Continuous functors between sites.

We define the notion of continuous functor between sites: these are functors F such that the precomposition with F.op preserves sheaves of types (and actually sheaves in any category).

Main definitions

• Functor.IsContinuous: a functor between sites is continuous if the precomposition with this functor preserves sheaves with values in the category Type t for a certain auxiliary universe t.
• Functor.sheafPushforwardContinuous: the induced functor Sheaf K A ⥤ Sheaf J A for a continuous functor G : (C, J) ⥤ (D, K). In case this is part of a morphism of sites, this would be understood as the pushforward functor even though it goes in the opposite direction as the functor G. (Here, the auxiliary universe t in the assumption that G is continuous is the one such that morphisms in the category A are in Type t.)
• Functor.PreservesOneHypercovers: a type-class expressing that a functor preserves 1-hypercovers of a certain size

Main result

• Functor.isContinuous_of_preservesOneHypercovers: if the topology on C is generated by 1-hypercovers of size w and that F : C ⥤ D preserves 1-hypercovers of size w, then F is continuous (for any auxiliary universe parameter t). This is an instance for w = max u₁ v₁ when C : Type u₁ and [Category.{v₁} C]

References

• https://stacks.math.columbia.edu/tag/00WU

def CategoryTheory.PreOneHypercover.map {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {X : C} (E : PreOneHypercover X) (F : Functor C D) : PreOneHypercover (F.obj X)

The image of a 1-pre-hypercover by a functor.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.PreOneHypercover.map_p₁ {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {X : C} (E : PreOneHypercover X) (F : Functor C D) (x✝ x✝¹ : E.I₀) (j : E.I₁ x✝ x✝¹) : (E.map F).p₁ j = F.map (E.p₁ j)

@[simp]

theorem CategoryTheory.PreOneHypercover.map_Y {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {X : C} (E : PreOneHypercover X) (F : Functor C D) (x✝ x✝¹ : E.I₀) (j : E.I₁ x✝ x✝¹) : (E.map F).Y j = F.obj (E.Y j)

@[simp]

theorem CategoryTheory.PreOneHypercover.map_I₀ {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {X : C} (E : PreOneHypercover X) (F : Functor C D) : (E.map F).I₀ = E.I₀

@[simp]

theorem CategoryTheory.PreOneHypercover.map_f {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {X : C} (E : PreOneHypercover X) (F : Functor C D) (i : E.I₀) : (E.map F).f i = F.map (E.f i)

@[simp]

theorem CategoryTheory.PreOneHypercover.map_I₁ {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {X : C} (E : PreOneHypercover X) (F : Functor C D) (i₁ i₂ : E.I₀) : (E.map F).I₁ i₁ i₂ = E.I₁ i₁ i₂

@[simp]

theorem CategoryTheory.PreOneHypercover.map_X {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {X : C} (E : PreOneHypercover X) (F : Functor C D) (i : E.I₀) : (E.map F).X i = F.obj (E.X i)

@[simp]

theorem CategoryTheory.PreOneHypercover.map_p₂ {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {X : C} (E : PreOneHypercover X) (F : Functor C D) (x✝ x✝¹ : E.I₀) (j : E.I₁ x✝ x✝¹) : (E.map F).p₂ j = F.map (E.p₂ j)

def CategoryTheory.PreOneHypercover.isLimitMapMultiforkEquiv {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {X : C} (E : PreOneHypercover X) (F : Functor C D) {A : Type u} [Category.{t, u} A] (P : Functor Dᵒᵖ A) : Limits.IsLimit ((E.map F).multifork P) ≃ Limits.IsLimit (E.multifork (F.op.comp P))

If F : C ⥤ D, P : Dᵒᵖ ⥤ A and E is a 1-pre-hypercover of an object of X, then (E.map F).multifork P is a limit iff E.multifork (F.op ⋙ P) is a limit.

Equations

• E.isLimitMapMultiforkEquiv F P = Equiv.refl (CategoryTheory.Limits.IsLimit ((E.map F).multifork P))

class CategoryTheory.GrothendieckTopology.OneHypercover.IsPreservedBy {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {J : GrothendieckTopology C} {X : C} (E : J.OneHypercover X) (F : Functor C D) (K : GrothendieckTopology D) : Prop

A 1-hypercover in C is preserved by a functor F : C ⥤ D if the mapped 1-pre-hypercover in D is a 1-hypercover for the given topology on D.

• mem₀ : (E.map F).sieve₀ ∈ K (F.obj X)
• mem₁ (i₁ i₂ : E.I₀) ⦃W : D⦄ (p₁ : W ⟶ F.obj (E.X i₁)) (p₂ : W ⟶ F.obj (E.X i₂)) (w : CategoryStruct.comp p₁ (F.map (E.f i₁)) = CategoryStruct.comp p₂ (F.map (E.f i₂))) : (E.map F).sieve₁ p₁ p₂ ∈ K W

def CategoryTheory.GrothendieckTopology.OneHypercover.map {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {J : GrothendieckTopology C} {X : C} (E : J.OneHypercover X) (F : Functor C D) (K : GrothendieckTopology D) [E.IsPreservedBy F K] : K.OneHypercover (F.obj X)

Given a 1-hypercover E : J.OneHypercover X of an object of C, a functor F : C ⥤ D such that E.IsPreversedBy F K for a Grothendieck topology K on D, this is the image of E by F, as a 1-hypercover of F.obj X for K.

Equations

• E.map F K = { toPreOneHypercover := E.map F, mem₀ := ⋯, mem₁ := ⋯ }

@[simp]

theorem CategoryTheory.GrothendieckTopology.OneHypercover.map_toPreOneHypercover {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {J : GrothendieckTopology C} {X : C} (E : J.OneHypercover X) (F : Functor C D) (K : GrothendieckTopology D) [E.IsPreservedBy F K] : (E.map F K).toPreOneHypercover = E.map F

instance CategoryTheory.GrothendieckTopology.OneHypercover.instIsPreservedById {C : Type u₁} [Category.{v₁, u₁} C] {J : GrothendieckTopology C} {X : C} (E : J.OneHypercover X) : E.IsPreservedBy (Functor.id C) J

@[reducible, inline]

abbrev CategoryTheory.Functor.PreservesOneHypercovers {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (J : GrothendieckTopology C) (K : GrothendieckTopology D) : Prop

The condition that a functor F : C ⥤ D sends 1-hypercovers for J : GrothendieckTopology C to 1-hypercovers for K : GrothendieckTopology D.

Equations

• F.PreservesOneHypercovers J K = ∀ {X : C} (E : J.OneHypercover X), E.IsPreservedBy F K

class CategoryTheory.Functor.IsContinuous {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (J : GrothendieckTopology C) (K : GrothendieckTopology D) : Prop

A functor F is continuous if the precomposition with F.op sends sheaves of Type t to sheaves.

• op_comp_isSheaf_of_types (G : Sheaf K (Type t)) : Presieve.IsSheaf J (F.op.comp G.val)

theorem CategoryTheory.Functor.op_comp_isSheaf_of_types {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.IsContinuous J K] (G : Sheaf K (Type t)) : Presieve.IsSheaf J (F.op.comp G.val)

@[deprecated CategoryTheory.Functor.op_comp_isSheaf_of_types (since := "2024-11-26")]

theorem CategoryTheory.Functor.op_comp_isSheafOfTypes {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.IsContinuous J K] (G : Sheaf K (Type t)) : Presieve.IsSheaf J (F.op.comp G.val)

Alias of CategoryTheory.Functor.op_comp_isSheaf_of_types.

theorem CategoryTheory.Functor.op_comp_isSheaf {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {A : Type u} [Category.{t, u} A] (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.IsContinuous J K] (G : Sheaf K A) : Presheaf.IsSheaf J (F.op.comp G.val)

theorem CategoryTheory.Functor.isContinuous_of_iso {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F₁ F₂ : Functor C D} (e : F₁ ≅ F₂) (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F₁.IsContinuous J K] : F₂.IsContinuous J K

instance CategoryTheory.Functor.isContinuous_id {C : Type u₁} [Category.{v₁, u₁} C] (J : GrothendieckTopology C) : (Functor.id C).IsContinuous J J

theorem CategoryTheory.Functor.isContinuous_comp {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {E : Type u₃} [Category.{v₃, u₃} E] (F₁ : Functor C D) (F₂ : Functor D E) (J : GrothendieckTopology C) (K : GrothendieckTopology D) (L : GrothendieckTopology E) [F₁.IsContinuous J K] [F₂.IsContinuous K L] : (F₁.comp F₂).IsContinuous J L

theorem CategoryTheory.Functor.isContinuous_comp' {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {E : Type u₃} [Category.{v₃, u₃} E] {F₁ : Functor C D} {F₂ : Functor D E} {F₁₂ : Functor C E} (e : F₁.comp F₂ ≅ F₁₂) (J : GrothendieckTopology C) (K : GrothendieckTopology D) (L : GrothendieckTopology E) [F₁.IsContinuous J K] [F₂.IsContinuous K L] : F₁₂.IsContinuous J L

theorem CategoryTheory.Functor.op_comp_isSheaf_of_preservesOneHypercovers {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {A : Type u} [Category.{t, u} A] (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.PreservesOneHypercovers J K] [J.IsGeneratedByOneHypercovers] (P : Functor Dᵒᵖ A) (hP : Presheaf.IsSheaf K P) : Presheaf.IsSheaf J (F.op.comp P)

theorem CategoryTheory.Functor.isContinuous_of_preservesOneHypercovers {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.PreservesOneHypercovers J K] [J.IsGeneratedByOneHypercovers] : F.IsContinuous J K

instance CategoryTheory.Functor.instIsContinuousOfPreservesOneHypercovers {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.PreservesOneHypercovers J K] : F.IsContinuous J K

def CategoryTheory.Functor.sheafPushforwardContinuous {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (A : Type u) [Category.{t, u} A] (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.IsContinuous J K] : Functor (Sheaf K A) (Sheaf J A)

The induced functor Sheaf K A ⥤ Sheaf J A given by F.op ⋙ _ if F is a continuous functor.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.Functor.sheafPushforwardContinuous_obj_val_map {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (A : Type u) [Category.{t, u} A] (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.IsContinuous J K] (ℱ : Sheaf K A) {X✝ Y✝ : Cᵒᵖ} (f : X✝ ⟶ Y✝) : ((F.sheafPushforwardContinuous A J K).obj ℱ).val.map f = ℱ.val.map (F.map f.unop).op

@[simp]

theorem CategoryTheory.Functor.sheafPushforwardContinuous_obj_val_obj {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (A : Type u) [Category.{t, u} A] (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.IsContinuous J K] (ℱ : Sheaf K A) (X : Cᵒᵖ) : ((F.sheafPushforwardContinuous A J K).obj ℱ).val.obj X = ℱ.val.obj (Opposite.op (F.obj (Opposite.unop X)))

@[simp]

theorem CategoryTheory.Functor.sheafPushforwardContinuous_map_val_app {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (A : Type u) [Category.{t, u} A] (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.IsContinuous J K] {X✝ Y✝ : Sheaf K A} (f : X✝ ⟶ Y✝) (X : Cᵒᵖ) : ((F.sheafPushforwardContinuous A J K).map f).val.app X = f.val.app (Opposite.op (F.obj (Opposite.unop X)))

def CategoryTheory.Functor.sheafPushforwardContinuousCompSheafToPresheafIso {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (A : Type u) [Category.{t, u} A] (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.IsContinuous J K] : (F.sheafPushforwardContinuous A J K).comp (sheafToPresheaf J A) ≅ (sheafToPresheaf K A).comp ((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj F.op)

The functor F.sheafPushforwardContinuous A J K : Sheaf K A ⥤ Sheaf J A is induced by the precomposition with F.op.

Equations

• F.sheafPushforwardContinuousCompSheafToPresheafIso A J K = CategoryTheory.Iso.refl ((F.sheafPushforwardContinuous A J K).comp (CategoryTheory.sheafToPresheaf J A))

@[simp]

theorem CategoryTheory.Functor.sheafPushforwardContinuousCompSheafToPresheafIso_inv_app_app {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (A : Type u) [Category.{t, u} A] (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.IsContinuous J K] (X : Sheaf K A) (X✝ : Cᵒᵖ) : ((F.sheafPushforwardContinuousCompSheafToPresheafIso A J K).inv.app X).app X✝ = CategoryStruct.id (X.val.obj (Opposite.op (F.obj (Opposite.unop X✝))))

@[simp]

theorem CategoryTheory.Functor.sheafPushforwardContinuousCompSheafToPresheafIso_hom_app_app {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (A : Type u) [Category.{t, u} A] (J : GrothendieckTopology C) (K : GrothendieckTopology D) [F.IsContinuous J K] (X : Sheaf K A) (X✝ : Cᵒᵖ) : ((F.sheafPushforwardContinuousCompSheafToPresheafIso A J K).hom.app X).app X✝ = CategoryStruct.id (X.val.obj (Opposite.op (F.obj (Opposite.unop X✝))))