Cover-preserving functors between sites.

We define cover-preserving functors between sites as functors that push covering sieves to covering sieves. A cover-preserving and compatible-preserving functor G : C ⥤ D then pulls sheaves on D back to sheaves on C via G.op ⋙ -.

Main definitions

• CategoryTheory.CoverPreserving: a functor between sites is cover-preserving if it pushes covering sieves to covering sieves
• CategoryTheory.CompatiblePreserving: a functor between sites is compatible-preserving if it pushes compatible families of elements to compatible families.
• CategoryTheory.pullbackSheaf: the pullback of a sheaf along a cover-preserving and compatible-preserving functor.
• CategoryTheory.Sites.pullback: the induced functor Sheaf K A ⥤ Sheaf J A for a cover-preserving and compatible-preserving functor G : (C, J) ⥤ (D, K).

Main results

• CategoryTheory.pullback_isSheaf_of_coverPreserving: If G : C ⥤ D is cover-preserving and compatible-preserving, then G ⋙ - (uᵖ) as a functor (Dᵒᵖ ⥤ A) ⥤ (Cᵒᵖ ⥤ A) of presheaves maps sheaves to sheaves.

References

• [Elephant]: Sketches of an Elephant, P. T. Johnstone: C2.3.
• https://stacks.math.columbia.edu/tag/00WW

structure CategoryTheory.CoverPreserving {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (J : CategoryTheory.GrothendieckTopology C) (K : CategoryTheory.GrothendieckTopology D) (G : CategoryTheory.Functor C D) : Prop

• cover_preserve : ∀ {U : C} {S : CategoryTheory.Sieve U}, S ∈ CategoryTheory.GrothendieckTopology.sieves J U → CategoryTheory.Sieve.functorPushforward G S ∈ CategoryTheory.GrothendieckTopology.sieves K (G.obj U)

A functor G : (C, J) ⥤ (D, K) between sites is cover-preserving if for all covering sieves R in C, R.functorPushforward G is a covering sieve in D.

theorem CategoryTheory.idCoverPreserving {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (J : CategoryTheory.GrothendieckTopology C) : CategoryTheory.CoverPreserving J J (CategoryTheory.Functor.id C)

The identity functor on a site is cover-preserving.

theorem CategoryTheory.CoverPreserving.comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {A : Type u₃} [CategoryTheory.Category.{v₃, u₃} A] (J : CategoryTheory.GrothendieckTopology C) (K : CategoryTheory.GrothendieckTopology D) {L : CategoryTheory.GrothendieckTopology A} {F : CategoryTheory.Functor C D} (hF : CategoryTheory.CoverPreserving J K F) {G : CategoryTheory.Functor D A} (hG : CategoryTheory.CoverPreserving K L G) : CategoryTheory.CoverPreserving J L (CategoryTheory.Functor.comp F G)

The composition of two cover-preserving functors is cover-preserving.

structure CategoryTheory.CompatiblePreserving {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (K : CategoryTheory.GrothendieckTopology D) (G : CategoryTheory.Functor C D) : Prop

• Compatible : ∀ (ℱ : CategoryTheory.SheafOfTypes K) {Z : C} {T : CategoryTheory.Presieve Z} {x : CategoryTheory.Presieve.FamilyOfElements (CategoryTheory.Functor.comp G.op ℱ.val) T}, CategoryTheory.Presieve.FamilyOfElements.Compatible x → ∀ {Y₁ Y₂ : C} {X : D} (f₁ : X ⟶ G.obj Y₁) (f₂ : X ⟶ G.obj Y₂) {g₁ : Y₁ ⟶ Z} {g₂ : Y₂ ⟶ Z} (hg₁ : T g₁) (hg₂ : T g₂), CategoryTheory.CategoryStruct.comp f₁ (G.map g₁) = CategoryTheory.CategoryStruct.comp f₂ (G.map g₂) → ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)

A functor G : (C, J) ⥤ (D, K) between sites is called compatible preserving if for each compatible family of elements at C and valued in G.op ⋙ ℱ, and each commuting diagram f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂, x g₁ and x g₂ coincide when restricted via fᵢ. This is actually stronger than merely preserving compatible families because of the definition of functorPushforward used.

theorem CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {K : CategoryTheory.GrothendieckTopology D} {G : CategoryTheory.Functor C D} (hG : CategoryTheory.CompatiblePreserving K G) (ℱ : CategoryTheory.SheafOfTypes K) {Z : C} {T : CategoryTheory.Presieve Z} {x : CategoryTheory.Presieve.FamilyOfElements (CategoryTheory.Functor.comp G.op ℱ.val) T} (h : CategoryTheory.Presieve.FamilyOfElements.Compatible x) : CategoryTheory.Presieve.FamilyOfElements.Compatible (CategoryTheory.Presieve.FamilyOfElements.functorPushforward G x)

CompatiblePreserving functors indeed preserve compatible families.

@[simp]

theorem CategoryTheory.CompatiblePreserving.apply_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {K : CategoryTheory.GrothendieckTopology D} {G : CategoryTheory.Functor C D} (hG : CategoryTheory.CompatiblePreserving K G) (ℱ : CategoryTheory.SheafOfTypes K) {Z : C} {T : CategoryTheory.Presieve Z} {x : CategoryTheory.Presieve.FamilyOfElements (CategoryTheory.Functor.comp G.op ℱ.val) T} (h : CategoryTheory.Presieve.FamilyOfElements.Compatible x) {Y : C} {f : Y ⟶ Z} (hf : T Y f) : CategoryTheory.Presieve.FamilyOfElements.functorPushforward G x (G.map f) (_ : CategoryTheory.Presieve.functorPushforward G T (G.map f)) = x Y f hf

theorem CategoryTheory.compatiblePreservingOfFlat {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₁} [CategoryTheory.Category.{v₁, u₁} D] (K : CategoryTheory.GrothendieckTopology D) (G : CategoryTheory.Functor C D) [CategoryTheory.RepresentablyFlat G] : CategoryTheory.CompatiblePreserving K G

theorem CategoryTheory.compatiblePreservingOfDownwardsClosed {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {K : CategoryTheory.GrothendieckTopology D} (F : CategoryTheory.Functor C D) [CategoryTheory.Full F] [CategoryTheory.Faithful F] (hF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)) : CategoryTheory.CompatiblePreserving K F

theorem CategoryTheory.pullback_isSheaf_of_coverPreserving {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {A : Type u₃} [CategoryTheory.Category.{v₃, u₃} A] {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} {G : CategoryTheory.Functor C D} (hG₁ : CategoryTheory.CompatiblePreserving K G) (hG₂ : CategoryTheory.CoverPreserving J K G) (ℱ : CategoryTheory.Sheaf K A) : CategoryTheory.Presheaf.IsSheaf J (CategoryTheory.Functor.comp G.op ℱ.val)

If G is cover-preserving and compatible-preserving, then G.op ⋙ _ pulls sheaves back to sheaves.

This result is basically https://stacks.math.columbia.edu/tag/00WW.

def CategoryTheory.pullbackSheaf {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {A : Type u₃} [CategoryTheory.Category.{v₃, u₃} A] {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} {G : CategoryTheory.Functor C D} (hG₁ : CategoryTheory.CompatiblePreserving K G) (hG₂ : CategoryTheory.CoverPreserving J K G) (ℱ : CategoryTheory.Sheaf K A) : CategoryTheory.Sheaf J A

The pullback of a sheaf along a cover-preserving and compatible-preserving functor.

@[simp]

theorem CategoryTheory.Sites.pullback_map_val {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (A : Type u₃) [CategoryTheory.Category.{v₃, u₃} A] {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} {G : CategoryTheory.Functor C D} (hG₁ : CategoryTheory.CompatiblePreserving K G) (hG₂ : CategoryTheory.CoverPreserving J K G) : ∀ {X Y : CategoryTheory.Sheaf K A} (f : X ⟶ Y), ((CategoryTheory.Sites.pullback A hG₁ hG₂).map f).val = ((CategoryTheory.whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).map f.val

@[simp]

theorem CategoryTheory.Sites.pullback_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (A : Type u₃) [CategoryTheory.Category.{v₃, u₃} A] {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} {G : CategoryTheory.Functor C D} (hG₁ : CategoryTheory.CompatiblePreserving K G) (hG₂ : CategoryTheory.CoverPreserving J K G) (ℱ : CategoryTheory.Sheaf K A) : (CategoryTheory.Sites.pullback A hG₁ hG₂).obj ℱ = CategoryTheory.pullbackSheaf hG₁ hG₂ ℱ

def CategoryTheory.Sites.pullback {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (A : Type u₃) [CategoryTheory.Category.{v₃, u₃} A] {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} {G : CategoryTheory.Functor C D} (hG₁ : CategoryTheory.CompatiblePreserving K G) (hG₂ : CategoryTheory.CoverPreserving J K G) : CategoryTheory.Functor (CategoryTheory.Sheaf K A) (CategoryTheory.Sheaf J A)

The induced functor from Sheaf K A ⥤ Sheaf J A given by G.op ⋙ _ if G is cover-preserving and compatible-preserving.