Subsheaf of types

We define the sub(pre)sheaf of a type valued presheaf.

Main results

• CategoryTheory.Subpresheaf : A subpresheaf of a presheaf of types.
• CategoryTheory.Subpresheaf.sheafify : The sheafification of a subpresheaf as a subpresheaf. Note that this is a sheaf only when the whole sheaf is.
• CategoryTheory.Subpresheaf.sheafify_isSheaf : The sheafification is a sheaf
• CategoryTheory.Subpresheaf.sheafifyLift : The descent of a map into a sheaf to the sheafification.
• CategoryTheory.GrothendieckTopology.imageSheaf : The image sheaf of a morphism.
• CategoryTheory.GrothendieckTopology.imageFactorization : The image sheaf as a Limits.imageFactorization.

def CategoryTheory.Subpresheaf.sheafify {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {F : Functor Cᵒᵖ (Type w)} (G : Subpresheaf F) : Subpresheaf F

The sheafification of a subpresheaf as a subpresheaf. Note that this is a sheaf only when the whole presheaf is a sheaf.

Equations

• CategoryTheory.Subpresheaf.sheafify J G = { obj := fun (U : Cᵒᵖ) => {s : F.obj U | G.sieveOfSection s ∈ J (Opposite.unop U)}, map := ⋯ }

theorem CategoryTheory.Subpresheaf.le_sheafify {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {F : Functor Cᵒᵖ (Type w)} (G : Subpresheaf F) : G ≤ sheafify J G

theorem CategoryTheory.Subpresheaf.eq_sheafify {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F : Functor Cᵒᵖ (Type w)} (G : Subpresheaf F) (h : Presieve.IsSheaf J F) (hG : Presieve.IsSheaf J G.toPresheaf) : G = sheafify J G

theorem CategoryTheory.Subpresheaf.sheafify_isSheaf {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F : Functor Cᵒᵖ (Type w)} (G : Subpresheaf F) (hF : Presieve.IsSheaf J F) : Presieve.IsSheaf J (sheafify J G).toPresheaf

theorem CategoryTheory.Subpresheaf.eq_sheafify_iff {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F : Functor Cᵒᵖ (Type w)} (G : Subpresheaf F) (h : Presieve.IsSheaf J F) : G = sheafify J G ↔ Presieve.IsSheaf J G.toPresheaf

theorem CategoryTheory.Subpresheaf.isSheaf_iff {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F : Functor Cᵒᵖ (Type w)} (G : Subpresheaf F) (h : Presieve.IsSheaf J F) : Presieve.IsSheaf J G.toPresheaf ↔ ∀ (U : Cᵒᵖ) (s : F.obj U), G.sieveOfSection s ∈ J (Opposite.unop U) → s ∈ G.obj U

theorem CategoryTheory.Subpresheaf.sheafify_sheafify {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F : Functor Cᵒᵖ (Type w)} (G : Subpresheaf F) (h : Presieve.IsSheaf J F) : sheafify J (sheafify J G) = sheafify J G

noncomputable def CategoryTheory.Subpresheaf.sheafifyLift {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Functor Cᵒᵖ (Type w)} (G : Subpresheaf F) (f : G.toPresheaf ⟶ F') (h : Presieve.IsSheaf J F') : (sheafify J G).toPresheaf ⟶ F'

The lift of a presheaf morphism onto the sheafification subpresheaf.

Equations

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

theorem CategoryTheory.Subpresheaf.to_sheafifyLift {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Functor Cᵒᵖ (Type w)} (G : Subpresheaf F) (f : G.toPresheaf ⟶ F') (h : Presieve.IsSheaf J F') : CategoryStruct.comp (homOfLe ⋯) (G.sheafifyLift f h) = f

theorem CategoryTheory.Subpresheaf.to_sheafify_lift_unique {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Functor Cᵒᵖ (Type w)} (G : Subpresheaf F) (h : Presieve.IsSheaf J F') (l₁ l₂ : (sheafify J G).toPresheaf ⟶ F') (e : CategoryStruct.comp (homOfLe ⋯) l₁ = CategoryStruct.comp (homOfLe ⋯) l₂) : l₁ = l₂

theorem CategoryTheory.Subpresheaf.sheafify_le {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F : Functor Cᵒᵖ (Type w)} (G G' : Subpresheaf F) (h : G ≤ G') (hF : Presieve.IsSheaf J F) (hG' : Presieve.IsSheaf J G'.toPresheaf) : sheafify J G ≤ G'

def CategoryTheory.Subpresheaf.toRangeSheafify {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {F F' : Functor Cᵒᵖ (Type w)} (f : F' ⟶ F) : F' ⟶ (sheafify J (range f)).toPresheaf

A morphism factors through the sheafification of the image presheaf.

Equations

• CategoryTheory.Subpresheaf.toRangeSheafify J f = CategoryTheory.CategoryStruct.comp (CategoryTheory.Subpresheaf.toRange f) (CategoryTheory.Subpresheaf.homOfLe ⋯)

@[simp]

theorem CategoryTheory.Subpresheaf.toRangeSheafify_app_coe {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {F F' : Functor Cᵒᵖ (Type w)} (f : F' ⟶ F) (X : Cᵒᵖ) (a✝ : F'.obj X) : ↑((toRangeSheafify J f).app X a✝) = ↑((toRange f).app X a✝)

def CategoryTheory.Sheaf.image {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Sheaf J (Type w)

The image sheaf of a morphism between sheaves, defined to be the sheafification of image_presheaf.

Equations

• CategoryTheory.Sheaf.image f = { val := (CategoryTheory.Subpresheaf.sheafify J (CategoryTheory.Subpresheaf.range f.val)).toPresheaf, cond := ⋯ }

@[simp]

theorem CategoryTheory.Sheaf.image_val {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : (image f).val = (Subpresheaf.sheafify J (Subpresheaf.range f.val)).toPresheaf

def CategoryTheory.Sheaf.toImage {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : F ⟶ image f

A morphism factors through the image sheaf.

Equations

• CategoryTheory.Sheaf.toImage f = { val := CategoryTheory.Subpresheaf.toRangeSheafify J f.val }

@[simp]

theorem CategoryTheory.Sheaf.toImage_val {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : (toImage f).val = Subpresheaf.toRangeSheafify J f.val

def CategoryTheory.Sheaf.imageι {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : image f ⟶ F'

The inclusion of the image sheaf to the target.

Equations

• CategoryTheory.Sheaf.imageι f = { val := (CategoryTheory.Subpresheaf.sheafify J (CategoryTheory.Subpresheaf.range f.val)).ι }

@[simp]

theorem CategoryTheory.Sheaf.imageι_val {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : (imageι f).val = (Subpresheaf.sheafify J (Subpresheaf.range f.val)).ι

@[simp]

theorem CategoryTheory.Sheaf.toImage_ι {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : CategoryStruct.comp (toImage f) (imageι f) = f

@[simp]

theorem CategoryTheory.Sheaf.toImage_ι_assoc {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') {Z : Sheaf J (Type w)} (h : F' ⟶ Z) : CategoryStruct.comp (toImage f) (CategoryStruct.comp (imageι f) h) = CategoryStruct.comp f h

instance CategoryTheory.instMonoSheafTypeImageι {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Mono (Sheaf.imageι f)

instance CategoryTheory.instEpiSheafTypeToImage {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Epi (Sheaf.toImage f)

def CategoryTheory.imageMonoFactorization {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Limits.MonoFactorisation f

The mono factorization given by image_sheaf for a morphism.

Equations

• CategoryTheory.imageMonoFactorization f = CategoryTheory.Limits.MonoFactorisation.mk (CategoryTheory.Sheaf.image f) (CategoryTheory.Sheaf.imageι f) (CategoryTheory.Sheaf.toImage f) ⋯

noncomputable def CategoryTheory.imageFactorization {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type (max v u))} (f : F ⟶ F') : Limits.ImageFactorisation f

The mono factorization given by image_sheaf for a morphism is an image.

Equations

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

instance CategoryTheory.instHasImagesSheafType {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} : Limits.HasImages (Sheaf J (Type (max v u)))

@[deprecated CategoryTheory.Subpresheaf.toRangeSheafify (since := "2025-01-25")]

def CategoryTheory.toImagePresheafSheafify {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {F F' : Functor Cᵒᵖ (Type w)} (f : F' ⟶ F) : F' ⟶ (Subpresheaf.sheafify J (Subpresheaf.range f)).toPresheaf

Alias of CategoryTheory.Subpresheaf.toRangeSheafify.

---

A morphism factors through the sheafification of the image presheaf.

Equations

• @CategoryTheory.toImagePresheafSheafify = @CategoryTheory.Subpresheaf.toRangeSheafify

@[deprecated CategoryTheory.Sheaf.image (since := "2025-01-25")]

def CategoryTheory.imageSheaf {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Sheaf J (Type w)

Alias of CategoryTheory.Sheaf.image.

---

The image sheaf of a morphism between sheaves, defined to be the sheafification of image_presheaf.

Equations

• @CategoryTheory.imageSheaf = @CategoryTheory.Sheaf.image

@[deprecated CategoryTheory.Sheaf.toImage (since := "2025-01-25")]

def CategoryTheory.toImageSheaf {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : F ⟶ Sheaf.image f

Alias of CategoryTheory.Sheaf.toImage.

---

A morphism factors through the image sheaf.

Equations

• @CategoryTheory.toImageSheaf = @CategoryTheory.Sheaf.toImage

@[deprecated CategoryTheory.Sheaf.imageι (since := "2025-01-25")]

def CategoryTheory.imageSheafι {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Sheaf.image f ⟶ F'

Alias of CategoryTheory.Sheaf.imageι.

---

The inclusion of the image sheaf to the target.

Equations

• @CategoryTheory.imageSheafι = @CategoryTheory.Sheaf.imageι

@[deprecated CategoryTheory.Sheaf.toImage_ι (since := "2025-01-25")]

theorem CategoryTheory.toImageSheaf_ι {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {F F' : Sheaf J (Type w)} (f : F ⟶ F') : CategoryStruct.comp (Sheaf.toImage f) (Sheaf.imageι f) = f

Alias of CategoryTheory.Sheaf.toImage_ι.