Subsheaf of types

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

Main results

• CategoryTheory.GrothendieckTopology.Subpresheaf : A subpresheaf of a presheaf of types.
• CategoryTheory.GrothendieckTopology.Subpresheaf.sheafify : The sheafification of a subpresheaf as a subpresheaf. Note that this is a sheaf only when the whole sheaf is.
• CategoryTheory.GrothendieckTopology.Subpresheaf.sheafify_isSheaf : The sheafification is a sheaf
• CategoryTheory.GrothendieckTopology.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.

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.ext_iff {C : Type u} : ∀ {inst : CategoryTheory.Category.{v, u} C} {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {x y : CategoryTheory.GrothendieckTopology.Subpresheaf F}, x = y ↔ x.obj = y.obj

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.ext {C : Type u} : ∀ {inst : CategoryTheory.Category.{v, u} C} {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {x y : CategoryTheory.GrothendieckTopology.Subpresheaf F}, x.obj = y.obj → x = y

structure CategoryTheory.GrothendieckTopology.Subpresheaf {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor Cᵒᵖ (Type w)) : Type (max u w)

A subpresheaf of a presheaf consists of a subset of F.obj U for every U, compatible with the restriction maps F.map i.

• obj : (U : Cᵒᵖ) → Set (F.obj U)

  If G is a sub-presheaf of F, then the sections of G on U forms a subset of sections of F on U.

• map : ∀ {U V : Cᵒᵖ} (i : U ⟶ V), self.obj U ⊆ F.map i ⁻¹' self.obj V

  If G is a sub-presheaf of F and i : U ⟶ V, then for each G-sections on U x, F i x is in F(V).

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.map {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (self : CategoryTheory.GrothendieckTopology.Subpresheaf F) {U : Cᵒᵖ} {V : Cᵒᵖ} (i : U ⟶ V) : self.obj U ⊆ F.map i ⁻¹' self.obj V

If G is a sub-presheaf of F and i : U ⟶ V, then for each G-sections on U x, F i x is in F(V).

instance CategoryTheory.GrothendieckTopology.instPartialOrderSubpresheaf {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} : PartialOrder (CategoryTheory.GrothendieckTopology.Subpresheaf F)

Equations

• CategoryTheory.GrothendieckTopology.instPartialOrderSubpresheaf = PartialOrder.lift CategoryTheory.GrothendieckTopology.Subpresheaf.obj ⋯

instance CategoryTheory.GrothendieckTopology.instTopSubpresheaf {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} : Top (CategoryTheory.GrothendieckTopology.Subpresheaf F)

Equations

• CategoryTheory.GrothendieckTopology.instTopSubpresheaf = { top := { obj := fun (x : Cᵒᵖ) => ⊤, map := ⋯ } }

instance CategoryTheory.GrothendieckTopology.instNonemptySubpresheaf {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} : Nonempty (CategoryTheory.GrothendieckTopology.Subpresheaf F)

Equations

• ⋯ = ⋯

@[simp]

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.toPresheaf_obj {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) (U : Cᵒᵖ) : G.toPresheaf.obj U = ↑(G.obj U)

@[simp]

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.toPresheaf_map_coe {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) : ∀ (x x_1 : Cᵒᵖ) (i : x ⟶ x_1) (x_2 : ↑(G.obj x)), ↑(G.toPresheaf.map i x_2) = F.map i ↑x_2

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

The subpresheaf as a presheaf.

Equations

• G.toPresheaf = { obj := fun (U : Cᵒᵖ) => ↑(G.obj U), map := fun (x x_1 : Cᵒᵖ) (i : x ⟶ x_1) (x_2 : ↑(G.obj x)) => ⟨F.map i ↑x_2, ⋯⟩, map_id := ⋯, map_comp := ⋯ }

instance CategoryTheory.GrothendieckTopology.instCoeHeadObjOppositeToPresheaf {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) {U : Cᵒᵖ} : CoeHead (G.toPresheaf.obj U) (F.obj U)

Equations

• CategoryTheory.GrothendieckTopology.instCoeHeadObjOppositeToPresheaf G = { coe := Subtype.val }

@[simp]

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.ι_app {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) : ∀ (x : Cᵒᵖ) (x_1 : G.toPresheaf.obj x), G.ι.app x x_1 = ↑x_1

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

The inclusion of a subpresheaf to the original presheaf.

Equations

• G.ι = { app := fun (x : Cᵒᵖ) (x_1 : G.toPresheaf.obj x) => ↑x_1, naturality := ⋯ }

instance CategoryTheory.GrothendieckTopology.instMonoFunctorOppositeTypeι {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) : CategoryTheory.Mono G.ι

Equations

• ⋯ = ⋯

@[simp]

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe_app_coe {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {G : CategoryTheory.GrothendieckTopology.Subpresheaf F} {G' : CategoryTheory.GrothendieckTopology.Subpresheaf F} (h : G ≤ G') (U : Cᵒᵖ) (x : G.toPresheaf.obj U) : ↑((CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe h).app U x) = ↑x

def CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {G : CategoryTheory.GrothendieckTopology.Subpresheaf F} {G' : CategoryTheory.GrothendieckTopology.Subpresheaf F} (h : G ≤ G') : G.toPresheaf ⟶ G'.toPresheaf

The inclusion of a subpresheaf to a larger subpresheaf

Equations

• CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe h = { app := fun (U : Cᵒᵖ) (x : G.toPresheaf.obj U) => ⟨↑x, ⋯⟩, naturality := ⋯ }

instance CategoryTheory.GrothendieckTopology.instMonoFunctorOppositeTypeHomOfLe {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {G : CategoryTheory.GrothendieckTopology.Subpresheaf F} {G' : CategoryTheory.GrothendieckTopology.Subpresheaf F} (h : G ≤ G') : CategoryTheory.Mono (CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe h)

Equations

• ⋯ = ⋯

@[simp]

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe_ι_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {G : CategoryTheory.GrothendieckTopology.Subpresheaf F} {G' : CategoryTheory.GrothendieckTopology.Subpresheaf F} (h : G ≤ G') {Z : CategoryTheory.Functor Cᵒᵖ (Type w)} (h : F ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe h✝) (CategoryTheory.CategoryStruct.comp G'.ι h) = CategoryTheory.CategoryStruct.comp G.ι h

@[simp]

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe_ι {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {G : CategoryTheory.GrothendieckTopology.Subpresheaf F} {G' : CategoryTheory.GrothendieckTopology.Subpresheaf F} (h : G ≤ G') : CategoryTheory.CategoryStruct.comp (CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe h) G'.ι = G.ι

instance CategoryTheory.GrothendieckTopology.instIsIsoFunctorOppositeTypeιTopSubpresheaf {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} : CategoryTheory.IsIso ⊤.ι

Equations

• ⋯ = ⋯

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.eq_top_iff_isIso {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) : G = ⊤ ↔ CategoryTheory.IsIso G.ι

@[simp]

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.lift_app_coe {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) (f : F' ⟶ F) (hf : ∀ (U : Cᵒᵖ) (x : F'.obj U), f.app U x ∈ G.obj U) (U : Cᵒᵖ) (x : F'.obj U) : ↑((G.lift f hf).app U x) = f.app U x

def CategoryTheory.GrothendieckTopology.Subpresheaf.lift {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) (f : F' ⟶ F) (hf : ∀ (U : Cᵒᵖ) (x : F'.obj U), f.app U x ∈ G.obj U) : F' ⟶ G.toPresheaf

If the image of a morphism falls in a subpresheaf, then the morphism factors through it.

Equations

• G.lift f hf = { app := fun (U : Cᵒᵖ) (x : F'.obj U) => ⟨f.app U x, ⋯⟩, naturality := ⋯ }

@[simp]

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.lift_ι_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) (f : F' ⟶ F) (hf : ∀ (U : Cᵒᵖ) (x : F'.obj U), f.app U x ∈ G.obj U) {Z : CategoryTheory.Functor Cᵒᵖ (Type w)} (h : F ⟶ Z) : CategoryTheory.CategoryStruct.comp (G.lift f hf) (CategoryTheory.CategoryStruct.comp G.ι h) = CategoryTheory.CategoryStruct.comp f h

@[simp]

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.lift_ι {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) (f : F' ⟶ F) (hf : ∀ (U : Cᵒᵖ) (x : F'.obj U), f.app U x ∈ G.obj U) : CategoryTheory.CategoryStruct.comp (G.lift f hf) G.ι = f

@[simp]

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.sieveOfSection_apply {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) {U : Cᵒᵖ} (s : F.obj U) (V : C) (f : V ⟶ Opposite.unop U) : (G.sieveOfSection s).arrows f = (F.map f.op s ∈ G.obj (Opposite.op V))

def CategoryTheory.GrothendieckTopology.Subpresheaf.sieveOfSection {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) {U : Cᵒᵖ} (s : F.obj U) : CategoryTheory.Sieve (Opposite.unop U)

Given a subpresheaf G of F, an F-section s on U, we may define a sieve of U consisting of all f : V ⟶ U such that the restriction of s along f is in G.

Equations

• G.sieveOfSection s = { arrows := fun (V : C) (f : V ⟶ Opposite.unop U) => F.map f.op s ∈ G.obj (Opposite.op V), downward_closed := ⋯ }

def CategoryTheory.GrothendieckTopology.Subpresheaf.familyOfElementsOfSection {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) {U : Cᵒᵖ} (s : F.obj U) : CategoryTheory.Presieve.FamilyOfElements G.toPresheaf (G.sieveOfSection s).arrows

Given an F-section s on U and a subpresheaf G, we may define a family of elements in G consisting of the restrictions of s

Equations

• G.familyOfElementsOfSection s i hi = ⟨F.map i.op s, hi⟩

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.family_of_elements_compatible {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) {U : Cᵒᵖ} (s : F.obj U) : (G.familyOfElementsOfSection s).Compatible

theorem CategoryTheory.GrothendieckTopology.Subpresheaf.nat_trans_naturality {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) (f : F' ⟶ G.toPresheaf) {U : Cᵒᵖ} {V : Cᵒᵖ} (i : U ⟶ V) (x : F'.obj U) : ↑(f.app V (F'.map i x)) = F.map i ↑(f.app U x)

def CategoryTheory.GrothendieckTopology.Subpresheaf.sheafify {C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (G : CategoryTheory.GrothendieckTopology.Subpresheaf F) : CategoryTheory.GrothendieckTopology.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.GrothendieckTopology.Subpresheaf.sheafify J G = { obj := fun (U : Cᵒᵖ) => {s : F.obj U | G.sieveOfSection s ∈ J (Opposite.unop U)}, map := ⋯ }

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

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

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

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

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

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

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

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

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

@[simp]

theorem CategoryTheory.GrothendieckTopology.imagePresheaf_obj {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} (f : F' ⟶ F) (U : Cᵒᵖ) : (CategoryTheory.GrothendieckTopology.imagePresheaf f).obj U = Set.range (f.app U)

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

The image presheaf of a morphism, whose components are the set-theoretic images.

Equations

• CategoryTheory.GrothendieckTopology.imagePresheaf f = { obj := fun (U : Cᵒᵖ) => Set.range (f.app U), map := ⋯ }

@[simp]

theorem CategoryTheory.GrothendieckTopology.top_subpresheaf_obj {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} (U : Cᵒᵖ) : ⊤.obj U = ⊤

@[simp]

theorem CategoryTheory.GrothendieckTopology.imagePresheaf_id {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} : CategoryTheory.GrothendieckTopology.imagePresheaf (CategoryTheory.CategoryStruct.id F) = ⊤

@[simp]

theorem CategoryTheory.GrothendieckTopology.toImagePresheaf_app_coe {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} (f : F' ⟶ F) (U : Cᵒᵖ) (x : F'.obj U) : ↑((CategoryTheory.GrothendieckTopology.toImagePresheaf f).app U x) = f.app U x

def CategoryTheory.GrothendieckTopology.toImagePresheaf {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} (f : F' ⟶ F) : F' ⟶ (CategoryTheory.GrothendieckTopology.imagePresheaf f).toPresheaf

A morphism factors through the image presheaf.

Equations

• CategoryTheory.GrothendieckTopology.toImagePresheaf f = (CategoryTheory.GrothendieckTopology.imagePresheaf f).lift f ⋯

@[simp]

theorem CategoryTheory.GrothendieckTopology.toImagePresheafSheafify_app_coe {C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} (f : F' ⟶ F) (X : Cᵒᵖ) : ∀ (a : F'.obj X), ↑((J.toImagePresheafSheafify f).app X a) = f.app X a

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

A morphism factors through the sheafification of the image presheaf.

Equations

• J.toImagePresheafSheafify f = CategoryTheory.CategoryStruct.comp (CategoryTheory.GrothendieckTopology.toImagePresheaf f) (CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe ⋯)

@[simp]

theorem CategoryTheory.GrothendieckTopology.toImagePresheaf_ι_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} (f : F' ⟶ F) {Z : CategoryTheory.Functor Cᵒᵖ (Type w)} (h : F ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.GrothendieckTopology.toImagePresheaf f) (CategoryTheory.CategoryStruct.comp (CategoryTheory.GrothendieckTopology.imagePresheaf f).ι h) = CategoryTheory.CategoryStruct.comp f h

@[simp]

theorem CategoryTheory.GrothendieckTopology.toImagePresheaf_ι {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} (f : F' ⟶ F) : CategoryTheory.CategoryStruct.comp (CategoryTheory.GrothendieckTopology.toImagePresheaf f) (CategoryTheory.GrothendieckTopology.imagePresheaf f).ι = f

theorem CategoryTheory.GrothendieckTopology.imagePresheaf_comp_le {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type w)} {F' : CategoryTheory.Functor Cᵒᵖ (Type w)} {F'' : CategoryTheory.Functor Cᵒᵖ (Type w)} (f₁ : F ⟶ F') (f₂ : F' ⟶ F'') : CategoryTheory.GrothendieckTopology.imagePresheaf (CategoryTheory.CategoryStruct.comp f₁ f₂) ≤ CategoryTheory.GrothendieckTopology.imagePresheaf f₂

instance CategoryTheory.GrothendieckTopology.isIso_toImagePresheaf {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type (max v w))} {F' : CategoryTheory.Functor Cᵒᵖ (Type (max v w))} (f : F ⟶ F') [hf : CategoryTheory.Mono f] : CategoryTheory.IsIso (CategoryTheory.GrothendieckTopology.toImagePresheaf f)

Equations

• ⋯ = ⋯

@[simp]

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

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

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

Equations

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

@[simp]

theorem CategoryTheory.GrothendieckTopology.toImageSheaf_val {C : Type u} [CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} {F : CategoryTheory.Sheaf J (Type w)} {F' : CategoryTheory.Sheaf J (Type w)} (f : F ⟶ F') : (CategoryTheory.GrothendieckTopology.toImageSheaf f).val = J.toImagePresheafSheafify f.val

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

A morphism factors through the image sheaf.

Equations

• CategoryTheory.GrothendieckTopology.toImageSheaf f = { val := J.toImagePresheafSheafify f.val }

@[simp]

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

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

The inclusion of the image sheaf to the target.

Equations

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

@[simp]

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

@[simp]

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

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

Equations

• ⋯ = ⋯

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

Equations

• ⋯ = ⋯

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

The mono factorization given by image_sheaf for a morphism.

Equations

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

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

Equations

• ⋯ = ⋯