
P′′ : Pushout (≈C₁.to ∘ C₁′.f₂) (≈C₂.to ∘ C₂′.f₁)
P′′ = extend-i₂-iso (extend-i₁-iso P′ (≅.sym ≈C₁.≅N)) (≅.sym ≈C₂.≅N)
P″ : Pushout C₁.f₂ C₂.f₁
P″ =
pushout-resp-≈
P′′
(Equiv.sym (switch-fromtoˡ ≈C₁.≅N ≈C₁.from∘f₂≈f₂))
(Equiv.sym (switch-fromtoˡ ≈C₂.≅N ≈C₂.from∘f₁≈f₁))
module P = Pushout P
module P′ = Pushout P′
≅P : P.Q ≅ P′.Q
≅P = up-to-iso P P″
module ≅P = _≅_ ≅P
open HomReasoning
abstract
from∘f₁≈f₁ : ≅P.from ∘ P.i₁ ∘ C₁.f₁ 𝒞.≈ P′.i₁ ∘ C₁′.f₁
from∘f₁≈f₁ = begin
≅P.from ∘ P.i₁ ∘ C₁.f₁ ≈⟨ pullˡ P.universal∘i₁≈h₁ ⟩
(P′.i₁ ∘ ≈C₁.from) ∘ C₁.f₁ ≈⟨ pullʳ ≈C₁.from∘f₁≈f₁ ⟩
P′.i₁ ∘ C₁′.f₁ ∎
from∘f₂≈f₂ : ≅P.from ∘ P.i₂ ∘ C₂.f₂ 𝒞.≈ P′.i₂ ∘ C₂′.f₂
from∘f₂≈f₂ = begin
≅P.from ∘ P.i₂ ∘ C₂.f₂ ≈⟨ pullˡ P.universal∘i₂≈h₂ ⟩
(P′.i₂ ∘ ≈C₂.from) ∘ C₂.f₂ ≈⟨ pullʳ ≈C₂.from∘f₂≈f₂ ⟩
P′.i₂ ∘ C₂′.f₂ ∎
Cospans : Category o (o ⊔ ℓ) (ℓ ⊔ e)
Cospans = record
{ Obj = Obj
; _⇒_ = Cospan
; _≈_ = _≈_
; id = identity
; _∘_ = flip compose
; assoc = compose-assoc
; sym-assoc = compose-sym-assoc
; identityˡ = compose-idˡ
; identityʳ = compose-idʳ
; identity² = compose-id²
; equiv = ≈-equiv
; ∘-resp-≈ = compose-equiv
}
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