1-hypercovers

Given a Grothendieck topology J on a category C, we define the type of 1-hypercovers of an object S : C. They consists of a covering family of morphisms X i ⟶ S indexed by a type I₀ and, for each tuple (i₁, i₂) of elements of I₀, a "covering Y j of the fibre product of X i₁ and X i₂ over S", a condition which is phrased here without assuming that the fibre product actually exist.

The definition OneHypercover.isLimitMultifork shows that if E is a 1-hypercover of S, and F is a sheaf, then F.obj (op S) identifies to the multiequalizer of suitable maps F.obj (op (E.X i)) ⟶ F.obj (op (E.Y j)).

structure CategoryTheory.PreOneHypercover {C : Type u} [Category.{v, u} C] (S : C) : Type (max (max u v) (w + 1))

The categorical data that is involved in a 1-hypercover of an object S. This consists of a family of morphisms f i : X i ⟶ S for i : I₀, and for each tuple (i₁, i₂) of elements in I₀, a family of objects Y j indexed by a type I₁ i₁ i₂, which are equipped with a map to the fibre product of X i₁ and X i₂, which is phrased here as the data of the two projections p₁ : Y j ⟶ X i₁, p₂ : Y j ⟶ X i₂ and the relation p₁ j ≫ f i₁ = p₂ j ≫ f i₂. (See GrothendieckTopology.OneHypercover for the topological conditions.)

• I₀ : Type w

  the index type of the covering of S

• X (i : self.I₀) : C

  the objects in the covering of S

• f (i : self.I₀) : self.X i ⟶ S

  the morphisms in the covering of S

• I₁ (i₁ i₂ : self.I₀) : Type w

  the index type of the coverings of the fibre products

• Y ⦃i₁ i₂ : self.I₀⦄ (j : self.I₁ i₁ i₂) : C

  the objects in the coverings of the fibre products

• p₁ ⦃i₁ i₂ : self.I₀⦄ (j : self.I₁ i₁ i₂) : self.Y j ⟶ self.X i₁

  the first projection Y j ⟶ X i₁

• p₂ ⦃i₁ i₂ : self.I₀⦄ (j : self.I₁ i₁ i₂) : self.Y j ⟶ self.X i₂

  the second projection Y j ⟶ X i₂

• w ⦃i₁ i₂ : self.I₀⦄ (j : self.I₁ i₁ i₂) : CategoryStruct.comp (self.p₁ j) (self.f i₁) = CategoryStruct.comp (self.p₂ j) (self.f i₂)

@[reducible, inline]

abbrev CategoryTheory.PreOneHypercover.HasPullbacks {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) : Prop

The assumption that the pullback of X i₁ and X i₂ over S exists for any i₁ and i₂.

Equations

• E.HasPullbacks = ∀ (i₁ i₂ : E.I₀), CategoryTheory.Limits.HasPullback (E.f i₁) (E.f i₂)

@[reducible, inline]

abbrev CategoryTheory.PreOneHypercover.sieve₀ {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) : Sieve S

The sieve of S that is generated by the morphisms f i : X i ⟶ S.

Equations

• E.sieve₀ = CategoryTheory.Sieve.ofArrows E.X E.f

def CategoryTheory.PreOneHypercover.sieve₁ {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) {i₁ i₂ : E.I₀} {W : C} (p₁ : W ⟶ E.X i₁) (p₂ : W ⟶ E.X i₂) : Sieve W

Given an object W equipped with morphisms p₁ : W ⟶ E.X i₁, p₂ : W ⟶ E.X i₂, this is the sieve of W which consists of morphisms g : Z ⟶ W such that there exists j and h : Z ⟶ E.Y j such that g ≫ p₁ = h ≫ E.p₁ j and g ≫ p₂ = h ≫ E.p₂ j. See lemmas sieve₁_eq_pullback_sieve₁' and sieve₁'_eq_sieve₁ for equational lemmas regarding this sieve.

Equations

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

@[simp]

theorem CategoryTheory.PreOneHypercover.sieve₁_apply {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) {i₁ i₂ : E.I₀} {W : C} (p₁ : W ⟶ E.X i₁) (p₂ : W ⟶ E.X i₂) (Z : C) (g : Z ⟶ W) : (E.sieve₁ p₁ p₂).arrows g = ∃ (j : E.I₁ i₁ i₂) (h : Z ⟶ E.Y j), CategoryStruct.comp g p₁ = CategoryStruct.comp h (E.p₁ j) ∧ CategoryStruct.comp g p₂ = CategoryStruct.comp h (E.p₂ j)

@[reducible, inline]

noncomputable abbrev CategoryTheory.PreOneHypercover.toPullback {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) {i₁ i₂ : E.I₀} (j : E.I₁ i₁ i₂) [Limits.HasPullback (E.f i₁) (E.f i₂)] : E.Y j ⟶ Limits.pullback (E.f i₁) (E.f i₂)

The obvious morphism E.Y j ⟶ pullback (E.f i₁) (E.f i₂) given by E : PreOneHypercover S.

Equations

• E.toPullback j = CategoryTheory.Limits.pullback.lift (E.p₁ j) (E.p₂ j) ⋯

noncomputable def CategoryTheory.PreOneHypercover.sieve₁' {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) (i₁ i₂ : E.I₀) [Limits.HasPullback (E.f i₁) (E.f i₂)] : Sieve (Limits.pullback (E.f i₁) (E.f i₂))

The sieve of pullback (E.f i₁) (E.f i₂) given by E : PreOneHypercover S.

Equations

• E.sieve₁' i₁ i₂ = CategoryTheory.Sieve.ofArrows E.Y fun (j : E.I₁ i₁ i₂) => E.toPullback j

theorem CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁' {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) {i₁ i₂ : E.I₀} [Limits.HasPullback (E.f i₁) (E.f i₂)] {W : C} (p₁ : W ⟶ E.X i₁) (p₂ : W ⟶ E.X i₂) (w : CategoryStruct.comp p₁ (E.f i₁) = CategoryStruct.comp p₂ (E.f i₂)) : E.sieve₁ p₁ p₂ = Sieve.pullback (Limits.pullback.lift p₁ p₂ w) (E.sieve₁' i₁ i₂)

theorem CategoryTheory.PreOneHypercover.sieve₁'_eq_sieve₁ {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) (i₁ i₂ : E.I₀) [Limits.HasPullback (E.f i₁) (E.f i₂)] : E.sieve₁' i₁ i₂ = E.sieve₁ (Limits.pullback.fst (E.f i₁) (E.f i₂)) (Limits.pullback.snd (E.f i₁) (E.f i₂))

@[reducible, inline]

abbrev CategoryTheory.PreOneHypercover.I₁' {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) : Type w

The sigma type of all E.I₁ i₁ i₂ for ⟨i₁, i₂⟩ : E.I₀ × E.I₀.

Equations

• E.I₁' = ((i : E.I₀ × E.I₀) × E.I₁ i.1 i.2)

def CategoryTheory.PreOneHypercover.multicospanShape {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) : Limits.MulticospanShape

The shape of the multiforks attached to E : PreOneHypercover S.

Equations

• E.multicospanShape = { L := E.I₀, R := E.I₁', fst := fun (j : E.I₁') => j.fst.1, snd := fun (j : E.I₁') => j.fst.2 }

@[simp]

theorem CategoryTheory.PreOneHypercover.multicospanShape_R {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) : E.multicospanShape.R = E.I₁'

@[simp]

theorem CategoryTheory.PreOneHypercover.multicospanShape_fst {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) (j : E.I₁') : E.multicospanShape.fst j = j.fst.1

@[simp]

theorem CategoryTheory.PreOneHypercover.multicospanShape_L {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) : E.multicospanShape.L = E.I₀

@[simp]

theorem CategoryTheory.PreOneHypercover.multicospanShape_snd {C : Type u} [Category.{v, u} C] {S : C} (E : PreOneHypercover S) (j : E.I₁') : E.multicospanShape.snd j = j.fst.2

def CategoryTheory.PreOneHypercover.multicospanIndex {C : Type u} [Category.{v, u} C] {A : Type u_1} [Category.{u_2, u_1} A] {S : C} (E : PreOneHypercover S) (F : Functor Cᵒᵖ A) : Limits.MulticospanIndex E.multicospanShape A

The diagram of the multifork attached to a presheaf F : Cᵒᵖ ⥤ A, S : C and E : PreOneHypercover S.

Equations

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

@[simp]

theorem CategoryTheory.PreOneHypercover.multicospanIndex_fst {C : Type u} [Category.{v, u} C] {A : Type u_1} [Category.{u_2, u_1} A] {S : C} (E : PreOneHypercover S) (F : Functor Cᵒᵖ A) (j : E.multicospanShape.R) : (E.multicospanIndex F).fst j = F.map (E.p₁ j.snd).op

@[simp]

theorem CategoryTheory.PreOneHypercover.multicospanIndex_left {C : Type u} [Category.{v, u} C] {A : Type u_1} [Category.{u_2, u_1} A] {S : C} (E : PreOneHypercover S) (F : Functor Cᵒᵖ A) (i : E.multicospanShape.L) : (E.multicospanIndex F).left i = F.obj (Opposite.op (E.X i))

@[simp]

theorem CategoryTheory.PreOneHypercover.multicospanIndex_right {C : Type u} [Category.{v, u} C] {A : Type u_1} [Category.{u_2, u_1} A] {S : C} (E : PreOneHypercover S) (F : Functor Cᵒᵖ A) (j : E.multicospanShape.R) : (E.multicospanIndex F).right j = F.obj (Opposite.op (E.Y j.snd))

@[simp]

theorem CategoryTheory.PreOneHypercover.multicospanIndex_snd {C : Type u} [Category.{v, u} C] {A : Type u_1} [Category.{u_2, u_1} A] {S : C} (E : PreOneHypercover S) (F : Functor Cᵒᵖ A) (j : E.multicospanShape.R) : (E.multicospanIndex F).snd j = F.map (E.p₂ j.snd).op

def CategoryTheory.PreOneHypercover.multifork {C : Type u} [Category.{v, u} C] {A : Type u_1} [Category.{u_2, u_1} A] {S : C} (E : PreOneHypercover S) (F : Functor Cᵒᵖ A) : Limits.Multifork (E.multicospanIndex F)

The multifork attached to a presheaf F : Cᵒᵖ ⥤ A, S : C and E : PreOneHypercover S.

Equations

• E.multifork F = CategoryTheory.Limits.Multifork.ofι (E.multicospanIndex F) (F.obj (Opposite.op S)) (fun (i₀ : E.multicospanShape.L) => F.map (E.f i₀).op) ⋯

structure CategoryTheory.GrothendieckTopology.OneHypercover {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) (S : C) extends CategoryTheory.PreOneHypercover S : Type (max (max u v) (w + 1))

The type of 1-hypercovers of an object S : C in a category equipped with a Grothendieck topology J. This can be constructed from a covering of S and a covering of the fibre products of the objects in this covering (see OneHypercover.mk').

• I₀ : Type w
• X (i : self.I₀) : C
• f (i : self.I₀) : self.X i ⟶ S
• I₁ (i₁ i₂ : self.I₀) : Type w
• Y ⦃i₁ i₂ : self.I₀⦄ (j : self.I₁ i₁ i₂) : C
• p₁ ⦃i₁ i₂ : self.I₀⦄ (j : self.I₁ i₁ i₂) : self.Y j ⟶ self.X i₁
• p₂ ⦃i₁ i₂ : self.I₀⦄ (j : self.I₁ i₁ i₂) : self.Y j ⟶ self.X i₂
• w ⦃i₁ i₂ : self.I₀⦄ (j : self.I₁ i₁ i₂) : CategoryStruct.comp (self.p₁ j) (self.f i₁) = CategoryStruct.comp (self.p₂ j) (self.f i₂)
• mem₀ : self.sieve₀ ∈ J S
• mem₁ (i₁ i₂ : self.I₀) ⦃W : C⦄ (p₁ : W ⟶ self.X i₁) (p₂ : W ⟶ self.X i₂) (w : CategoryStruct.comp p₁ (self.f i₁) = CategoryStruct.comp p₂ (self.f i₂)) : self.sieve₁ p₁ p₂ ∈ J W

theorem CategoryTheory.GrothendieckTopology.OneHypercover.mem_sieve₁' {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {S : C} (E : J.OneHypercover S) (i₁ i₂ : E.I₀) [Limits.HasPullback (E.f i₁) (E.f i₂)] : E.sieve₁' i₁ i₂ ∈ J (Limits.pullback (E.f i₁) (E.f i₂))

def CategoryTheory.GrothendieckTopology.OneHypercover.mk' {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {S : C} (E : PreOneHypercover S) [E.HasPullbacks] (mem₀ : E.sieve₀ ∈ J S) (mem₁' : ∀ (i₁ i₂ : E.I₀), E.sieve₁' i₁ i₂ ∈ J (Limits.pullback (E.f i₁) (E.f i₂))) : J.OneHypercover S

In order to check that a certain data is a 1-hypercover of S, it suffices to check that the data provides a covering of S and of the fibre products.

Equations

• CategoryTheory.GrothendieckTopology.OneHypercover.mk' E mem₀ mem₁' = { toPreOneHypercover := E, mem₀ := mem₀, mem₁ := ⋯ }

@[simp]

theorem CategoryTheory.GrothendieckTopology.OneHypercover.mk'_toPreOneHypercover {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {S : C} (E : PreOneHypercover S) [E.HasPullbacks] (mem₀ : E.sieve₀ ∈ J S) (mem₁' : ∀ (i₁ i₂ : E.I₀), E.sieve₁' i₁ i₂ ∈ J (Limits.pullback (E.f i₁) (E.f i₂))) : (mk' E mem₀ mem₁').toPreOneHypercover = E

noncomputable def CategoryTheory.GrothendieckTopology.OneHypercover.multiforkLift {C : Type u} [Category.{v, u} C] {A : Type u_1} [Category.{u_2, u_1} A] {J : GrothendieckTopology C} {S : C} {E : J.OneHypercover S} {F : Sheaf J A} (c : Limits.Multifork (E.multicospanIndex F.val)) : c.pt ⟶ F.val.obj (Opposite.op S)

Auxiliary definition of isLimitMultifork.

Equations

• CategoryTheory.GrothendieckTopology.OneHypercover.multiforkLift c = ⋯.amalgamateOfArrows E.f ⋯ c.ι ⋯

theorem CategoryTheory.GrothendieckTopology.OneHypercover.multiforkLift_map {C : Type u} [Category.{v, u} C] {A : Type u_1} [Category.{u_3, u_1} A] {J : GrothendieckTopology C} {S : C} {E : J.OneHypercover S} {F : Sheaf J A} (c : Limits.Multifork (E.multicospanIndex F.val)) (i₀ : E.I₀) : CategoryStruct.comp (multiforkLift c) (F.val.map (E.f i₀).op) = c.ι i₀

theorem CategoryTheory.GrothendieckTopology.OneHypercover.multiforkLift_map_assoc {C : Type u} [Category.{v, u} C] {A : Type u_1} [Category.{u_3, u_1} A] {J : GrothendieckTopology C} {S : C} {E : J.OneHypercover S} {F : Sheaf J A} (c : Limits.Multifork (E.multicospanIndex F.val)) (i₀ : E.I₀) {Z : A} (h : F.val.obj (Opposite.op (E.X i₀)) ⟶ Z) : CategoryStruct.comp (multiforkLift c) (CategoryStruct.comp (F.val.map (E.f i₀).op) h) = CategoryStruct.comp (c.ι i₀) h

noncomputable def CategoryTheory.GrothendieckTopology.OneHypercover.isLimitMultifork {C : Type u} [Category.{v, u} C] {A : Type u_1} [Category.{u_2, u_1} A] {J : GrothendieckTopology C} {S : C} (E : J.OneHypercover S) (F : Sheaf J A) : Limits.IsLimit (E.multifork F.val)

If E : J.OneHypercover S and F : Sheaf J A, then F.obj (op S) is a multiequalizer of suitable maps F.obj (op (E.X i)) ⟶ F.obj (op (E.Y j)) induced by E.p₁ j and E.p₂ j.

Equations

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

def CategoryTheory.GrothendieckTopology.Cover.preOneHypercover {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) : PreOneHypercover X

The tautological 1-pre-hypercover induced by S : J.Cover X. Its index type I₀ is given by S.Arrow (i.e. all the morphisms in the sieve S), while I₁ is given by all possible pullback cones.

Equations

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

@[simp]

theorem CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_p₁ {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) (x✝ x✝¹ : S.Arrow) (r : x✝.Relation x✝¹) : S.preOneHypercover.p₁ r = r.g₁

@[simp]

theorem CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_Y {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) (x✝ x✝¹ : S.Arrow) (r : x✝.Relation x✝¹) : S.preOneHypercover.Y r = r.Z

@[simp]

theorem CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_I₀ {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) : S.preOneHypercover.I₀ = S.Arrow

@[simp]

theorem CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_p₂ {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) (x✝ x✝¹ : S.Arrow) (r : x✝.Relation x✝¹) : S.preOneHypercover.p₂ r = r.g₂

@[simp]

theorem CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_I₁ {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) (f₁ f₂ : S.Arrow) : S.preOneHypercover.I₁ f₁ f₂ = f₁.Relation f₂

@[simp]

theorem CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_X {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) (f : S.Arrow) : S.preOneHypercover.X f = f.Y

@[simp]

theorem CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_f {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) (f : S.Arrow) : S.preOneHypercover.f f = f.f

@[simp]

theorem CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_sieve₀ {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) : S.preOneHypercover.sieve₀ = ↑S

theorem CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_sieve₁ {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) (f₁ f₂ : S.Arrow) {W : C} (p₁ : W ⟶ f₁.Y) (p₂ : W ⟶ f₂.Y) (w : CategoryStruct.comp p₁ f₁.f = CategoryStruct.comp p₂ f₂.f) : S.preOneHypercover.sieve₁ p₁ p₂ = ⊤

def CategoryTheory.GrothendieckTopology.Cover.oneHypercover {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) : J.OneHypercover X

The tautological 1-hypercover induced by S : J.Cover X. Its index type I₀ is given by S.Arrow (i.e. all the morphisms in the sieve S), while I₁ is given by all possible pullback cones.

Equations

• S.oneHypercover = { toPreOneHypercover := S.preOneHypercover, mem₀ := ⋯, mem₁ := ⋯ }

@[simp]

theorem CategoryTheory.GrothendieckTopology.Cover.oneHypercover_toPreOneHypercover {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {X : C} (S : J.Cover X) : S.oneHypercover.toPreOneHypercover = S.preOneHypercover