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

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

CategoryTheory.PreOneHypercover (F.obj X)

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

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

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theorem CategoryTheory.PreOneHypercover.map_p₁ {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {X : C} (E : CategoryTheory.PreOneHypercover X) (F : CategoryTheory.Functor C D) :

∀ (x x_1 : E.I₀) (j : E.I₁ x x_1), (E.map F).p₁ j = F.map (E.p₁ j)

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

∀ (x x_1 : E.I₀) (j : E.I₁ x x_1), (E.map F).Y j = F.obj (E.Y j)

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

(E.map F).I₀ = E.I₀

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

(E.map F).f i = F.map (E.f i)

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

(E.map F).I₁ i₁ i₂ = E.I₁ i₁ i₂

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

(E.map F).X i = F.obj (E.X i)

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theorem CategoryTheory.PreOneHypercover.map_p₂ {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {X : C} (E : CategoryTheory.PreOneHypercover X) (F : CategoryTheory.Functor C D) :

∀ (x x_1 : E.I₀) (j : E.I₁ x x_1), (E.map F).p₂ j = F.map (E.p₂ j)

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

CategoryTheory.Limits.IsLimit ((E.map F).multifork P) ≃ CategoryTheory.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.

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

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class CategoryTheory.GrothendieckTopology.OneHypercover.IsPreservedBy {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {J : CategoryTheory.GrothendieckTopology C} {X : C} (E : J.OneHypercover X) (F : CategoryTheory.Functor C D) (K : CategoryTheory.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)), CategoryTheory.CategoryStruct.comp p₁ (F.map (E.f i₁)) = CategoryTheory.CategoryStruct.comp p₂ (F.map (E.f i₂)) → (E.map F).sieve₁ p₁ p₂ ∈ K W

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theorem CategoryTheory.GrothendieckTopology.OneHypercover.IsPreservedBy.mem₀ {C : Type u₁} :

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theorem CategoryTheory.GrothendieckTopology.OneHypercover.IsPreservedBy.mem₁ {C : Type u₁} :

∀ {inst : CategoryTheory.Category.{v₁, u₁} C} {D : Type u₂} {inst_1 : CategoryTheory.Category.{v₂, u₂} D} {J : CategoryTheory.GrothendieckTopology C} {X : C} {E : J.OneHypercover X} {F : CategoryTheory.Functor C D} {K : CategoryTheory.GrothendieckTopology D} [self : E.IsPreservedBy F K] (i₁ i₂ : E.I₀) ⦃W : D⦄ (p₁ : W ⟶ F.obj (E.X i₁)) (p₂ : W ⟶ F.obj (E.X i₂)), CategoryTheory.CategoryStruct.comp p₁ (F.map (E.f i₁)) = CategoryTheory.CategoryStruct.comp p₂ (F.map (E.f i₂)) → (E.map F).sieve₁ p₁ p₂ ∈ K W

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

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

(E.map F K).toPreOneHypercover = E.map F

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

E.IsPreservedBy (CategoryTheory.Functor.id C) J

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⋯ = ⋯

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@[reducible, inline]

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

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

Prop

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

op_comp_isSheafOfTypes : ∀ (G : CategoryTheory.SheafOfTypes K), CategoryTheory.Presieve.IsSheaf J (F.op.comp G.val)

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theorem CategoryTheory.Functor.IsContinuous.op_comp_isSheafOfTypes {C : Type u₁} :

∀ {inst : CategoryTheory.Category.{v₁, u₁} C} {D : Type u₂} {inst_1 : CategoryTheory.Category.{v₂, u₂} D} {F : CategoryTheory.Functor C D} {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} [self : F.IsContinuous J K] (G : CategoryTheory.SheafOfTypes K), CategoryTheory.Presieve.IsSheaf J (F.op.comp G.val)

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

CategoryTheory.Presieve.IsSheaf J (F.op.comp G.val)

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

CategoryTheory.Presheaf.IsSheaf J (F.op.comp G.val)

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

F₂.IsContinuous J K

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

(CategoryTheory.Functor.id C).IsContinuous J J

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⋯ = ⋯

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

(F₁.comp F₂).IsContinuous J L

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

F₁₂.IsContinuous J L

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

CategoryTheory.Presheaf.IsSheaf J (F.op.comp P)

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

F.IsContinuous J K

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

F.IsContinuous J K

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⋯ = ⋯

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

CategoryTheory.Functor (CategoryTheory.Sheaf K A) (CategoryTheory.Sheaf J A)

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

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

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

∀ {X Y : CategoryTheory.Sheaf K A} (f : X ⟶ Y) (X_1 : Cᵒᵖ), ((F.sheafPushforwardContinuous A J K).map f).val.app X_1 = f.val.app (Opposite.op (F.obj (Opposite.unop X_1)))

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@[simp]

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

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@[simp]

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

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

(F.sheafPushforwardContinuous A J K).comp (CategoryTheory.sheafToPresheaf J A) ≅ (CategoryTheory.sheafToPresheaf K A).comp ((CategoryTheory.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))

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

((F.sheafPushforwardContinuousCompSheafToPresheafIso A J K).hom.app X✝).app X = CategoryTheory.CategoryStruct.id (X✝.val.obj (Opposite.op (F.obj (Opposite.unop X))))

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

((F.sheafPushforwardContinuousCompSheafToPresheafIso A J K).inv.app X✝).app X = CategoryTheory.CategoryStruct.id (X✝.val.obj (Opposite.op (F.obj (Opposite.unop X))))

Documentation

Mathlib.CategoryTheory.Sites.Continuous

Continuous functors between sites. #

Main definitions #

Main result #

References #