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{-# OPTIONS --without-K --safe #-}
open import Level using (Level; suc; _⊔_)
module Category.Cartesian.Instance.FinitelyCocompletes {o ℓ e : Level} where
import Categories.Morphism as Morphism
import Categories.Morphism.Reasoning as ⇒-Reasoning
open import Categories.Category.BinaryProducts using (BinaryProducts)
open import Categories.Category.Cartesian.Bundle using (CartesianCategory)
open import Categories.Diagram.Coequalizer using (IsCoequalizer)
open import Categories.Functor.Bifunctor using (flip-bifunctor)
open import Categories.Functor.Core using (Functor)
open import Categories.NaturalTransformation.NaturalIsomorphism using (_≃_; niHelper)
open import Categories.NaturalTransformation.NaturalIsomorphism.Properties using (pointwise-iso)
open import Categories.Object.Coproduct using (IsCoproduct; IsCoproduct⇒Coproduct; Coproduct)
open import Categories.Object.Initial using (IsInitial)
open import Data.Product using (_,_; swap) renaming (_×_ to _×′_)
open import Function using () renaming (_∘_ to _∘′_)
open import Category.Cocomplete.Finitely.Bundle using (FinitelyCocompleteCategory)
open import Category.Instance.FinitelyCocompletes {o} {ℓ} {e} using (FinitelyCocompletes; FinitelyCocompletes-Cartesian; _×₁_)
open import Functor.Exact using (IsRightExact; RightExactFunctor)
open import Functor.Exact.Instance.Swap using (Swap)
FinitelyCocompletes-CC : CartesianCategory (suc (o ⊔ ℓ ⊔ e)) (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e)
FinitelyCocompletes-CC = record
{ U = FinitelyCocompletes
; cartesian = FinitelyCocompletes-Cartesian
}
module FinCoCom = CartesianCategory FinitelyCocompletes-CC
open BinaryProducts (FinCoCom.products) using (_×_; π₁; π₂; _⁂_; assocˡ) -- hiding (unique)
module _ (𝒞 : FinitelyCocompleteCategory o ℓ e) where
private
module 𝒞 = FinitelyCocompleteCategory 𝒞
module 𝒞×𝒞 = FinitelyCocompleteCategory (𝒞 × 𝒞)
open 𝒞 using ([_,_]; +-unique; i₁; i₂; _∘_; _+_; module Equiv; _⇒_; _+₁_; -+-)
open Equiv
private
module -+- = Functor -+-
flip-IsInitial
: {(A , B) : 𝒞×𝒞.Obj}
→ IsInitial 𝒞×𝒞.U (A , B)
→ IsInitial 𝒞×𝒞.U (B , A)
flip-IsInitial isInitial = let open IsInitial isInitial in record
{ ! = swap !
; !-unique = swap ∘′ !-unique ∘′ swap
}
flip-IsCoproduct
: {(A₁ , A₂) (B₁ , B₂) (C₁ , C₂) : 𝒞×𝒞.Obj}
{(i₁-₁ , i₁-₂) : (A₁ ⇒ C₁) ×′ (A₂ ⇒ C₂)}
{(i₂-₁ , i₂-₂) : (B₁ ⇒ C₁) ×′ (B₂ ⇒ C₂)}
→ IsCoproduct 𝒞×𝒞.U (i₁-₁ , i₁-₂) (i₂-₁ , i₂-₂)
→ IsCoproduct 𝒞×𝒞.U (i₁-₂ , i₁-₁) (i₂-₂ , i₂-₁)
flip-IsCoproduct isCoproduct = let module + = IsCoproduct isCoproduct in record
{ [_,_] = λ x y → swap (+.[ swap x , swap y ])
; inject₁ = swap +.inject₁
; inject₂ = swap +.inject₂
; unique = λ x y → swap (+.unique (swap x) (swap y))
}
flip-IsCoequalizer
: {(A₁ , A₂) (B₁ , B₂) (C₁ , C₂) : 𝒞×𝒞.Obj}
{(f₁ , f₂) (g₁ , g₂) : (A₁ ⇒ B₁) ×′ (A₂ ⇒ B₂)}
{(h₁ , h₂) : (B₁ ⇒ C₁) ×′ (B₂ ⇒ C₂)}
→ IsCoequalizer 𝒞×𝒞.U (f₁ , f₂) (g₁ , g₂) (h₁ , h₂)
→ IsCoequalizer 𝒞×𝒞.U (f₂ , f₁) (g₂ , g₁) (h₂ , h₁)
flip-IsCoequalizer isCoeq = let open IsCoequalizer isCoeq in record
{ equality = swap equality
; coequalize = swap ∘′ coequalize ∘′ swap
; universal = swap universal
; unique = swap ∘′ unique ∘′ swap
}
+-resp-⊥
: {(A , B) : 𝒞×𝒞.Obj}
→ IsInitial 𝒞×𝒞.U (A , B)
→ IsInitial 𝒞.U (A + B)
+-resp-⊥ {A , B} A,B-isInitial = record
{ ! = [ A-isInitial.! , B-isInitial.! ]
; !-unique = λ f → +-unique (sym (A-isInitial.!-unique (f ∘ i₁))) (sym (B-isInitial.!-unique (f ∘ i₂)))
}
where
open IsRightExact
open RightExactFunctor
module A-isInitial = IsInitial (F-resp-⊥ (isRightExact (π₁ {𝒞} {𝒞})) A,B-isInitial)
module B-isInitial = IsInitial (F-resp-⊥ (isRightExact (π₂ {𝒞} {𝒞})) A,B-isInitial)
+-resp-+
: {(A₁ , A₂) (B₁ , B₂) (C₁ , C₂) : 𝒞×𝒞.Obj}
{(i₁-₁ , i₁-₂) : (A₁ ⇒ C₁) ×′ (A₂ ⇒ C₂)}
{(i₂-₁ , i₂-₂) : (B₁ ⇒ C₁) ×′ (B₂ ⇒ C₂)}
→ IsCoproduct 𝒞×𝒞.U (i₁-₁ , i₁-₂) (i₂-₁ , i₂-₂)
→ IsCoproduct 𝒞.U (i₁-₁ +₁ i₁-₂) (i₂-₁ +₁ i₂-₂)
+-resp-+ {A₁ , A₂} {B₁ , B₂} {C₁ , C₂} {i₁-₁ , i₁-₂} {i₂-₁ , i₂-₂} isCoproduct = record
{ [_,_] = λ { h₁ h₂ → [ Coprod₁.[ h₁ ∘ i₁ , h₂ ∘ i₁ ] , Coprod₂.[ h₁ ∘ i₂ , h₂ ∘ i₂ ] ] }
; inject₁ = inject₁
; inject₂ = inject₂
; unique = unique
}
where
open IsRightExact
open RightExactFunctor
module Coprod₁ = Coproduct (IsCoproduct⇒Coproduct 𝒞.U (F-resp-+ (isRightExact (π₁ {𝒞} {𝒞})) isCoproduct))
module Coprod₂ = Coproduct (IsCoproduct⇒Coproduct 𝒞.U (F-resp-+ (isRightExact (π₂ {𝒞} {𝒞})) isCoproduct))
open 𝒞 using ([]-cong₂; []∘+₁; +-g-η; +₁∘i₁; +₁∘i₂)
open 𝒞 using (Obj; _≈_; module HomReasoning; assoc)
open HomReasoning
open ⇒-Reasoning 𝒞.U
inject₁
: {X : Obj}
{h₁ : A₁ + A₂ ⇒ X}
{h₂ : B₁ + B₂ ⇒ X}
→ [ Coprod₁.[ h₁ ∘ i₁ , h₂ ∘ i₁ ] , Coprod₂.[ h₁ ∘ i₂ , h₂ ∘ i₂ ] ] ∘ (i₁-₁ +₁ i₁-₂) ≈ h₁
inject₁ {_} {h₁} {h₂} = begin
[ Coprod₁.[ h₁ ∘ i₁ , h₂ ∘ i₁ ] , Coprod₂.[ h₁ ∘ i₂ , h₂ ∘ i₂ ] ] ∘ (i₁-₁ +₁ i₁-₂) ≈⟨ []∘+₁ ⟩
[ Coprod₁.[ h₁ ∘ i₁ , h₂ ∘ i₁ ] ∘ i₁-₁ , Coprod₂.[ h₁ ∘ i₂ , h₂ ∘ i₂ ] ∘ i₁-₂ ] ≈⟨ []-cong₂ Coprod₁.inject₁ Coprod₂.inject₁ ⟩
[ h₁ ∘ i₁ , h₁ ∘ i₂ ] ≈⟨ +-g-η ⟩
h₁ ∎
inject₂
: {X : Obj}
{h₁ : A₁ + A₂ ⇒ X}
{h₂ : B₁ + B₂ ⇒ X}
→ [ Coprod₁.[ h₁ ∘ i₁ , h₂ ∘ i₁ ] , Coprod₂.[ h₁ ∘ i₂ , h₂ ∘ i₂ ] ] ∘ (i₂-₁ +₁ i₂-₂) ≈ h₂
inject₂ {_} {h₁} {h₂} = begin
[ Coprod₁.[ h₁ ∘ i₁ , h₂ ∘ i₁ ] , Coprod₂.[ h₁ ∘ i₂ , h₂ ∘ i₂ ] ] ∘ (i₂-₁ +₁ i₂-₂) ≈⟨ []∘+₁ ⟩
[ Coprod₁.[ h₁ ∘ i₁ , h₂ ∘ i₁ ] ∘ i₂-₁ , Coprod₂.[ h₁ ∘ i₂ , h₂ ∘ i₂ ] ∘ i₂-₂ ] ≈⟨ []-cong₂ Coprod₁.inject₂ Coprod₂.inject₂ ⟩
[ h₂ ∘ i₁ , h₂ ∘ i₂ ] ≈⟨ +-g-η ⟩
h₂ ∎
unique
: {X : Obj}
{i : C₁ + C₂ ⇒ X}
{h₁ : A₁ + A₂ ⇒ X}
{h₂ : B₁ + B₂ ⇒ X}
(eq₁ : i ∘ (i₁-₁ +₁ i₁-₂) ≈ h₁)
(eq₂ : i ∘ (i₂-₁ +₁ i₂-₂) ≈ h₂)
→ [ Coprod₁.[ h₁ ∘ i₁ , h₂ ∘ i₁ ] , Coprod₂.[ h₁ ∘ i₂ , h₂ ∘ i₂ ] ] ≈ i
unique {X} {i} {h₁} {h₂} eq₁ eq₂ = begin
[ Coprod₁.[ h₁ ∘ i₁ , h₂ ∘ i₁ ] , Coprod₂.[ h₁ ∘ i₂ , h₂ ∘ i₂ ] ] ≈⟨ []-cong₂ (Coprod₁.unique eq₁-₁ eq₂-₁) (Coprod₂.unique eq₁-₂ eq₂-₂) ⟩
[ i ∘ i₁ , i ∘ i₂ ] ≈⟨ +-g-η ⟩
i ∎
where
eq₁-₁ : (i ∘ i₁) ∘ i₁-₁ ≈ h₁ ∘ i₁
eq₁-₁ = begin
(i ∘ i₁) ∘ i₁-₁ ≈⟨ pushʳ +₁∘i₁ ⟨
i ∘ (i₁-₁ +₁ i₁-₂) ∘ i₁ ≈⟨ pullˡ eq₁ ⟩
h₁ ∘ i₁ ∎
eq₂-₁ : (i ∘ i₁) ∘ i₂-₁ ≈ h₂ ∘ i₁
eq₂-₁ = begin
(i ∘ i₁) ∘ i₂-₁ ≈⟨ pushʳ +₁∘i₁ ⟨
i ∘ (i₂-₁ +₁ i₂-₂) ∘ i₁ ≈⟨ pullˡ eq₂ ⟩
h₂ ∘ i₁ ∎
eq₁-₂ : (i ∘ i₂) ∘ i₁-₂ ≈ h₁ ∘ i₂
eq₁-₂ = begin
(i ∘ i₂) ∘ i₁-₂ ≈⟨ pushʳ +₁∘i₂ ⟨
i ∘ (i₁-₁ +₁ i₁-₂) ∘ i₂ ≈⟨ pullˡ eq₁ ⟩
h₁ ∘ i₂ ∎
eq₂-₂ : (i ∘ i₂) ∘ i₂-₂ ≈ h₂ ∘ i₂
eq₂-₂ = begin
(i ∘ i₂) ∘ i₂-₂ ≈⟨ pushʳ +₁∘i₂ ⟨
i ∘ (i₂-₁ +₁ i₂-₂) ∘ i₂ ≈⟨ pullˡ eq₂ ⟩
h₂ ∘ i₂ ∎
+-resp-coeq
: {(A₁ , A₂) (B₁ , B₂) (C₁ , C₂) : 𝒞×𝒞.Obj}
{(f₁ , f₂) (g₁ , g₂) : (A₁ ⇒ B₁) ×′ (A₂ ⇒ B₂)}
{(h₁ , h₂) : (B₁ ⇒ C₁) ×′ (B₂ ⇒ C₂)}
→ IsCoequalizer 𝒞×𝒞.U (f₁ , f₂) (g₁ , g₂) (h₁ , h₂)
→ IsCoequalizer 𝒞.U (f₁ +₁ f₂) (g₁ +₁ g₂) (h₁ +₁ h₂)
+-resp-coeq {A₁ , A₂} {B₁ , B₂} {C₁ , C₂} {f₁ , f₂} {g₁ , g₂} {h₁ , h₂} isCoeq = record
{ equality = sym -+-.homomorphism ○ []-cong₂ (refl⟩∘⟨ Coeq₁.equality) (refl⟩∘⟨ Coeq₂.equality) ○ -+-.homomorphism
; coequalize = coequalize
; universal = universal _
; unique = uniq _
}
where
open IsRightExact
open RightExactFunctor
module Coeq₁ = IsCoequalizer (F-resp-coeq (isRightExact (π₁ {𝒞} {𝒞})) isCoeq)
module Coeq₂ = IsCoequalizer (F-resp-coeq (isRightExact (π₂ {𝒞} {𝒞})) isCoeq)
open 𝒞 using ([]-cong₂; +₁∘i₁; +₁∘i₂; []∘+₁; +-g-η)
open 𝒞 using (Obj; _≈_; module HomReasoning; assoc; sym-assoc)
open 𝒞.HomReasoning
open ⇒-Reasoning 𝒞.U
module _ {X : Obj} {k : B₁ + B₂ ⇒ X} (eq : k ∘ (f₁ +₁ f₂) ≈ k ∘ (g₁ +₁ g₂)) where
eq₁ : (k ∘ i₁) ∘ f₁ ≈ (k ∘ i₁) ∘ g₁
eq₁ = begin
(k ∘ i₁) ∘ f₁ ≈⟨ pushʳ +₁∘i₁ ⟨
k ∘ (f₁ +₁ f₂) ∘ i₁ ≈⟨ extendʳ eq ⟩
k ∘ (g₁ +₁ g₂) ∘ i₁ ≈⟨ pushʳ +₁∘i₁ ⟩
(k ∘ i₁) ∘ g₁ ∎
eq₂ : (k ∘ i₂) ∘ f₂ ≈ (k ∘ i₂) ∘ g₂
eq₂ = begin
(k ∘ i₂) ∘ f₂ ≈⟨ pushʳ +₁∘i₂ ⟨
k ∘ (f₁ +₁ f₂) ∘ i₂ ≈⟨ extendʳ eq ⟩
k ∘ (g₁ +₁ g₂) ∘ i₂ ≈⟨ pushʳ +₁∘i₂ ⟩
(k ∘ i₂) ∘ g₂ ∎
coequalize : C₁ + C₂ ⇒ X
coequalize = [ Coeq₁.coequalize eq₁ , Coeq₂.coequalize eq₂ ]
universal : k ≈ coequalize ∘ (h₁ +₁ h₂)
universal = begin
k ≈⟨ +-g-η ⟨
[ k ∘ i₁ , k ∘ i₂ ] ≈⟨ []-cong₂ Coeq₁.universal Coeq₂.universal ⟩
[ Coeq₁.coequalize eq₁ ∘ h₁ , Coeq₂.coequalize eq₂ ∘ h₂ ] ≈⟨ []∘+₁ ⟨
coequalize ∘ (h₁ +₁ h₂) ∎
uniq : {i : C₁ + C₂ ⇒ X} → k ≈ i ∘ (h₁ +₁ h₂) → i ≈ coequalize
uniq {i} eq′ = begin
i ≈⟨ +-g-η ⟨
[ i ∘ i₁ , i ∘ i₂ ] ≈⟨ []-cong₂ (Coeq₁.unique eq₁′) (Coeq₂.unique eq₂′) ⟩
[ Coeq₁.coequalize eq₁ , Coeq₂.coequalize eq₂ ] ∎
where
eq₁′ : k ∘ i₁ ≈ (i ∘ i₁) ∘ h₁
eq₁′ = eq′ ⟩∘⟨refl ○ extendˡ +₁∘i₁
eq₂′ : k ∘ i₂ ≈ (i ∘ i₂) ∘ h₂
eq₂′ = eq′ ⟩∘⟨refl ○ extendˡ +₁∘i₂
module _ {𝒞 : FinitelyCocompleteCategory o ℓ e} where
open FinCoCom using (_⇒_; _∘_; id)
module 𝒞 = FinitelyCocompleteCategory 𝒞
-+- : 𝒞 × 𝒞 ⇒ 𝒞
-+- = record
{ F = 𝒞.-+-
; isRightExact = record
{ F-resp-⊥ = +-resp-⊥ 𝒞
; F-resp-+ = +-resp-+ 𝒞
; F-resp-coeq = +-resp-coeq 𝒞
}
}
module x+y = RightExactFunctor -+-
↔-+- : 𝒞 × 𝒞 ⇒ 𝒞
↔-+- = record
{ F = flip-bifunctor 𝒞.-+-
; isRightExact = record
{ F-resp-⊥ = λ x → +-resp-⊥ 𝒞 (flip-IsInitial 𝒞 x)
; F-resp-+ = λ x → +-resp-+ 𝒞 (flip-IsCoproduct 𝒞 x)
; F-resp-coeq = λ x → (+-resp-coeq 𝒞 (flip-IsCoequalizer 𝒞 x))
}
}
module y+x = RightExactFunctor ↔-+-
[x+y]+z : (𝒞 × 𝒞) × 𝒞 ⇒ 𝒞
[x+y]+z = -+- ∘ (-+- ×₁ id)
module [x+y]+z = RightExactFunctor [x+y]+z
x+[y+z] : (𝒞 × 𝒞) × 𝒞 ⇒ 𝒞
x+[y+z] = -+- ∘ (id ×₁ -+-) ∘ assocˡ
module x+[y+z] = RightExactFunctor x+[y+z]
assoc-≃ : [x+y]+z.F ≃ x+[y+z].F
assoc-≃ = pointwise-iso (λ { ((X , Y) , Z) → ≅.sym (𝒞.+-assoc {X} {Y} {Z})}) commute
where
open 𝒞
module 𝒞×𝒞×𝒞 = FinitelyCocompleteCategory ((𝒞 × 𝒞) × 𝒞)
open Morphism U using (_≅_; module ≅)
module +-assoc {X} {Y} {Z} = _≅_ (≅.sym (+-assoc {X} {Y} {Z}))
open import Categories.Object.Duality 𝒞.U using (Coproduct⇒coProduct)
op-binaryProducts : BinaryProducts op
op-binaryProducts = record { product = Coproduct⇒coProduct coproduct }
open BinaryProducts op-binaryProducts using () renaming (assocʳ∘⁂ to +₁∘assocˡ)
open Equiv
commute
: {((X , Y) , Z) : 𝒞×𝒞×𝒞.Obj}
{((X′ , Y′) , Z′) : 𝒞×𝒞×𝒞.Obj}
→ (F : ((X , Y) , Z) 𝒞×𝒞×𝒞.⇒ ((X′ , Y′) , Z′))
→ (+-assoc.from 𝒞.∘ [x+y]+z.₁ F)
≈ (x+[y+z].₁ F 𝒞.∘ +-assoc.from)
commute {(X , Y) , Z} {(X′ , Y′) , Z′} ((F , G) , H) = sym +₁∘assocˡ
|
