{-# OPTIONS --safe #-}
module Cubical.Categories.Site.Sheaf where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Structure
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Function using (_∘_)
open import Cubical.Foundations.Equiv
open import Cubical.Data.Sigma
open import Cubical.Functions.Embedding
open import Cubical.Categories.Category
open import Cubical.Categories.Site.Cover
open import Cubical.Categories.Site.Sieve
open import Cubical.Categories.Site.Coverage
open import Cubical.Categories.Presheaf
open import Cubical.Categories.Functor
open import Cubical.Categories.Constructions.FullSubcategory
open import Cubical.Categories.Yoneda
module _
{ℓ ℓ' : Level}
{C : Category ℓ ℓ'}
{ℓP : Level}
(P : Presheaf C ℓP)
where
open Category C hiding (_∘_)
module _
{c : ob}
{ℓ'' : Level}
(cov : Cover C ℓ'' c)
where
FamilyOnCover : Type (ℓ-max ℓP ℓ'')
FamilyOnCover = (i : ⟨ cov ⟩) → ⟨ P ⟅ patchObj cov i ⟆ ⟩
isCompatibleFamily : FamilyOnCover → Type (ℓ-max (ℓ-max (ℓ-max ℓ ℓ') ℓP) ℓ'')
isCompatibleFamily fam =
(i : ⟨ cov ⟩) →
(j : ⟨ cov ⟩) →
(d : ob) →
(f : Hom[ d , patchObj cov i ]) →
(g : Hom[ d , patchObj cov j ]) →
f ⋆ patchArr cov i ≡ g ⋆ patchArr cov j →
(P ⟪ f ⟫) (fam i) ≡ (P ⟪ g ⟫) (fam j)
isPropIsCompatibleFamily : (fam : FamilyOnCover) → isProp (isCompatibleFamily fam)
isPropIsCompatibleFamily fam =
isPropΠ6 λ _ _ d _ _ _ → str (P ⟅ d ⟆) _ _
CompatibleFamily : Type (ℓ-max (ℓ-max (ℓ-max ℓ ℓ') ℓP) ℓ'')
CompatibleFamily = Σ FamilyOnCover isCompatibleFamily
isSetCompatibleFamily : isSet CompatibleFamily
isSetCompatibleFamily =
isSetΣSndProp
(isSetΠ (λ i → str (P ⟅ patchObj cov i ⟆)))
isPropIsCompatibleFamily
CompatibleFamily≡ : (fam fam' : CompatibleFamily)
→ (∀ i → fam .fst i ≡ fam' .fst i)
→ fam ≡ fam'
CompatibleFamily≡ fam fam' p = Σ≡Prop isPropIsCompatibleFamily (funExt p)
elementToCompatibleFamily : ⟨ P ⟅ c ⟆ ⟩ → CompatibleFamily
elementToCompatibleFamily x =
(λ i → (P ⟪ patchArr cov i ⟫) x) ,
λ i j d f g eq → cong (λ f → f x) (
P ⟪ f ⟫ ∘ P ⟪ patchArr cov i ⟫ ≡⟨ sym (F-seq (patchArr cov i) f ) ⟩
P ⟪ f ⋆ patchArr cov i ⟫ ≡⟨ cong (P ⟪_⟫) eq ⟩
P ⟪ g ⋆ patchArr cov j ⟫ ≡⟨ F-seq (patchArr cov j) g ⟩
P ⟪ g ⟫ ∘ P ⟪ patchArr cov j ⟫ ∎ )
where
open Functor P
hasAmalgamationPropertyForCover : Type (ℓ-max (ℓ-max (ℓ-max ℓ ℓ') ℓP) ℓ'')
hasAmalgamationPropertyForCover =
isEquiv elementToCompatibleFamily
isPropHasAmalgamationPropertyForCover : isProp hasAmalgamationPropertyForCover
isPropHasAmalgamationPropertyForCover =
isPropIsEquiv _
module _
{ℓ ℓ' ℓcov ℓpat : Level}
{C : Category ℓ ℓ'}
(J : Coverage C ℓcov ℓpat)
{ℓP : Level}
(P : Presheaf C ℓP)
where
open Coverage J
isSeparated : Type (ℓ-max (ℓ-max (ℓ-max ℓ ℓcov) ℓpat) ℓP)
isSeparated =
(c : _) →
(cov : ⟨ covers c ⟩) →
(x : ⟨ P ⟅ c ⟆ ⟩) →
(y : ⟨ P ⟅ c ⟆ ⟩) →
( (patch : _) →
let f = patchArr (str (covers c) cov) patch
in (P ⟪ f ⟫) x ≡ (P ⟪ f ⟫) y ) →
x ≡ y
isPropIsSeparated : isProp isSeparated
isPropIsSeparated = isPropΠ5 (λ c _ _ _ _ → str (P ⟅ c ⟆) _ _)
isSheaf : Type (ℓ-max (ℓ-max (ℓ-max (ℓ-max ℓ ℓ') ℓcov) ℓpat) ℓP)
isSheaf =
(c : _) →
(cov : ⟨ covers c ⟩) →
hasAmalgamationPropertyForCover P (str (covers c) cov)
isPropIsSheaf : isProp isSheaf
isPropIsSheaf = isPropΠ2 λ c cov → isPropHasAmalgamationPropertyForCover P (str (covers c) cov)
isSheaf→isSeparated : isSheaf → isSeparated
isSheaf→isSeparated isSheafP c cov x y locallyEqual =
isEmbedding→Inj (isEquiv→isEmbedding (isSheafP c cov)) x y
(Σ≡Prop
(isPropIsCompatibleFamily {C = C} P _)
(funExt locallyEqual))
module _
{ℓ ℓ' ℓcov ℓpat : Level}
{C : Category ℓ ℓ'}
(J : Coverage C ℓcov ℓpat)
(ℓF : Level)
where
Sheaf : Type (ℓ-max (ℓ-max (ℓ-max (ℓ-max ℓ ℓ') ℓcov) ℓpat) (ℓ-suc ℓF))
Sheaf = Σ[ P ∈ Presheaf C ℓF ] isSheaf J P
SheafCategory :
Category
(ℓ-max (ℓ-max (ℓ-max (ℓ-max ℓ ℓ') ℓcov) ℓpat) (ℓ-suc ℓF))
(ℓ-max (ℓ-max ℓ ℓ') ℓF)
SheafCategory = FullSubcategory (PresheafCategory C ℓF) (isSheaf J)
module _
{ℓ ℓ' ℓcov ℓpat : Level}
{C : Category ℓ ℓ'}
(J : Coverage C ℓcov ℓpat)
where
isSubcanonical : Type (ℓ-max (ℓ-max (ℓ-max ℓ ℓ') ℓcov) ℓpat)
isSubcanonical = ∀ c → isSheaf J (yo c)
isPropIsSubcanonical : isProp isSubcanonical
isPropIsSubcanonical = isPropΠ λ c → isPropIsSheaf J (yo c)