mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Sites.IsSheafFor

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
I : Type u_1
B B' : C
p : B ⟶ B'
P : (Over B')ᵒᵖ ⥤ Type w
X : Over B
Y : I → Over B
f : (i : I) → Y i ⟶ X
⊢ IsSheafFor ((Over.map p).op ⋙ P) (ofArrows Y f) ↔
    IsSheafFor P (ofArrows (fun i ↦ (Over.map p).obj (Y i)) fun i ↦ (Over.map p).map (f i))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Subfunctor.Image

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Type w
G : Subfunctor F
⊢ G.preimage (𝟙 F) = G

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Subfunctor.Image

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F F' F'' : C ⥤ Type w
G : Subfunctor F
f : F'' ⟶ F'
g : F' ⟶ F
⊢ G.preimage (f ≫ g) = (G.preimage g).preimage f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Subfunctor.Image

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F F' : C ⥤ Type w
G : Subfunctor F
f : F ⟶ F'
G' : Subfunctor F'
⊢ G.image f ≤ G' ↔ G ≤ G'.preimage f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Subfunctor.Image

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F F' F'' : C ⥤ Type w
G : Subfunctor F
p : F' ⟶ F
⊢ range ((G.preimage p).ι ≫ p) ≤ G

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Subfunctor.Image

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F F' : C ⥤ Type w
G : Subfunctor F
p : F' ⟶ F
⊢ G.preimage p = ⊤ ↔ range p ≤ G

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Sheaf

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
A : Type u₂
inst✝⁶ : Category.{v₂, u₂} A
A' : Type u₂
inst✝⁵ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝⁴ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U✝ : C
R✝ : Presieve U✝
P✝ : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
inst✝³ : HasProducts A
inst✝² : HasProducts A'
inst✝¹ : HasPullbacks C
P : Cᵒᵖ ⥤ A
s : A ⥤ Type (max v₁ u₁)
inst✝ : ∀ (J : Type (max v₁ u₁)), PreservesLimitsOfShape (Discrete J) s
U : C
R : Presieve U
⊢ parallelPair (s.map (firstMap R P)) (s.map (secondMap R P)) ≅
    parallelPair (Equalizer.Presieve.firstMap (P ⋙ s) R) (Equalizer.Presieve.secondMap (P ⋙ s) R)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Subfunctor.Image

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F F' : C ⥤ Type w
G : Subfunctor F
p : F' ⟶ F
hp : Epi p
⊢ (G.preimage p).image p = G

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Subfunctor.Sieves

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
U : Cᵒᵖ
s : F.obj U
V W : C
i : V ⟶ unop U
hi : (ConcreteCategory.hom (F.map i.op)) s ∈ G.obj (op V)
j : W ⟶ V
⊢ (ConcreteCategory.hom (F.map (j ≫ i).op)) s ∈ G.obj (op W)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Subfunctor.Sieves

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
U : Cᵒᵖ
s : F.obj U
⊢ (G.familyOfElementsOfSection s).Compatible

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subfunctor F
⊢ ∀ {U V : Cᵒᵖ} (i : U ⟶ V),
    {s | G.sieveOfSection s ∈ J (unop U)} ⊆ ⇑(ConcreteCategory.hom (F.map i)) ⁻¹' {s | G.sieveOfSection s ∈ J (unop V)}

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
⊢ G ≤ sheafify J G

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.IsSheafFor

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
B B' : C
p : B ⟶ B'
P : (Over B')ᵒᵖ ⥤ Type w
X : Over B
R : Sieve X
X' : Over B'
e : (Over.map p).obj X ≅ X'
⊢ IsSheafFor ((Over.map p).op ⋙ P) R.arrows ↔
    IsSheafFor P (Sieve.pullback e.inv (Sieve.functorPushforward (Over.map p) R)).arrows

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Sheaf

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
A : Type u₂
inst✝⁶ : Category.{v₂, u₂} A
A' : Type u₂
inst✝⁵ : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst✝⁴ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U✝ : C
R✝ : Presieve U✝
P✝ : Cᵒᵖ ⥤ A
P' : Cᵒᵖ ⥤ A'
inst✝³ : HasProducts A
inst✝² : HasProducts A'
inst✝¹ : HasPullbacks C
P : Cᵒᵖ ⥤ A
s : A ⥤ Type (max v₁ u₁)
inst✝ : ∀ (J : Type (max v₁ u₁)), PreservesLimitsOfShape (Discrete J) s
U : C
R : Presieve U
⊢ IsLimit (s.mapCone (Fork.ofι (forkMap R P) ⋯)) ≃ IsLimit (Fork.ofι (Equalizer.forkMap (P ⋙ s) R) ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
h : Presieve.IsSheaf J F
hG : Presieve.IsSheaf J G.toFunctor
U : Cᵒᵖ
s : F.obj U
hs : s ∈ (sheafify J G).obj U
this : ↑(⋯.amalgamate (G.familyOfElementsOfSection s) ⋯) = s
⊢ s ∈ G.obj U

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
h : Presieve.IsSheaf J F
hG : Presieve.IsSheaf J G.toFunctor
⊢ G = sheafify J G

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Sheaf

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A' : Type u₂
inst✝² : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology C
P' : Cᵒᵖ ⥤ A'
inst✝¹ : HasProducts A'
inst✝ : HasPullbacks C
⊢ IsSheaf J P' ↔ IsSheaf' J P'

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.IsSheafFor

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
inst✝ : (ofArrows X π).HasPairwisePullbacks
x : (i : I) → P.obj (op (X i))
⊢ Compatible P π x ↔ PullbackCompatible P π x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Sheaf

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
B : Type u₃
inst✝¹ : Category.{v₃, u₃} B
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
s : A ⥤ B
inst✝ : ReflectsLimitsOfSize.{v₁, max v₁ u₁, v₂, v₃, u₂, u₃} s
h : IsSheaf J (P ⋙ s)
⊢ IsSheaf J P

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.IsSheafFor

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
inst✝ : (ofArrows X π).HasPairwisePullbacks
⊢ IsSheafFor P (ofArrows X π) ↔
    ∀ (x : (i : I) → P.obj (op (X i))),
      Arrows.PullbackCompatible P π x → ∃! t, ∀ (i : I), (ConcreteCategory.hom (P.map (π i).op)) t = x i

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Sheaf

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
A : Type u₂
inst✝² : Category.{v₂, u₂} A
B : Type u₃
inst✝¹ : Category.{v₃, u₃} B
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
s : A ⥤ B
inst✝ : PreservesLimitsOfSize.{v₁, max v₁ u₁, v₂, v₃, u₂, u₃} s
h : IsSheaf J P
⊢ IsSheaf J (P ⋙ s)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Sheaf

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
A : Type u₂
inst✝⁴ : Category.{v₂, u₂} A
B : Type u₃
inst✝³ : Category.{v₃, u₃} B
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
s : A ⥤ B
inst✝² : HasLimitsOfSize.{v₁, max v₁ u₁, v₂, u₂} A
inst✝¹ : PreservesLimitsOfSize.{v₁, max v₁ u₁, v₂, v₃, u₂, u₃} s
inst✝ : s.ReflectsIsomorphisms
⊢ IsSheaf J P ↔ IsSheaf J (P ⋙ s)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Sheaf

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
A' : Type u₂
inst✝³ : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology C
P' : Cᵒᵖ ⥤ A'
s : A' ⥤ Type (max v₁ u₁)
inst✝² : HasLimits A'
inst✝¹ : PreservesLimits s
inst✝ : s.ReflectsIsomorphisms
⊢ IsSheaf J P' ↔ IsSheaf J (P' ⋙ s)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Whiskering

Goal state before:
pp: C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
A : Type u₂
inst✝⁵ : Category.{v₂, u₂} A
B : Type u₃
inst✝⁴ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
F G H : A ⥤ B
η : F ⟶ G
γ : G ⟶ H
inst✝³ : J.HasSheafCompose F
inst✝² : J.HasSheafCompose G
inst✝¹ : J.HasSheafCompose H
inst✝ : F.ReflectsIsomorphisms
G₁ G₂ : Sheaf J A
f : G₁ ⟶ G₂
x✝ : IsIso ((sheafCompose J F).map f)
⊢ IsIso f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.IsSheafFor

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
X Y : C
f : X ⟶ Y
⊢ IsSeparatedFor P (singleton f) ↔ Function.Injective ⇑(ConcreteCategory.hom (P.map f.op))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Whiskering

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
A : Type u₂
inst✝⁴ : Category.{v₂, u₂} A
B : Type u₃
inst✝³ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
F G H : A ⥤ B
η : F ⟶ G
γ : G ⟶ H
inst✝² : J.HasSheafCompose F
inst✝¹ : J.HasSheafCompose G
inst✝ : J.HasSheafCompose H
P : Cᵒᵖ ⥤ A
X : C
S : J.Cover X
⊢ ∀ {X_1 Y : WalkingMulticospan S.shape} (f : X_1 ⟶ Y),
    (S.index (P ⋙ F)).multicospan.map f ≫
        (match Y with
          | WalkingMulticospan.left a => Iso.refl ((S.index (P ⋙ F)).multicospan.obj (WalkingMulticospan.left a))
          | WalkingMulticospan.right a =>
            Iso.refl ((S.index (P ⋙ F)).multicospan.obj (WalkingMulticospan.right a))).hom =
      (match X_1 with
          | WalkingMulticospan.left a => Iso.refl ((S.index (P ⋙ F)).multicospan.obj (WalkingMulticospan.left a))
          | WalkingMulticospan.right a =>
            Iso.refl ((S.index (P ⋙ F)).multicospan.obj (WalkingMulticospan.right a))).hom ≫
        ((S.index P).multicospan ⋙ F).map f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Whiskering

Goal state before:
pp: C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
A : Type u₂
inst✝⁵ : Category.{v₂, u₂} A
B : Type u₃
inst✝⁴ : Category.{v₃, u₃} B
J : GrothendieckTopology C
U : C
R : Presieve U
F✝ G H : A ⥤ B
η : F✝ ⟶ G
γ : G ⟶ H
inst✝³ : J.HasSheafCompose F✝
inst✝² : J.HasSheafCompose G
inst✝¹ : J.HasSheafCompose H
F : A ⥤ B
inst✝ : ∀ (X : C) (S : J.Cover X) (P : Cᵒᵖ ⥤ A), PreservesLimit (S.index P).multicospan F
P : Cᵒᵖ ⥤ A
hP : Presheaf.IsSheaf J P
⊢ Presheaf.IsSheaf J (P ⋙ F)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Whiskering

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
A : Type u₂
inst✝³ : Category.{v₂, u₂} A
J : GrothendieckTopology C
FA : A → A → Type u_1
CA : A → Type u_2
inst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝¹ : ConcreteCategory A FA
inst✝ : J.HasSheafCompose (forget A)
F : Sheaf J A
⊢ Presheaf.IsSeparated J F.obj

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.IsSheafFor

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
X Y : C
f : X ⟶ Y
⊢ IsSheafFor P (singleton f) ↔
    ∀ (x : P.obj (op X)),
      (∀ {Z : C} (p₁ p₂ : Z ⟶ X),
          p₁ ≫ f = p₂ ≫ f → (ConcreteCategory.hom (P.map p₁.op)) x = (ConcreteCategory.hom (P.map p₂.op)) x) →
        ∃! y, (ConcreteCategory.hom (P.map f.op)) y = x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
hF : Presieve.IsSheaf J F
U : C
S : Sieve U
hS : S ∈ J U
x : Presieve.FamilyOfElements (sheafify J G).toFunctor S.arrows
hx : x.Compatible
S' : Sieve U := Sieve.bind S.arrows fun Y f hf ↦ G.sieveOfSection ↑(x f hf)
W : (V : C) → (i : V ⟶ U) → S'.arrows i → C
i₁ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → V ⟶ W V i hi
i₂ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → W V i hi ⟶ U
hi₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), S.arrows (i₂ V i hi)
h₁ :
  ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i),
    (ConcreteCategory.hom (F.map (i₁ V i hi).op)) ↑(x (i₂ V i hi) ⋯) ∈ G.obj (op V)
h₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), i₁ V i hi ≫ i₂ V i hi = i
x'' : Presieve.FamilyOfElements F S'.arrows :=
  fun V i hi ↦ (ConcreteCategory.hom (F.map (i₁ V i hi).op)) ↑(x (i₂ V i hi) ⋯)
⊢ ∀ (s : (sheafify J G).toFunctor.obj (op U)), x''.IsAmalgamation ↑s → x.IsAmalgamation s

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.ConcreteSheafification

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : J.Cover X
x : Meq P S
y : Meq P T
⊢ mk x = mk y ↔ ∃ W h1 h2, x.refine h1 = y.refine h2

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.IsSheafFor

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
P : Cᵒᵖ ⥤ Type u_1
U : Sieve X
B : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → U.arrows f → Sieve Y
hU : IsSheafFor P U.arrows
hB : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), IsSheafFor P (B hf).arrows
hB' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (h : U.arrows f) ⦃Z : C⦄ (g : Z ⟶ Y), IsSeparatedFor P (Sieve.pullback g (B h)).arrows
s : FamilyOfElements P (Sieve.bind U.arrows B).arrows
hs : s.Compatible
y : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → (hf : U.arrows f) → FamilyOfElements P (B hf).arrows := fun Y f hf Z g hg ↦ s (g ≫ f) ⋯
⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), (y hf).Compatible

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
hF : Presieve.IsSheaf J F
U : C
S : Sieve U
hS : S ∈ J U
x : Presieve.FamilyOfElements (sheafify J G).toFunctor S.arrows
hx : x.Compatible
S' : Sieve U := Sieve.bind S.arrows fun Y f hf ↦ G.sieveOfSection ↑(x f hf)
W : (V : C) → (i : V ⟶ U) → S'.arrows i → C
i₁ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → V ⟶ W V i hi
i₂ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → W V i hi ⟶ U
hi₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), S.arrows (i₂ V i hi)
h₁ :
  ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i),
    (ConcreteCategory.hom (F.map (i₁ V i hi).op)) ↑(x (i₂ V i hi) ⋯) ∈ G.obj (op V)
h₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), i₁ V i hi ≫ i₂ V i hi = i
x'' : Presieve.FamilyOfElements F S'.arrows :=
  fun V i hi ↦ (ConcreteCategory.hom (F.map (i₁ V i hi).op)) ↑(x (i₂ V i hi) ⋯)
H : ∀ (s : (sheafify J G).toFunctor.obj (op U)), x''.IsAmalgamation ↑s → x.IsAmalgamation s
V₁ V₂ V₃ : C
g₁ : V₃ ⟶ V₁
g₂ : V₃ ⟶ V₂
g₃ : V₁ ⟶ U
g₄ : V₂ ⟶ U
S₁ : S'.arrows g₃
S₂ : S'.arrows g₄
e : g₁ ≫ g₃ = g₂ ≫ g₄
⊢ (g₁ ≫ i₁ V₁ g₃ S₁) ≫ i₂ V₁ g₃ S₁ = (g₂ ≫ i₁ V₂ g₄ S₂) ≫ i₂ V₂ g₄ S₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
hF : Presieve.IsSheaf J F
U : C
S : Sieve U
hS : S ∈ J U
x : Presieve.FamilyOfElements (sheafify J G).toFunctor S.arrows
hx : x.Compatible
S' : Sieve U := Sieve.bind S.arrows fun Y f hf ↦ G.sieveOfSection ↑(x f hf)
W : (V : C) → (i : V ⟶ U) → S'.arrows i → C
i₁ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → V ⟶ W V i hi
i₂ : (V : C) → (i : V ⟶ U) → (hi : S'.arrows i) → W V i hi ⟶ U
hi₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), S.arrows (i₂ V i hi)
h₁ :
  ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i),
    (ConcreteCategory.hom (F.map (i₁ V i hi).op)) ↑(x (i₂ V i hi) ⋯) ∈ G.obj (op V)
h₂ : ∀ (V : C) (i : V ⟶ U) (hi : S'.arrows i), i₁ V i hi ≫ i₂ V i hi = i
x'' : Presieve.FamilyOfElements F S'.arrows :=
  fun V i hi ↦ (ConcreteCategory.hom (F.map (i₁ V i hi).op)) ↑(x (i₂ V i hi) ⋯)
H : ∀ (s : (sheafify J G).toFunctor.obj (op U)), x''.IsAmalgamation ↑s → x.IsAmalgamation s
⊢ x''.Compatible

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.ConcreteSheafification

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S Sx : J.Cover X
x : Meq P Sx
Sy : J.Cover X
y : Meq P Sy
h : ∀ (I : S.Arrow), mk (x.pullback I.f) = mk (y.pullback I.f)
W : (I : S.Arrow) → J.Cover I.Y
h1 : (I : S.Arrow) → W I ⟶ (J.pullback I.f).obj Sx
h2 : (I : S.Arrow) → W I ⟶ (J.pullback I.f).obj Sy
hh : ∀ (I : S.Arrow), (x.pullback I.f).refine (h1 I) = (y.pullback I.f).refine (h2 I)
B : J.Cover X := S.bind W
⊢ B ≤ Sx

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.ConcreteSheafification

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S Sx : J.Cover X
x : Meq P Sx
Sy : J.Cover X
y : Meq P Sy
h : ∀ (I : S.Arrow), mk (x.pullback I.f) = mk (y.pullback I.f)
W : (I : S.Arrow) → J.Cover I.Y
h1 : (I : S.Arrow) → W I ⟶ (J.pullback I.f).obj Sx
h2 : (I : S.Arrow) → W I ⟶ (J.pullback I.f).obj Sy
hh : ∀ (I : S.Arrow), (x.pullback I.f).refine (h1 I) = (y.pullback I.f).refine (h2 I)
B : J.Cover X := S.bind W
ex : B ⟶ Sx := homOfLE ⋯
⊢ B ≤ Sy

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.IsSheafFor

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
P : Cᵒᵖ ⥤ Type u_1
U : Sieve X
B : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → U.arrows f → Sieve Y
hU : IsSheafFor P U.arrows
hB : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), IsSheafFor P (B hf).arrows
hB' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (h : U.arrows f) ⦃Z : C⦄ (g : Z ⟶ Y), IsSeparatedFor P (Sieve.pullback g (B h)).arrows
s : FamilyOfElements P (Sieve.bind U.arrows B).arrows
hs : s.Compatible
y : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → (hf : U.arrows f) → FamilyOfElements P (B hf).arrows := fun Y f hf Z g hg ↦ s (g ≫ f) ⋯
hy : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), (y hf).Compatible
t : FamilyOfElements P U.arrows := fun Y f hf ↦ ⋯.amalgamate (y hf) ⋯
ht : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), (y hf).IsAmalgamation (t f hf)
Z W : C
f : Z ⟶ X
h : W ⟶ Z
hf : U.arrows f
Y : C
l : Y ⟶ W
hl : (B ⋯).arrows l
M : C
m : M ⟶ Y
hm : (Sieve.pullback (l ≫ h) (B hf)).arrows m
⊢ (Sieve.bind U.arrows B).arrows (m ≫ l ≫ h ≫ f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
hF : Presieve.IsSheaf J F
⊢ Presieve.IsSheaf J (sheafify J G).toFunctor

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
h : Presieve.IsSheaf J F
⊢ Presieve.IsSheaf J G.toFunctor ↔ ∀ (U : Cᵒᵖ) (s : F.obj U), G.sieveOfSection s ∈ J (unop U) → s ∈ G.obj U

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.ConcreteSheafification

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S : J.Cover X
x y : ToType ((J.plusObj P).obj (op X))
h :
  ∀ (I : S.Arrow),
    (ConcreteCategory.hom ((J.plusObj P).map I.f.op)) x = (ConcreteCategory.hom ((J.plusObj P).map I.f.op)) y
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subfunctor F
f : G.toFunctor ⟶ F'
h : Presieve.IsSheaf J F'
⊢ ∀ ⦃X Y : Cᵒᵖ⦄ (f_1 : X ⟶ Y),
    ((sheafify J G).toFunctor.map f_1 ≫ ↾fun s ↦ ⋯.amalgamate ((G.familyOfElementsOfSection ↑s).map f) ⋯) =
      (↾fun s ↦ ⋯.amalgamate ((G.familyOfElementsOfSection ↑s).map f) ⋯) ≫ F'.map f_1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.ConcreteSheafification

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
  ∀ (X : C) (S : J.Cover X) (x y : ToType (P.obj (op X))),
    (∀ (I : S.Arrow), (ConcreteCategory.hom (P.map I.f.op)) x = (ConcreteCategory.hom (P.map I.f.op)) y) → x = y
X : C
⊢ Function.Injective ⇑(ConcreteCategory.hom ((J.toPlus P).app (op X)))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' : Cᵒᵖ ⥤ Type w
G : Subfunctor F
f : G.toFunctor ⟶ F'
h : Presieve.IsSheaf J F'
⊢ homOfLe ⋯ ≫ G.sheafifyLift f h = f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' : Cᵒᵖ ⥤ Type w
G : Subfunctor F
h : Presieve.IsSheaf J F'
l₁ l₂ : (sheafify J G).toFunctor ⟶ F'
e : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂
⊢ l₁ = l₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.IsSheafFor

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
P : Cᵒᵖ ⥤ Type u_1
U : Sieve X
B : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → U.arrows f → Sieve Y
hU : IsSheafFor P U.arrows
hB : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), IsSheafFor P (B hf).arrows
hB' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (h : U.arrows f) ⦃Z : C⦄ (g : Z ⟶ Y), IsSeparatedFor P (Sieve.pullback g (B h)).arrows
s : FamilyOfElements P (Sieve.bind U.arrows B).arrows
hs : s.Compatible
y : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → (hf : U.arrows f) → FamilyOfElements P (B hf).arrows := fun Y f hf Z g hg ↦ s (g ≫ f) ⋯
hy : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), (y hf).Compatible
t : FamilyOfElements P U.arrows := fun Y f hf ↦ ⋯.amalgamate (y hf) ⋯
ht : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), (y hf).IsAmalgamation (t f hf)
⊢ t.Compatible

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.CompatiblePlus

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
E : Type u_2
inst✝³ : Category.{v_2, u_2} E
F : D ⥤ E
inst✝² : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝¹ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
X : C
W : (J.Cover X)ᵒᵖ
⊢ (J.diagram P X ⋙ F).obj W ≅ (J.diagram (P ⋙ F) X).obj W

Tactic:
by

Goal state after: