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{-# OPTIONS --without-K --safe #-}
module Functor.Exact where
import Function.Base as Function
open import Categories.Category.Core using (Category)
open import Categories.Category.Cocomplete.Finitely using (FinitelyCocomplete)
open import Categories.Diagram.Coequalizer using (Coequalizer; IsCoequalizer; IsCoequalizer⇒Coequalizer)
open import Categories.Diagram.Pushout using (IsPushout; Pushout)
open import Categories.Diagram.Pushout.Properties using (Coproduct×Coequalizer⇒Pushout; up-to-iso)
open import Categories.Functor using (Functor; _∘F_) renaming (id to idF)
open import Categories.Functor.Properties using ([_]-resp-square; [_]-resp-≅)
open import Categories.Object.Coproduct using (IsCoproduct; Coproduct; IsCoproduct⇒Coproduct; Coproduct⇒IsCoproduct)
open import Categories.Object.Initial using (IsInitial)
open import Categories.NaturalTransformation.NaturalIsomorphism using (_≃_)
open import Category.Cocomplete.Finitely.Bundle using (FinitelyCocompleteCategory)
open import Function.Base using (id)
open import Level
module _ {o ℓ e : Level} {𝒞 : Category o ℓ e} where
open Category 𝒞
Coequalizer-resp-≈
: {A B C : Obj}
{f f′ g g′ : A ⇒ B}
{arr : B ⇒ C}
→ f ≈ f′
→ g ≈ g′
→ IsCoequalizer 𝒞 f g arr
→ IsCoequalizer 𝒞 f′ g′ arr
Coequalizer-resp-≈ ≈f ≈g ce = record
{ equality = refl⟩∘⟨ sym ≈f ○ equality ○ refl⟩∘⟨ ≈g
; coequalize = λ { eq → coequalize (refl⟩∘⟨ ≈f ○ eq ○ refl⟩∘⟨ sym ≈g) }
; universal = universal
; unique = unique
}
where
open HomReasoning
open Equiv
open IsCoequalizer ce
IsPushout⇒Pushout
: {A B C D : Obj}
{f : A ⇒ B} {g : A ⇒ C} {i₁ : B ⇒ D} {i₂ : C ⇒ D}
→ IsPushout 𝒞 f g i₁ i₂
→ Pushout 𝒞 f g
IsPushout⇒Pushout isP = record { i₁ = _ ; i₂ = _ ; isPushout = isP }
module _ {o ℓ e : Level} {𝒞 𝒟 : Category o ℓ e} (F : Functor 𝒞 𝒟) where
open Functor F
open Category 𝒞
PreservesCoequalizer : Set (o ⊔ ℓ ⊔ e)
PreservesCoequalizer = {A B C : Obj} {f g : A ⇒ B} {h : B ⇒ C} → IsCoequalizer 𝒞 f g h → IsCoequalizer 𝒟 (F₁ f) (F₁ g) (F₁ h)
PreservesCoproduct : Set (o ⊔ ℓ ⊔ e)
PreservesCoproduct = {A B C : Obj} {i₁ : A ⇒ C} {i₂ : B ⇒ C} → IsCoproduct 𝒞 i₁ i₂ → IsCoproduct 𝒟 (F₁ i₁) (F₁ i₂)
PreservesInitial : Set (o ⊔ ℓ ⊔ e)
PreservesInitial = {A : Obj} → IsInitial 𝒞 A → IsInitial 𝒟 (F₀ A)
PreservesPushouts : Set (o ⊔ ℓ ⊔ e)
PreservesPushouts = {A B C D : Obj} {f : A ⇒ B} {g : A ⇒ C} {i₁ : B ⇒ D} {i₂ : C ⇒ D} → IsPushout 𝒞 f g i₁ i₂ → IsPushout 𝒟 (F₁ f) (F₁ g) (F₁ i₁) (F₁ i₂)
module _ {o ℓ e : Level} (𝒞 𝒟 : FinitelyCocompleteCategory o ℓ e) where
open FinitelyCocompleteCategory using (U)
record IsRightExact (F : Functor (U 𝒞) (U 𝒟)) : Set (o ⊔ ℓ ⊔ e) where
field
F-resp-⊥ : PreservesInitial F
F-resp-+ : PreservesCoproduct F
F-resp-coeq : PreservesCoequalizer F
open FinitelyCocompleteCategory 𝒞 hiding (U)
open Functor F
F-resp-pushout : PreservesPushouts F
F-resp-pushout {A} {B} {C} {D} {f} {g} {i₁} {i₂} P = record
{ commute = [ F ]-resp-square P.commute
; universal = λ { eq → F-P′.universal eq ∘′ F-≅D.from }
; universal∘i₁≈h₁ = assoc′ ○′ refl⟩∘⟨′ [ F ]-resp-square P.universal∘i₁≈h₁ ○′ F-P′.universal∘i₁≈h₁
; universal∘i₂≈h₂ = assoc′ ○′ refl⟩∘⟨′ [ F ]-resp-square P.universal∘i₂≈h₂ ○′ F-P′.universal∘i₂≈h₂
; unique-diagram = λ { eq₁ eq₂ →
insertʳ′ F-≅D.isoˡ ○′
F-P′.unique-diagram
(assoc′ ○′
refl⟩∘⟨′ (Equiv′.sym (insertˡ′ F-≅D.isoˡ ○′ (refl⟩∘⟨′ [ F ]-resp-square P.universal∘i₁≈h₁))) ○′
eq₁ ○′
refl⟩∘⟨′ (insertˡ′ F-≅D.isoˡ ○′ (refl⟩∘⟨′ [ F ]-resp-square P.universal∘i₁≈h₁)) ○′
sym-assoc′)
(assoc′ ○′
refl⟩∘⟨′ (Equiv′.sym (insertˡ′ F-≅D.isoˡ ○′ (refl⟩∘⟨′ [ F ]-resp-square P.universal∘i₂≈h₂))) ○′
eq₂ ○′
refl⟩∘⟨′ (insertˡ′ F-≅D.isoˡ ○′ (refl⟩∘⟨′ [ F ]-resp-square P.universal∘i₂≈h₂)) ○′
sym-assoc′) ⟩∘⟨refl′ ○′
cancelʳ′ F-≅D.isoˡ }
}
where
module P = IsPushout P
cp : Coproduct (U 𝒞) B C
cp = coproduct
open Coproduct cp using (inject₁; inject₂; [_,_]; g-η; []-cong₂) renaming (i₁ to ι₁; i₂ to ι₂; A+B to B+C)
ce : Coequalizer (U 𝒞) (ι₁ ∘ f) (ι₂ ∘ g)
ce = coequalizer (ι₁ ∘ f) (ι₂ ∘ g)
open Coequalizer ce using (equality; coequalize) renaming (arr to i₁-i₂; obj to D′; universal to univ; unique to uniq)
open HomReasoning
open import Categories.Morphism.Reasoning (U 𝒞)
open import Categories.Morphism.Reasoning (U 𝒟) using () renaming (pullʳ to pullʳ′; insertʳ to insertʳ′; cancelʳ to cancelʳ′; insertˡ to insertˡ′; extendˡ to extendˡ′)
import Categories.Morphism as Morphism
open Morphism (U 𝒞) using (_≅_)
open Morphism (U 𝒟) using () renaming (_≅_ to _≅′_)
P′ : IsPushout (U 𝒞) f g (i₁-i₂ ∘ ι₁) (i₁-i₂ ∘ ι₂)
P′ = Pushout.isPushout (Coproduct×Coequalizer⇒Pushout (U 𝒞) cp ce)
open Category (U 𝒟) using () renaming (_∘_ to _∘′_; module HomReasoning to 𝒟-Reasoning; assoc to assoc′; sym-assoc to sym-assoc′; module Equiv to Equiv′)
open 𝒟-Reasoning using () renaming (_○_ to _○′_; refl⟩∘⟨_ to refl⟩∘⟨′_; _⟩∘⟨refl to _⟩∘⟨refl′)
≅D : D ≅ D′
≅D = up-to-iso (U 𝒞) (IsPushout⇒Pushout P) (IsPushout⇒Pushout P′)
F-≅D : F₀ D ≅′ F₀ D′
F-≅D = [ F ]-resp-≅ ≅D
module F-≅D = _≅′_ F-≅D
F-cp : IsCoproduct (U 𝒟) (F₁ ι₁) (F₁ ι₂)
F-cp = F-resp-+ (Coproduct⇒IsCoproduct (U 𝒞) cp)
F-ce : IsCoequalizer (U 𝒟) (F₁ ι₁ ∘′ F₁ f) (F₁ ι₂ ∘′ F₁ g) (F₁ i₁-i₂)
F-ce = Coequalizer-resp-≈ homomorphism homomorphism (F-resp-coeq (Coequalizer.isCoequalizer ce))
F-P′ : IsPushout (U 𝒟) (F₁ f) (F₁ g) (F₁ i₁-i₂ ∘′ F₁ ι₁) (F₁ i₁-i₂ ∘′ F₁ ι₂)
F-P′ = Pushout.isPushout (Coproduct×Coequalizer⇒Pushout (U 𝒟) (IsCoproduct⇒Coproduct (U 𝒟) F-cp) (IsCoequalizer⇒Coequalizer (U 𝒟) F-ce))
module F-P′ = IsPushout F-P′
record RightExactFunctor : Set (o ⊔ ℓ ⊔ e) where
constructor rightexact
field
F : Functor (U 𝒞) (U 𝒟)
isRightExact : IsRightExact F
open Functor F public
open IsRightExact isRightExact public
module _ where
open FinitelyCocompleteCategory using (U)
∘-resp-IsRightExact
: {o ℓ e : Level}
{𝒞 𝒟 ℰ : FinitelyCocompleteCategory o ℓ e}
{F : Functor (U 𝒞) (U 𝒟)}
{G : Functor (U 𝒟) (U ℰ)}
→ IsRightExact 𝒞 𝒟 F
→ IsRightExact 𝒟 ℰ G
→ IsRightExact 𝒞 ℰ (G ∘F F)
∘-resp-IsRightExact F-isRightExact G-isRightExact = record
{ F-resp-⊥ = G.F-resp-⊥ ∘ F.F-resp-⊥
; F-resp-+ = G.F-resp-+ ∘ F.F-resp-+
; F-resp-coeq = G.F-resp-coeq ∘ F.F-resp-coeq
}
where
open Function using (_∘_)
module F = IsRightExact F-isRightExact
module G = IsRightExact G-isRightExact
∘-RightExactFunctor
: {o ℓ e : Level}
→ {A B C : FinitelyCocompleteCategory o ℓ e}
→ RightExactFunctor B C → RightExactFunctor A B → RightExactFunctor A C
∘-RightExactFunctor F G = record
{ F = F.F ∘F G.F
; isRightExact = ∘-resp-IsRightExact G.isRightExact F.isRightExact
}
where
module F = RightExactFunctor F
module G = RightExactFunctor G
idF-RightExact : {o ℓ e : Level} → {𝒞 : FinitelyCocompleteCategory o ℓ e} → IsRightExact 𝒞 𝒞 idF
idF-RightExact = record
{ F-resp-⊥ = id
; F-resp-+ = id
; F-resp-coeq = id
}
idREF : {o ℓ e : Level} → {𝒞 : FinitelyCocompleteCategory o ℓ e} → RightExactFunctor 𝒞 𝒞
idREF = record
{ F = idF
; isRightExact = idF-RightExact
}
|
