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

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁷ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝⁶ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝⁵ : ConcreteCategory A FA
inst✝⁴ : HasWeakSheafify J A
inst✝³ : (forget A).ReflectsIsomorphisms
inst✝² : J.HasSheafCompose (forget A)
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ A), Presheaf.IsLocallyInjective J (CategoryTheory.toSheafify J P)
inst✝ : ∀ (P : Cᵒᵖ ⥤ A), Presheaf.IsLocallySurjective J (CategoryTheory.toSheafify J P)
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
⊢ J.W f ↔ Presheaf.IsLocallyInjective J f ∧ Presheaf.IsLocallySurjective J f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁹ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝⁸ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝⁷ : ConcreteCategory A FA
D : Type w
inst✝⁶ : Category.{w', w} D
FD : D → D → Type u_2
CD : D → Type (max u v)
inst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁴ : ConcreteCategory D FD
inst✝³ : HasWeakSheafify J D
inst✝² : J.HasSheafCompose (forget D)
inst✝¹ : J.PreservesSheafification (forget D)
inst✝ : (forget D).ReflectsIsomorphisms
⊢ J.WEqualsLocallyBijective D

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
inst✝¹ : HasWeakSheafify J A
inst✝ : J.WEqualsLocallyBijective A
P Q : Cᵒᵖ ⥤ A
φ : P ⟶ Q
⊢ Sheaf.IsLocallyInjective ((presheafToSheaf J A).map φ) ↔ IsLocallyInjective J φ

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
inst✝¹ : HasWeakSheafify J A
inst✝ : J.WEqualsLocallyBijective A
P Q : Cᵒᵖ ⥤ A
φ : P ⟶ Q
⊢ Sheaf.IsLocallySurjective ((presheafToSheaf J A).map φ) ↔ IsLocallySurjective J φ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R : Sheaf J RingCat
X Y : Cᵒᵖ
hX : Limits.IsInitial X
hY : Limits.IsInitial Y
φ : X ≅ Y
⊢ forgetToSheafModuleCat R X hX ≅
    forgetToSheafModuleCat R Y hY ⋙ sheafCompose J (ModuleCat.restrictScalars (RingCat.Hom.hom (R.obj.map φ.hom)))

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))
⊢ imageSieve f₁ x ∈ J 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 : IsLocallySurjective J f₃
inst✝ : IsLocallyInjective J f₂
⊢ IsLocallySurjective J f₁

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R : Sheaf J RingCat
⊢ ∀ (P Q R_1 : SheafOfModules R) (f f' : P ⟶ Q) (g : Q ⟶ R_1), (f + f') ≫ g = f ≫ g + f' ≫ g

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R : Sheaf J RingCat
⊢ ∀ (P Q R_1 : SheafOfModules R) (f : P ⟶ Q) (g g' : Q ⟶ R_1), f ≫ (g + g') = f ≫ g + f ≫ g'

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R : Sheaf J RingCat
M N : SheafOfModules R
s : M.sections
p : M ⟶ N
⊢ N.unitHomEquiv (M.unitHomEquiv.symm s ≫ p) = N.unitHomEquiv (N.unitHomEquiv.symm (sectionsMap p 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 : IsLocallyInjective J f₁
inst✝ : IsLocallySurjective J f₁
⊢ IsLocallyInjective J (f₁ ≫ f₂) ↔ IsLocallyInjective J f₂

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 f₂
inst✝ : IsLocallySurjective J f₂
⊢ IsLocallySurjective J (f₁ ≫ f₂) ↔ IsLocallySurjective J f₁

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : 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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst✝ : HasFiniteCoproducts C
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
Hc : IsColimit c
j : Fin (n + 1)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
⊢ F = Discrete.functor 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
F₁ F₂ : Cᵒᵖ ⥤ Type w
f : F₁ ⟶ F₂
X : C
⊢ ∀ (s : ToType ((Subfunctor.sheafify J (Subfunctor.range f)).toFunctor.obj (op X))),
    imageSieve (Subfunctor.toRangeSheafify J f) s ∈ J X

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 : Cᵒᵖ ⥤ Type (max u v)
⊢ J.toSheafify F ≫
      J.sheafifyLift (Subfunctor.toRangeSheafify J (J.toSheafify F)) ⋯ ≫
        (Subfunctor.sheafify J (Subfunctor.range (J.toSheafify F))).ι =
    J.toSheafify F ≫ 𝟙 (J.sheafify 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 : Cᵒᵖ ⥤ Type (max u v)
⊢ (Subfunctor.sheafify J (Subfunctor.range (J.toSheafify F))).ι ≫
      J.sheafifyLift (Subfunctor.toRangeSheafify J (J.toSheafify F)) ⋯ =
    𝟙 (Subfunctor.sheafify J (Subfunctor.range (J.toSheafify F))).toFunctor

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
P : Cᵒᵖ ⥤ Type (max u v)
U✝ : C
x : ToType ((J.plusObj P).obj (op U✝))
⊢ imageSieve (J.toPlus P) x ∈ J U✝

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
P : Cᵒᵖ ⥤ Type (max u v)
⊢ IsLocallySurjective J (J.toSheafify P)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁸ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝⁷ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝⁶ : ConcreteCategory A FA
D : Type u_2
inst✝⁵ : Category.{v_1, u_2} D
FD : D → D → Type u_3
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)
⊢ IsLocallySurjective J (toSheafify J P)

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
F₁ F₂ : Sheaf J A
φ : F₁ ⟶ F₂
⊢ Presheaf.IsLocallySurjective J ((sheafToPresheaf J A).map φ) ↔ IsLocallySurjective φ

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 : Sheaf J A
φ : F₁ ⟶ F₂
ψ : F₂ ⟶ F₃
inst✝ : IsIso φ
⊢ IsLocallySurjective φ

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
F₁ F₂ F₃ : Sheaf J A
φ : F₁ ⟶ F₂
ψ : F₂ ⟶ F₃
F G : Sheaf J (Type w)
f : F ⟶ G
⊢ IsLocallySurjective (toImage f)

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 : Sheaf J (Type w)
f : F ⟶ G
⊢ IsLocallySurjective f ↔ IsIso (imageι f)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R✝ : Sheaf J RingCat
R : Cᵒᵖ ⥤ RingCat
M₁ M₂ : PresheafOfModules R
f : M₁ ⟶ M₂
N : PresheafOfModules R
hN : Presheaf.IsSheaf J N.presheaf
inst✝² : J.WEqualsLocallyBijective AddCommGrpCat
inst✝¹ : IsLocallySurjective J f
inst✝ : IsLocallyInjective J f
ψ : M₁ ⟶ N
⊢ ∀ (X : Cᵒᵖ) (r : ↑(R.obj X)) (m : ↑(M₂.obj X)),
    (ConcreteCategory.hom (((ObjectProperty.isLocal.homEquiv ⋯ N.presheaf hN).symm ((toPresheaf R).map ψ)).app X))
        (r • m) =
      r •
        (ConcreteCategory.hom (((ObjectProperty.isLocal.homEquiv ⋯ N.presheaf hN).symm ((toPresheaf R).map ψ)).app X)) m

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 : Sheaf J A
φ✝ : F₁✝ ⟶ F₂✝
ψ : F₂✝ ⟶ F₃
inst✝¹ : J.HasSheafCompose (forget A)
F₁ F₂ : Sheaf J (Type w)
φ : F₁ ⟶ F₂
inst✝ : IsLocallySurjective φ
Z : Sheaf J (Type w)
f₁ f₂ : F₂ ⟶ Z
h : φ ≫ f₁ = φ ≫ f₂
⊢ f₁ = f₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf.ChangeOfRings

Goal state before:
pp: C : Type u'
inst✝¹ : Category.{v', u'} C
J : GrothendieckTopology C
R R' : Cᵒᵖ ⥤ RingCat
α : R ⟶ R'
M₁ M₂ : PresheafOfModules R'
hM₂ : Presheaf.IsSheaf J M₂.presheaf
inst✝ : Presheaf.IsLocallySurjective J α
g : (restrictScalars α).obj M₁ ⟶ (restrictScalars α).obj M₂
X : Cᵒᵖ
r' : ↑(R'.obj X)
m : ↑(M₁.obj X)
⊢ (ConcreteCategory.hom (((toPresheaf R).map g).app X)) (r' • m) =
    r' • (ConcreteCategory.hom (((toPresheaf R).map g).app X)) m

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
φ : F ⟶ G
inst✝ : HasSheafify J (Type w)
⊢ IsLocallySurjective φ ↔ Epi φ

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : 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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst✝ : HasFiniteCoproducts C
⊢ IsVanKampenColimit (extendCofan c₁ c₂)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
S : C
ι : Type u_2
inst✝¹ : Small.{w, u_2} ι
X : ι → C
f : (i : ι) → X i ⟶ S
inst✝ : LocallySmall.{w, v, u} C
c : Cofan fun i ↦ shrinkYoneda.{w, v, u}.obj (X i)
hc : IsColimit c
U : C
g : U ⟶ S
⊢ imageSieve (Cofan.IsColimit.desc hc fun i ↦ shrinkYoneda.{w, v, u}.map (f i)) (shrinkYonedaObjObjEquiv.symm g) =
    Sieve.pullback g (Sieve.ofArrows X f)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
inst✝¹ : LocallySmall.{w, v, u} C
S : C
ι : Type u_2
inst✝ : Small.{w, u_2} ι
X : ι → C
f : (i : ι) → X i ⟶ S
c : Cofan fun i ↦ shrinkYoneda.{w, v, u}.obj (X i)
hc : IsColimit c
⊢ Sieve.ofArrows X f ∈ J S ↔
    Presheaf.IsLocallySurjective J (Cofan.IsColimit.desc hc fun i ↦ shrinkYoneda.{w, v, u}.map (f i))

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ IsPullback (if h : j = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i))
    (if h : j = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X j)) (c.inj i) (c.inj j)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasInitial C
ι : Type u_3
F : Discrete ι ⥤ C
c : Cocone F
hc : IsVanKampenColimit c
i j : Discrete ι
hi : i ≠ j
f : ι → C := F.obj ∘ Discrete.mk
⊢ ∀ (X Y : Discrete ι) (f_1 : X ⟶ Y), F.map f_1 ≍ (Discrete.functor f).map f_1

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasInitial C
ι : Type u_3
F : Discrete ι ⥤ C
c : Cocone F
hc : IsVanKampenColimit c
i j : Discrete ι
hi : i ≠ j
⊢ IsPullback (initial.to (F.obj i)) (initial.to (F.obj j)) (c.ι.app i) (c.ι.app j)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasInitial C
ι : Type u_3
F : Discrete ι ⥤ C
c : Cocone F
hc : IsVanKampenColimit c
i : Discrete ι
f : ι → C := F.obj ∘ Discrete.mk
⊢ ∀ (X Y : Discrete ι) (f_1 : X ⟶ Y), F.map f_1 ≍ (Discrete.functor f).map f_1

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
S : C
ι : Type u_2
inst✝ : Small.{max w v, u_2} ι
X : ι → C
f : (i : ι) → X i ⟶ S
c : Cofan fun i ↦ uliftYoneda.{w, v, u}.obj (X i)
hc : IsColimit c
e : (Discrete.functor fun i ↦ uliftYoneda.{w, v, u}.obj (X i)) ≅
  Discrete.functor fun i ↦ shrinkYoneda.{max w v, v, u}.obj (X i) :=
  Discrete.natIso fun i ↦ uliftYonedaIsoShrinkYoneda.app (X i.as)
hc' : IsColimit ((Cocone.precompose e.inv).obj c) := (IsColimit.precomposeInvEquiv e c).invFun hc
i : ι
⊢ c.inj i ≫ (Cofan.IsColimit.desc hc fun i ↦ uliftYoneda.{w, v, u}.map (f i)) ≫ uliftYonedaIsoShrinkYoneda.hom.app S =
    c.inj i ≫ Cofan.IsColimit.desc hc' fun i ↦ shrinkYoneda.{max v w, v, u}.map (f i)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasInitial C
ι : Type u_3
i : Discrete ι
f : ι → C
c : Cocone (Discrete.functor f)
hc : IsVanKampenColimit c
⊢ 𝟙 ((Discrete.functor f).obj i) ≫ 𝟙 ((Discrete.functor f).obj i) =
    eqToHom ⋯ ≫ if h : i.as = i.as then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (f i.as)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasInitial C
ι : Type u_3
F : Discrete ι ⥤ C
c : Cocone F
hc : IsVanKampenColimit c
i : Discrete ι
⊢ Mono (c.ι.app i)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
ι : Type u_3
S B : C
X : ι → C
a : Cofan X
hau : IsUniversalColimit a
f : (i : ι) → X i ⟶ S
u : a.pt ⟶ S
v : B ⟶ S
s : (i : ι) → PullbackCone v (f i)
hs : (i : ι) → IsLimit (s i)
t : PullbackCone v u
ht : IsLimit t
d : Cofan fun i ↦ (s i).pt
e : d.pt ≅ t.pt
hu : ∀ (i : ι), a.inj i ≫ u = f i
he₁ : ∀ (i : ι), d.inj i ≫ e.hom ≫ t.fst = (s i).fst
he₂ : ∀ (i : ι), d.inj i ≫ e.hom ≫ t.snd = (s i).snd ≫ a.inj i
i : ι
⊢ (s i).fst ≫ v = ((s i).snd ≫ a.inj i) ≫ u

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
ι : Type u_3
S B : C
X : ι → C
a : Cofan X
hau : IsUniversalColimit a
f : (i : ι) → X i ⟶ S
u : a.pt ⟶ S
v : B ⟶ S
s : (i : ι) → PullbackCone v (f i)
hs : (i : ι) → IsLimit (s i)
t : PullbackCone v u
ht : IsLimit t
d : Cofan fun i ↦ (s i).pt
e : d.pt ≅ t.pt
hu : ∀ (i : ι), a.inj i ≫ u = f i
he₁ : ∀ (i : ι), d.inj i ≫ e.hom ≫ t.fst = (s i).fst
he₂ : ∀ (i : ι), d.inj i ≫ e.hom ≫ t.snd = (s i).snd ≫ a.inj i
p : ι
⊢ (d.inj p ≫ e.hom) ≫ t.fst =
    (Cofan.mk t.pt fun i ↦ PullbackCone.IsLimit.lift ht (s i).fst ((s i).snd ≫ a.inj i) ⋯).inj p ≫ t.fst

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
ι : Type u_3
S B : C
X : ι → C
a : Cofan X
hau : IsUniversalColimit a
f : (i : ι) → X i ⟶ S
u : a.pt ⟶ S
v : B ⟶ S
s : (i : ι) → PullbackCone v (f i)
hs : (i : ι) → IsLimit (s i)
t : PullbackCone v u
ht : IsLimit t
d : Cofan fun i ↦ (s i).pt
e : d.pt ≅ t.pt
hu : ∀ (i : ι), a.inj i ≫ u = f i
he₁ : ∀ (i : ι), d.inj i ≫ e.hom ≫ t.fst = (s i).fst
he₂ : ∀ (i : ι), d.inj i ≫ e.hom ≫ t.snd = (s i).snd ≫ a.inj i
p : ι
⊢ (d.inj p ≫ e.hom) ≫ t.snd =
    (Cofan.mk t.pt fun i ↦ PullbackCone.IsLimit.lift ht (s i).fst ((s i).snd ≫ a.inj i) ⋯).inj p ≫ t.snd

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
S : C
ι : Type u_2
inst✝ : Small.{max w v, u_2} ι
X : ι → C
f : (i : ι) → X i ⟶ S
c : Cofan fun i ↦ uliftYoneda.{w, v, u}.obj (X i)
hc : IsColimit c
⊢ Sieve.ofArrows X f ∈ J S ↔
    Presheaf.IsLocallySurjective J (Cofan.IsColimit.desc hc fun i ↦ uliftYoneda.{w, v, u}.map (f i))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf.Limits

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
R : Cᵒᵖ ⥤ RingCat
F : D ⥤ PresheafOfModules R
inst✝¹ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
c : Cone F
inst✝ : HasLimitsOfShape D AddCommGrpCat
hc : IsLimit c
hF : ∀ (j : D), Presheaf.IsSheaf J (F.obj j).presheaf
⊢ Presheaf.IsSheaf J c.pt.presheaf

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf.Limits

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
R : Sheaf J RingCat
F : D ⥤ SheafOfModules R
inst✝¹ :
  ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ CategoryTheory.forget (ModuleCat ↑(R.obj.obj X))).sections
inst✝ : HasLimitsOfShape D AddCommGrpCat
X : Cᵒᵖ
⊢ Small.{v, max u₂ v}
    ↑(((F ⋙ forget R) ⋙ PresheafOfModules.evaluation R.obj X) ⋙
          CategoryTheory.forget (ModuleCat ↑(R.obj.obj X))).sections

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf.Limits

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
R : Sheaf J RingCat
F : D ⥤ SheafOfModules R
inst✝¹ :
  ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ CategoryTheory.forget (ModuleCat ↑(R.obj.obj X))).sections
inst✝ : HasLimitsOfShape D AddCommGrpCat
X : Cᵒᵖ
⊢ PreservesLimit F (evaluation R X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
inst✝² : Balanced C
F : C ⥤ D
inst✝¹ : F.ReflectsMonomorphisms
inst✝ : F.ReflectsEpimorphisms
A✝ B✝ : C
f : A✝ ⟶ B✝
hf : IsIso (F.map f)
⊢ IsIso f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝³ : F.ReflectsIsomorphisms
inst✝² : F.PreservesEpimorphisms
inst✝¹ : F.PreservesMonomorphisms
inst✝ : Balanced D
X✝ Y✝ : C
f : X✝ ⟶ Y✝
x✝¹ : Mono f
x✝ : Epi f
⊢ IsIso f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
ι : Type u_3
S B : C
X : ι → C
a : Cofan X
hau : IsUniversalColimit a
f : (i : ι) → X i ⟶ S
u : a.pt ⟶ S
v : B ⟶ S
s : (i : ι) → PullbackCone v (f i)
hs : (i : ι) → IsLimit (s i)
t : PullbackCone v u
ht : IsLimit t
d : Cofan fun i ↦ (s i).pt
e : d.pt ≅ t.pt
hu : ∀ (i : ι), a.inj i ≫ u = f i
he₁ : ∀ (i : ι), d.inj i ≫ e.hom ≫ t.fst = (s i).fst
he₂ : ∀ (i : ι), d.inj i ≫ e.hom ≫ t.snd = (s i).snd ≫ a.inj i
iso : d ≅ Cofan.mk t.pt fun i ↦ PullbackCone.IsLimit.lift ht (s i).fst ((s i).snd ≫ a.inj i) ⋯ := Cofan.ext e ⋯
j : Discrete ι
⊢ PullbackCone.IsLimit.lift ht (s j.as).fst ((s j.as).snd ≫ a.inj j.as) ⋯ ≫ t.snd = (s j.as).snd ≫ a.inj j.as

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
ι : Type u_3
S B : C
X : ι → C
a : Cofan X
hau : IsUniversalColimit a
f : (i : ι) → X i ⟶ S
u : a.pt ⟶ S
v : B ⟶ S
s : (i : ι) → PullbackCone v (f i)
hs : (i : ι) → IsLimit (s i)
t : PullbackCone v u
ht : IsLimit t
d : Cofan fun i ↦ (s i).pt
e : d.pt ≅ t.pt
hu : ∀ (i : ι), a.inj i ≫ u = f i
he₁ : ∀ (i : ι), d.inj i ≫ e.hom ≫ t.fst = (s i).fst
he₂ : ∀ (i : ι), d.inj i ≫ e.hom ≫ t.snd = (s i).snd ≫ a.inj i
⊢ Nonempty (IsColimit d)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
ι : Type u_3
S B : C
X : ι → C
a : Cofan X
hau : IsUniversalColimit a
f : (i : ι) → X i ⟶ S
u : a.pt ⟶ S
v : B ⟶ S
P : ι → C
q₁ : (i : ι) → P i ⟶ B
q₂ : (i : ι) → P i ⟶ X i
hP : ∀ (i : ι), IsPullback (q₁ i) (q₂ i) v (f i)
d : Cofan P
hd : IsColimit d
hu : ∀ (i : ι), a.inj i ≫ u = f i
inst✝ : HasPullback v u
i : ι
⊢ q₁ i ≫ v = (q₂ i ≫ a.inj i) ≫ u

Tactic:
by

Goal state after: