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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝³ : Category.{v_1, u_1} A
inst✝² : Category.{v_2, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst✝¹ : J.HasSheafCompose F
inst✝ : J.PreservesSheafification F
⊢ IsIso (sheafComposeNatTrans J F adj₁ adj₂)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ F₃ : Cᵒᵖ ⥤ D
φ : F₁ ⟶ F₂
ψ : F₂ ⟶ F₃
P : Cᵒᵖ ⥤ Type (max u v)
X : Cᵒᵖ
x y : ToType (P.obj X)
h : (ConcreteCategory.hom ((J.toPlus P).app X)) x = (ConcreteCategory.hom ((J.toPlus P).app X)) y
⊢ equalizerSieve x y ∈ J (unop X)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : Category.{v_2, u_2} B
F : A ⥤ B
inst✝³ : HasWeakSheafify J A
inst✝² : HasWeakSheafify J B
inst✝¹ : J.HasSheafCompose F
inst✝ : J.PreservesSheafification F
P : Cᵒᵖ ⥤ A
⊢ whiskerRight (toSheafify J P) F ≫ (sheafifyComposeIso J F P).inv = toSheafify J (P ⋙ F)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ F₃ : Cᵒᵖ ⥤ D
φ : F₁ ⟶ F₂
ψ : F₂ ⟶ F₃
P : Cᵒᵖ ⥤ Type (max u v)
⊢ IsLocallyInjective J (J.toSheafify P)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
D : Type u'
inst✝⁷ : Category.{v', u'} D
FD : D → D → Type u_1
CD✝ : D → Type w
inst✝⁶ : (X Y : D) → FunLike (FD X Y) (CD✝ X) (CD✝ Y)
inst✝⁵ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ F₃ : Cᵒᵖ ⥤ D
φ : F₁ ⟶ F₂
ψ : F₂ ⟶ F₃
CD : D → Type (max u v)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝³ : ConcreteCategory D FD
P : Cᵒᵖ ⥤ D
inst✝² : HasWeakSheafify J D
inst✝¹ : J.HasSheafCompose (forget D)
inst✝ : J.PreservesSheafification (forget D)
⊢ IsLocallyInjective J (toSheafify J P)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ : Sheaf J D
φ : F₁ ⟶ F₂
⊢ Presheaf.IsLocallyInjective J ((sheafToPresheaf J D).map φ) ↔ IsLocallyInjective φ

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type u_3
E : Type u_4
inst✝¹⁵ : Category.{v_3, u_3} D
inst✝¹⁴ : Category.{v_4, u_4} E
F : D ⥤ E
inst✝¹³ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝¹² : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝¹¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹⁰ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁸ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
FD : D → D → Type u_5
CD : D → Type u_6
FE : E → E → Type u_7
CE : E → Type u_8
inst✝⁷ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁶ : (X Y : E) → FunLike (FE X Y) (CE X) (CE Y)
instCCD : ConcreteCategory D FD
instCCE : ConcreteCategory E FE
inst✝⁵ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁴ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget E)
inst✝³ : PreservesLimitsOfSize.{max v u, max v u, v_3, u_6, u_3, u_6 + 1} (forget D)
inst✝² : PreservesLimitsOfSize.{max v u, max v u, v_4, u_8, u_4, u_8 + 1} (forget E)
inst✝¹ : (forget D).ReflectsIsomorphisms
inst✝ : (forget E).ReflectsIsomorphisms
P : Cᵒᵖ ⥤ D
this :
  (sheafComposeNatTrans J F (plusPlusAdjunction J D) (plusPlusAdjunction J E)).app P =
    { hom := (J.sheafifyCompIso F P).inv }
⊢ (sheafToPresheaf J E).map ((sheafComposeNatTrans J F (plusPlusAdjunction J D) (plusPlusAdjunction J E)).app P) =
    (J.sheafifyCompIso F P).inv

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝² : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝¹ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ : Sheaf J D
φ : F₁ ⟶ F₂
inst✝ : IsIso φ
⊢ IsLocallyInjective φ

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝² : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝¹ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ : Sheaf J D
φ : F₁ ⟶ F₂
inst✝ : J.HasSheafCompose (forget D)
⊢ Presieve.IsSeparated J (F₁.obj ⋙ forget D)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝³ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝² : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ : Sheaf J D
φ : F₁ ⟶ F₂
inst✝¹ : J.HasSheafCompose (forget D)
inst✝ : IsLocallyInjective φ
⊢ Mono φ

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type u_3
E : Type u_4
inst✝¹⁵ : Category.{v_3, u_3} D
inst✝¹⁴ : Category.{v_4, u_4} E
F : D ⥤ E
inst✝¹³ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝¹² : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝¹¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹⁰ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁸ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
FD : D → D → Type u_5
CD : D → Type u_6
FE : E → E → Type u_7
CE : E → Type u_8
inst✝⁷ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁶ : (X Y : E) → FunLike (FE X Y) (CE X) (CE Y)
instCCD : ConcreteCategory D FD
instCCE : ConcreteCategory E FE
inst✝⁵ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁴ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget E)
inst✝³ : PreservesLimitsOfSize.{max v u, max v u, v_3, u_6, u_3, u_6 + 1} (forget D)
inst✝² : PreservesLimitsOfSize.{max v u, max v u, v_4, u_8, u_4, u_8 + 1} (forget E)
inst✝¹ : (forget D).ReflectsIsomorphisms
inst✝ : (forget E).ReflectsIsomorphisms
P : Cᵒᵖ ⥤ D
⊢ (sheafToPresheaf J E).map ((sheafComposeNatTrans J F (plusPlusAdjunction J D) (plusPlusAdjunction J E)).app P) =
    (J.sheafifyCompIso F P).inv

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝² : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝¹ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ : Sheaf J D
φ : F₁ ⟶ F₂
inst✝ : J.HasSheafCompose (forget D)
F G : Sheaf J (Type w)
f : F ⟶ G
⊢ IsLocallyInjective (imageι f)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹⁸ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝¹⁷ : Category.{v_1, u_1} A
inst✝¹⁶ : Category.{v_2, u_2} B
F✝ : A ⥤ B
D : Type u_3
E : Type u_4
inst✝¹⁵ : Category.{v_3, u_3} D
inst✝¹⁴ : Category.{v_4, u_4} E
F : D ⥤ E
inst✝¹³ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝¹² : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝¹¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹⁰ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁸ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
FD : D → D → Type u_5
CD : D → Type u_6
FE : E → E → Type u_7
CE : E → Type u_8
inst✝⁷ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁶ : (X Y : E) → FunLike (FE X Y) (CE X) (CE Y)
instCCD : ConcreteCategory D FD
instCCE : ConcreteCategory E FE
inst✝⁵ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁴ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget E)
inst✝³ : PreservesLimitsOfSize.{max v u, max v u, v_3, u_6, u_3, u_6 + 1} (forget D)
inst✝² : PreservesLimitsOfSize.{max v u, max v u, v_4, u_8, u_4, u_8 + 1} (forget E)
inst✝¹ : (forget D).ReflectsIsomorphisms
inst✝ : (forget E).ReflectsIsomorphisms
P : Cᵒᵖ ⥤ D
⊢ IsIso ((sheafComposeNatTrans J F (plusPlusAdjunction J D) (plusPlusAdjunction J E)).app P)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹⁸ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝¹⁷ : Category.{v_1, u_1} A
inst✝¹⁶ : Category.{v_2, u_2} B
F✝ : A ⥤ B
D : Type u_3
E : Type u_4
inst✝¹⁵ : Category.{v_3, u_3} D
inst✝¹⁴ : Category.{v_4, u_4} E
F : D ⥤ E
inst✝¹³ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝¹² : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝¹¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹⁰ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁸ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
FD : D → D → Type u_5
CD : D → Type u_6
FE : E → E → Type u_7
CE : E → Type u_8
inst✝⁷ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁶ : (X Y : E) → FunLike (FE X Y) (CE X) (CE Y)
instCCD : ConcreteCategory D FD
instCCE : ConcreteCategory E FE
inst✝⁵ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁴ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget E)
inst✝³ : PreservesLimitsOfSize.{max v u, max v u, v_3, u_6, u_3, u_6 + 1} (forget D)
inst✝² : PreservesLimitsOfSize.{max v u, max v u, v_4, u_8, u_4, u_8 + 1} (forget E)
inst✝¹ : (forget D).ReflectsIsomorphisms
inst✝ : (forget E).ReflectsIsomorphisms
⊢ J.PreservesSheafification F

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝ : ConcreteCategory A FA
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
U : C
s : ToType (G.obj (op U))
⊢ ∀ {Y Z : C} {f_1 : Y ⟶ U},
    (∃ t, (ConcreteCategory.hom (f.app (op Y))) t = (ConcreteCategory.hom (G.map f_1.op)) s) →
      ∀ (g : Z ⟶ Y), ∃ t, (ConcreteCategory.hom (f.app (op Z))) t = (ConcreteCategory.hom (G.map (g ≫ f_1).op)) s

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
f : F ⟶ G
h₁ : ∀ (X : Cᵒᵖ), Function.Injective ⇑(ConcreteCategory.hom (f.hom.app X))
right✝ : IsLocallySurjective f
this : ∀ (X : Cᵒᵖ), Function.Surjective ⇑(ConcreteCategory.hom (f.hom.app X))
⊢ IsIso f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
A : Type u'
inst✝² : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝ : ConcreteCategory A FA
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
U : C
s : ToType (G.obj (op U))
V : C
g : V ⟶ U
⊢ Sieve.pullback g (imageSieve f s) = imageSieve f ((ConcreteCategory.hom (G.map g.op)) s)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
A : Type u'
inst✝² : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝ : ConcreteCategory A FA
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
U : C
s : ToType (F.obj (op U))
⊢ imageSieve f ((ConcreteCategory.hom (f.app (op U))) s) = ⊤

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝ : ConcreteCategory A FA
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
⊢ IsLocallySurjective J f ↔ Subfunctor.sheafify J (Subfunctor.range (whiskerRight f (forget A))) = ⊤

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
⊢ IsLocallySurjective J f ↔ Subfunctor.sheafify J (Subfunctor.range f) = ⊤

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝ : ConcreteCategory A FA
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
⊢ IsLocallySurjective J f ↔ IsLocallySurjective J (whiskerRight f (forget A))

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝ : ConcreteCategory A FA
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
H : ∀ (U : Cᵒᵖ), Function.Surjective ⇑(ConcreteCategory.hom (f.app U))
U : C
s : ToType (G.obj (op U))
⊢ imageSieve f s ∈ J U

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝³ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝¹ : ConcreteCategory A FA
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
inst✝ : IsIso f
⊢ IsLocallySurjective J f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
inst✝¹ : IsLocallySurjective J f₁
inst✝ : IsLocallySurjective J f₂
U✝ : C
s : ToType (F₃.obj (op U✝))
⊢ (Sieve.bind (imageSieve f₂ s).arrows fun x x_1 h ↦ imageSieve f₁ (Exists.choose h)) ≤ imageSieve (f₁ ≫ f₂) s

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
inst✝¹ : IsLocallySurjective J f₁
inst✝ : IsLocallySurjective J f₂
U✝ : C
s : ToType (F₃.obj (op U✝))
⊢ imageSieve (f₁ ≫ f₂) s ∈ J U✝

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝³ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝¹ : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
inst✝ : IsLocallySurjective J (f₁ ≫ f₂)
X : C
x : ToType (F₃.obj (op X))
Y : C
g : Y ⟶ X
hg : (imageSieve (f₁ ≫ f₂) x).arrows g
⊢ (ConcreteCategory.hom (f₂.app (op Y))) ((ConcreteCategory.hom (f₁.app (op Y))) (localPreimage (f₁ ≫ f₂) x g hg)) =
    (ConcreteCategory.hom (F₃.map g.op)) x

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝³ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝¹ : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
inst✝ : IsLocallySurjective J (f₁ ≫ f₂)
X : C
x : ToType (F₃.obj (op X))
⊢ imageSieve f₂ x ∈ J X

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝³ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝¹ : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
f₃ : F₁ ⟶ F₃
fac : f₁ ≫ f₂ = f₃
inst✝ : IsLocallySurjective J f₃
⊢ IsLocallySurjective J f₂

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝³ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝¹ : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
f₃ : F₁ ⟶ F₃
fac : f₁ ≫ f₂ = f₃
inst✝ : IsLocallySurjective J f₁
⊢ IsLocallySurjective J f₃ ↔ IsLocallySurjective J f₂

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝ : ConcreteCategory A FA
K : GrothendieckTopology C
hJK : J ≤ K
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
h : IsLocallySurjective J f
U✝ : C
s : ToType (G.obj (op U✝))
⊢ imageSieve f s ∈ K U✝

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
inst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)
inst✝ : IsLocallySurjective J f₁
X : Cᵒᵖ
x₁ x₂ : ToType (F₂.obj X)
h : (ConcreteCategory.hom (f₂.app X)) x₁ = (ConcreteCategory.hom (f₂.app X)) x₂
S : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂
⊢ S ∈ J (unop X)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
f : F ⟶ G
h₁ : ∀ (X : Cᵒᵖ), Function.Injective ⇑(ConcreteCategory.hom (f.hom.app X))
right✝ : IsLocallySurjective f
X : Cᵒᵖ
s : G.obj.obj X
H : Presieve.IsSheafFor F.obj (Presheaf.imageSieve f.hom s).arrows
t : Presieve.FamilyOfElements F.obj (Presheaf.imageSieve f.hom s).arrows :=
  fun Y g hg ↦ Presheaf.localPreimage f.hom s g hg
⊢ t.Compatible

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
f : F ⟶ G
⊢ IsLocallyInjective f ∧ IsLocallySurjective f ↔ IsIso f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
inst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)
inst✝ : IsLocallySurjective J f₁
X : Cᵒᵖ
x₁ x₂ : ToType (F₂.obj X)
h : (ConcreteCategory.hom (f₂.app X)) x₁ = (ConcreteCategory.hom (f₂.app X)) x₂
⊢ equalizerSieve x₁ x₂ ∈ J (unop X)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
f₃ : F₁ ⟶ F₃
fac : f₁ ≫ f₂ = f₃
inst✝¹ : IsLocallyInjective J f₃
inst✝ : IsLocallySurjective J f₁
⊢ IsLocallyInjective J f₂

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F G : Sheaf J A
f : F ⟶ G
inst✝¹ : (forget A).ReflectsIsomorphisms
inst✝ : J.HasSheafCompose (forget A)
⊢ IsLocallyInjective f ∧ IsLocallySurjective f ↔ IsIso f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝³ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝¹ : ConcreteCategory A FA
inst✝ : J.WEqualsLocallyBijective A
X Y : Cᵒᵖ ⥤ A
f : X ⟶ Y
⊢ J.W f ↔ Presheaf.IsLocallyInjective J f ∧ Presheaf.IsLocallySurjective J f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
inst✝¹ : IsLocallySurjective J (f₁ ≫ f₂)
inst✝ : IsLocallyInjective J f₂
X : C
x : ToType (F₂.obj (op X))
S : Sieve X := imageSieve (f₁ ≫ f₂) ((ConcreteCategory.hom (f₂.app (op X))) x)
T : ⦃Y : C⦄ → (f : Y ⟶ X) → S.arrows f → Sieve Y :=
  fun Y f hf ↦
    equalizerSieve
      ((ConcreteCategory.hom (f₁.app (op Y))) (localPreimage (f₁ ≫ f₂) ((ConcreteCategory.hom (f₂.app (op X))) x) f hf))
      ((ConcreteCategory.hom (F₂.map f.op)) x)
Y Z : C
a : Y ⟶ Z
g : Z ⟶ unop (op X)
hg : S.arrows g
ha : (T g hg).arrows a
⊢ (ConcreteCategory.hom (f₁.app (op Y)))
      ((ConcreteCategory.hom (F₁.map a.op)) (localPreimage (f₁ ≫ f₂) ((ConcreteCategory.hom (f₂.app (op X))) x) g hg)) =
    (ConcreteCategory.hom (F₂.map (a ≫ g).op)) x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.VanKampen

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i ↦ f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
⊢ IsUniversalColimit (extendCofan c₁ c₂)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁵ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝³ : ConcreteCategory A FA
inst✝² : J.WEqualsLocallyBijective A
X Y : Cᵒᵖ ⥤ A
f : X ⟶ Y
inst✝¹ : Presheaf.IsLocallyInjective J f
inst✝ : Presheaf.IsLocallySurjective J f
⊢ J.W f

Tactic:
by

Goal state after: