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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
inst✝⁴ : LocallySmall.{w, v, u} C
P : ObjectProperty J.Point
inst✝³ : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
inst✝² : J.WEqualsLocallyBijective (Type w)
inst✝¹ : HasSheafify J (Type w)
hP : P.IsConservativeFamilyOfPoints
X : C
inst✝ : HasSheafify (J.over X) (Type w)
Y : C
f : Y ⟶ X
ι : Type (max u v)
Z : ι → C
g : (i : ι) → Z i ⟶ (Over.mk f).left
hS :
  ∀ (Φ : (ofObj fun ψ ↦ ψ.fst.obj.over ψ.snd).FullSubcategory) (x : Φ.obj.fiber.obj (Over.mk f)),
    ∃ Y_1 g_1,
      ∃ (_ : ((Sieve.overEquiv (Over.mk f)).symm (Sieve.ofArrows Z g)).arrows g_1),
        ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g_1)) y = x
Φ : P.FullSubcategory
y : Φ.obj.fiber.obj (Over.mk f).left
⊢ (Over.forget X ⋙ ⟨Φ, (ConcreteCategory.hom (Φ.obj.fiber.map f)) y⟩.fst.obj.fiber).obj (Over.mk f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : LocallySmall.{w, v, u} C
N : Type u'
inst✝¹ : Category.{v', u'} N
p : N ⥤ C
inst✝ : InitiallySmall N
U V : N
g : V ⟶ U
X : C
f : p.obj U ⟶ X
⊢ fiberMk (p.map g ≫ f) = fiberMk f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : LocallySmall.{w, v, u} C
N : Type u'
inst✝¹ : Category.{v', u'} N
p : N ⥤ C
inst✝ : InitiallySmall N
U V : N
g : V ⟶ U
⊢ fiberMk (p.map g) = fiberMk (𝟙 (p.obj U))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Map

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
J : GrothendieckTopology C
Φ : J.Point
F : C ⥤ D
K : GrothendieckTopology D
inst✝³ : F.IsCocontinuous J K
inst✝² : LocallySmall.{w, v', u'} D
A : Type u''
inst✝¹ : Category.{v'', u''} A
inst✝ : HasColimitsOfSize.{w, w, v'', u''} A
P : Dᵒᵖ ⥤ A
X : C
x : Φ.fiber.obj X
⊢ Φ.toPresheafFiber X x (F.op ⋙ P) ≫ (Φ.presheafFiberMapObjIso F K P).inv = Φ.toPresheafFiberMap F K P X x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Over

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
inst✝⁴ : LocallySmall.{w, v, u} C
P : ObjectProperty J.Point
inst✝³ : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
inst✝² : J.WEqualsLocallyBijective (Type w)
inst✝¹ : HasSheafify J (Type w)
hP : P.IsConservativeFamilyOfPoints
X : C
inst✝ : HasSheafify (J.over X) (Type w)
Y : C
f : Y ⟶ X
ι : Type (max u v)
Z : ι → C
g : (i : ι) → Z i ⟶ (Over.mk f).left
hS :
  ∀ (Φ : (ofObj fun ψ ↦ ψ.fst.obj.over ψ.snd).FullSubcategory) (x : Φ.obj.fiber.obj (Over.mk f)),
    ∃ Y_1 g_1,
      ∃ (_ : ((Sieve.overEquiv (Over.mk f)).symm (Sieve.ofArrows Z g)).arrows g_1),
        ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g_1)) y = x
Φ : P.FullSubcategory
y : Φ.obj.fiber.obj (Over.mk f).left
⊢ y ∈
    (FunctorToTypes.fromOverSubfunctor ⟨Φ, (ConcreteCategory.hom (Φ.obj.fiber.map f)) y⟩.fst.obj.fiber
          ⟨Φ, (ConcreteCategory.hom (Φ.obj.fiber.map f)) y⟩.snd).obj
      (Over.mk f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : LocallySmall.{w, v, u} C
N : Type u'
inst✝¹ : Category.{v', u'} N
p : N ⥤ C
inst✝ : InitiallySmall N
U : N
X : C
f : p.obj U ⟶ X
Y : C
g : X ⟶ Y
⊢ (hom
        ((p.op.whiskerLeft (shrinkYoneda.{w, v, u}.map g)).app (op U) ≫
          colimit.ι (p.op ⋙ shrinkYoneda.{w, v, u}.obj Y) (op U)))
      (shrinkYonedaObjObjEquiv.symm f) =
    fiberMk (f ≫ g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : LocallySmall.{w, v, u} C
X U : C
R : Sieve U
hR : R ∈ ⊥ U
x : (shrinkYoneda.{w, v, u}.flip.obj (op X)).obj U
⊢ R = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : LocallySmall.{w, v, u} C
X U : C
x : (shrinkYoneda.{w, v, u}.flip.obj (op X)).obj U
hR : ⊤ ∈ ⊥ U
⊢ ⊤.arrows (𝟙 U)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : LocallySmall.{w, v, u} C
X U : C
x : (shrinkYoneda.{w, v, u}.flip.obj (op X)).obj U
hR : ⊤ ∈ ⊥ U
⊢ (ConcreteCategory.hom ((shrinkYoneda.{w, v, u}.flip.obj (op X)).map (𝟙 U))) x = x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : LocallySmall.{w, v, u} C
X U : C
R : Sieve U
hR : R ∈ ⊥ U
x : (shrinkYoneda.{w, v, u}.flip.obj (op X)).obj U
⊢ ∃ Y f, ∃ (_ : R.arrows f), ∃ y, (ConcreteCategory.hom ((shrinkYoneda.{w, v, u}.flip.obj (op X)).map f)) y = x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Presheaf

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : LocallySmall.{w, v, u} C
X : C
A : Type u_1
inst✝¹ : Category.{u_2, u_1} A
inst✝ : HasColimitsOfSize.{w, w, u_2, u_1} A
⊢ IsIso ((pointBot X).toPresheafFiberNatTrans X (shrinkYonedaObjObjEquiv.symm (𝟙 X)))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Presheaf

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : LocallySmall.{w, v, u} C
inst✝ : Small.{w, u} C
⊢ ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} (pointsBot C)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : LocallySmall.{w, v, u} C
N : Type u'
inst✝¹ : Category.{v', u'} N
p : N ⥤ C
inst✝ : InitiallySmall N
J : GrothendieckTopology C
U V : N
f : U ⟶ V
⊢ (hom ((fiber p).map (p.map f))) ((fiber p).elementsMk (p.obj U) (fiberMk (𝟙 (p.obj U)))).snd =
    ((fiber p).elementsMk (p.obj V) (fiberMk (𝟙 (p.obj V)))).snd

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : LocallySmall.{w, v, u} C
N : Type u'
inst✝² : Category.{v', u'} N
p : N ⥤ C
inst✝¹ : InitiallySmall N
J : GrothendieckTopology C
inst✝ : IsCofiltered N
X : C
U : N
f : p.obj U ⟶ X
⊢ (hom ((fiber p).map f)) ((functor p).obj U).snd = ⟨X, fiberMk f⟩.snd

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Over

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
inst✝⁴ : LocallySmall.{w, v, u} C
P : ObjectProperty J.Point
inst✝³ : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
inst✝² : J.WEqualsLocallyBijective (Type w)
inst✝¹ : HasSheafify J (Type w)
hP : P.IsConservativeFamilyOfPoints
X : C
inst✝ : HasSheafify (J.over X) (Type w)
Y✝ : C
f : Y✝ ⟶ X
ι : Type (max u v)
Z : ι → C
g : (i : ι) → Z i ⟶ (Over.mk f).left
hS :
  ∀ (Φ : (ofObj fun ψ ↦ ψ.fst.obj.over ψ.snd).FullSubcategory) (x : Φ.obj.fiber.obj (Over.mk f)),
    ∃ Y g_1,
      ∃ (_ : ((Sieve.overEquiv (Over.mk f)).symm (Sieve.ofArrows Z g)).arrows g_1),
        ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g_1)) y = x
Φ : P.FullSubcategory
y : Φ.obj.fiber.obj (Over.mk f).left
T : Over X
a : T ⟶ Over.mk f
Y : C
i : ι
b : (Over.forget X).obj T ⟶ Z i
hb : b ≫ g i = (Over.forget X).map a
z : (Over.forget X ⋙ ⟨Φ, (ConcreteCategory.hom (Φ.obj.fiber.map f)) y⟩.fst.obj.fiber).obj T
hz₁ :
  z ∈
    (FunctorToTypes.fromOverSubfunctor ⟨Φ, (ConcreteCategory.hom (Φ.obj.fiber.map f)) y⟩.fst.obj.fiber
          ⟨Φ, (ConcreteCategory.hom (Φ.obj.fiber.map f)) y⟩.snd).obj
      T
hz₂ :
  ↑((ConcreteCategory.hom
          ({
                    obj :=
                      ⟨Φ, (ConcreteCategory.hom (Φ.obj.fiber.map f)) y⟩.fst.obj.over
                        ⟨Φ, (ConcreteCategory.hom (Φ.obj.fiber.map f)) y⟩.snd,
                    property := ⋯ }.obj.fiber.map
            a))
        ⟨z, hz₁⟩) =
    ↑⟨y, ⋯⟩
⊢ (ConcreteCategory.hom (Φ.obj.fiber.map (b ≫ g i))) z = y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : LocallySmall.{w, v, u} C
X : C
S : Sieve X
hS :
  ∀ (Φ : (pointsBot C).FullSubcategory) (x : Φ.obj.fiber.obj X),
    ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x
Y : C
a : Y ⟶ X
ha : S.arrows a
b : X ⟶ Y
hb :
  (ConcreteCategory.hom ((shrinkYoneda.{w, v, u}.map a).app (op X))) (shrinkYonedaObjObjEquiv.symm b) =
    shrinkYonedaObjObjEquiv.symm (𝟙 X)
⊢ shrinkYonedaObjObjEquiv.symm (b ≫ a) = shrinkYonedaObjObjEquiv.symm (𝟙 X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Over

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
inst✝⁴ : LocallySmall.{w, v, u} C
P : ObjectProperty J.Point
inst✝³ : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
inst✝² : J.WEqualsLocallyBijective (Type w)
inst✝¹ : HasSheafify J (Type w)
hP : P.IsConservativeFamilyOfPoints
X : C
inst✝ : HasSheafify (J.over X) (Type w)
Y : Over X
S : Sieve Y
hS :
  ∀ (Φ : (ofObj fun ψ ↦ ψ.fst.obj.over ψ.snd).FullSubcategory) (x : Φ.obj.fiber.obj Y),
    ∃ Y_1 g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x
⊢ S ∈ (J.over X) Y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Descent.DescentData

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
S' : C
p : S' ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S'
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j ≫ p
D : F.DescentData f
⊢ ∀ ⦃Y : C⦄ (q : Y ⟶ S) ⦃i₁ i₂ : ι⦄ (f₁ : Y ⟶ X i₁) (f₂ : Y ⟶ X i₂) (hf₁ : f₁ ≫ f i₁ = q) (hf₂ : f₂ ≫ f i₂ = q),
    (F.map f₁.op.toLoc).toFunctor.map
          ((Cat.Hom.toNatIso (F.mapId { as := op (X ((fun x ↦ x) i₁)) })).app (D.obj ((fun x ↦ x) i₁))).hom ≫
        ((𝟭 (F.DescentData f)).obj D).hom q f₁ f₂ ⋯ ⋯ =
      ((pullFunctor F ⋯).obj D).hom q f₁ f₂ ⋯ ⋯ ≫
        (F.map f₂.op.toLoc).toFunctor.map
          ((Cat.Hom.toNatIso (F.mapId { as := op (X ((fun x ↦ x) i₂)) })).app (D.obj ((fun x ↦ x) i₂))).hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : LocallySmall.{w, v, u} C
X : C
S : Sieve X
hS :
  ∀ (Φ : (pointsBot C).FullSubcategory) (x : Φ.obj.fiber.obj X),
    ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x
⊢ S ∈ ⊥ X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Over

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
inst✝⁴ : LocallySmall.{w, v, u} C
Φ : J.Point
X✝ : C
x : Φ.fiber.obj X✝
inst✝³ : J.HasEnoughPoints
inst✝² : J.WEqualsLocallyBijective (Type w)
inst✝¹ : HasSheafify J (Type w)
X : C
inst✝ : HasSheafify (J.over X) (Type w)
⊢ (J.over X).HasEnoughPoints

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
h : K.Generates J
F : Cᵒᵖ ⥤ Type w
H : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
inst✝ : ∀ (Z : C), _root_.Small.{max u v, w} (F.obj (Opposite.op Z))
X✝ : C
S : Sieve X✝
hS : S ∈ J X✝
F' : Cᵒᵖ ⥤ Type (max u v) := FunctorToTypes.shrink.{max u v, w, v, u} F
e : (X : C) → F.obj (Opposite.op X) ≃ F'.obj (Opposite.op X) := fun X ↦ equivShrink (F.obj (Opposite.op X))
X Y : C
f : X ⟶ Y
x : F.obj (Opposite.op Y)
⊢ (e X) ((ConcreteCategory.hom (F.map f.op)) x) = (ConcreteCategory.hom (F'.map f.op)) ((e Y) x)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : LocallySmall.{w, v, u} C
N : Type u'
inst✝² : Category.{v', u'} N
p : N ⥤ C
inst✝¹ : InitiallySmall N
J : GrothendieckTopology C
inst✝ : IsCofiltered N
X : C
V U : N
f : p.obj U ⟶ X
φ₁ : ((functor p).obj V).fst ⟶ ⟨X, fiberMk f⟩.fst
hφ₁ : (hom ((fiber p).map φ₁)) ((functor p).obj V).snd = ⟨X, fiberMk f⟩.snd
φ₂ : ((functor p).obj V).fst ⟶ ⟨X, fiberMk f⟩.fst
hφ₂ : (hom ((fiber p).map φ₂)) ((functor p).obj V).snd = ⟨X, fiberMk f⟩.snd
W : N
g : W ⟶ V
hg : p.map g ≫ φ₁ = p.map g ≫ φ₂
⊢ (functor p).map g ≫ ⟨φ₁, hφ₁⟩ = (functor p).map g ≫ ⟨φ₂, hφ₂⟩

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Descent.DescentData

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
S' : C
p : S' ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S'
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j ≫ p
S'' : C
q : S'' ⟶ S'
ι'' : Type t''
X'' : ι'' → C
f'' : (k : ι'') → X'' k ⟶ S''
β : ι'' → ι'
q' : (k : ι'') → X'' k ⟶ X' (β k)
w' : ∀ (k : ι''), q' k ≫ f' (β k) = f'' k ≫ q
r : S'' ⟶ S
r' : (k : ι'') → X'' k ⟶ X (α (β k))
hr : autoParam (q ≫ p = r) pullFunctorCompIso._auto_1
hr' : autoParam (∀ (k : ι''), q' k ≫ p' (β k) = r' k) pullFunctorCompIso._auto_3
k : ι''
⊢ r' k ≫ f ((α ∘ β) k) = f'' k ≫ r

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Descent.DescentData

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
S' : C
p : S' ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S'
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j ≫ p
S'' : C
q : S'' ⟶ S'
ι'' : Type t''
X'' : ι'' → C
f'' : (k : ι'') → X'' k ⟶ S''
β : ι'' → ι'
q' : (k : ι'') → X'' k ⟶ X' (β k)
w' : ∀ (k : ι''), q' k ≫ f' (β k) = f'' k ≫ q
r : S'' ⟶ S
r' : (k : ι'') → X'' k ⟶ X (α (β k))
hr : q ≫ p = r
hr' : ∀ (k : ι''), q' k ≫ p' (β k) = r' k
D : F.DescentData f
x✝ : ι''
⊢ (p' (β x✝)).op.toLoc ≫ (q' x✝).op.toLoc = (r' x✝).op.toLoc

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : LocallySmall.{w, v, u} C
N : Type u'
inst✝² : Category.{v', u'} N
p : N ⥤ C
inst✝¹ : InitiallySmall N
J : GrothendieckTopology C
inst✝ : IsCofiltered N
⊢ (functor p).Initial

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
h : K.Generates J
F : Cᵒᵖ ⥤ Type w
H : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
inst✝ : ∀ (Z : C), _root_.Small.{max u v, w} (F.obj (Opposite.op Z))
⊢ Presieve.IsSheaf J F

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
F : Cᵒᵖ ⥤ Type w
𝒮 : (Z : C) → Set (F.obj (Opposite.op Z))
X : C
R : Presieve X
h : R ∈ K.coverings X
h' : Presieve.IsSheafFor F R
⊢ Presieve.IsSheafFor (K.subsheafify 𝒮).toFunctor R

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : LocallySmall.{w, v, u} C
N : Type u'
inst✝² : Category.{v', u'} N
p : N ⥤ C
inst✝¹ : InitiallySmall N
J : GrothendieckTopology C
inst✝ : IsCofiltered N
hp : ∀ ⦃X : C⦄, ∀ R ∈ J X, ∀ ⦃U : N⦄ (f : p.obj U ⟶ X), ∃ Y g, ∃ (_ : R.arrows g), ∃ V q a, a ≫ g = p.map q ≫ f
X : C
R : Sieve X
hR : R ∈ J X
U : N
f : p.obj U ⟶ X
Y : C
g : Y ⟶ X
hg : R.arrows g
V : N
q : V ⟶ U
a : p.obj V ⟶ Y
ha : a ≫ g = p.map q ≫ f
⊢ (hom ((ofIsCofiltered.fiber p).map g)) (fiberMk a) = fiberMk f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : LocallySmall.{w, v, u} C
N : Type u'
inst✝² : Category.{v', u'} N
p : N ⥤ C
inst✝¹ : InitiallySmall N
J : GrothendieckTopology C
inst✝ : IsCofiltered N
hp : ∀ ⦃X : C⦄, ∀ R ∈ J X, ∀ ⦃U : N⦄ (f : p.obj U ⟶ X), ∃ Y g, ∃ (_ : R.arrows g), ∃ V q a, a ≫ g = p.map q ≫ f
X : C
R : Sieve X
hR : R ∈ J X
x : (ofIsCofiltered.fiber p).obj X
⊢ ∃ Y f, ∃ (_ : R.arrows f), ∃ y, (hom ((ofIsCofiltered.fiber p).map f)) y = x

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
F : Cᵒᵖ ⥤ Type w
hF : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
𝒮 : (Z : C) → Set (F.obj (Opposite.op Z))
h : ∀ (Z : C), Small.{max u v, w} ↑(𝒮 Z)
X : C
ι : C → Type (max u v) := fun X ↦ Shrink.{max u v, w} ↑(𝒮 X)
t : (X : C) → ι X → F.obj (Opposite.op X) := fun X i ↦ ↑((equivShrink ↑(𝒮 X)).symm i)
x : F.obj (Opposite.op X)
Z✝ : C
a✝ : F.obj (Opposite.op Z✝)
ha : a✝ ∈ 𝒮 Z✝
⊢ Witness.eval hF ι t (Witness.base Z✝ ((equivShrink ↑(𝒮 Z✝)) ⟨a✝, ha⟩)) = some a✝

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : LocallySmall.{w, v, u} C
N : Type u'
inst✝⁴ : Category.{v', u'} N
p : N ⥤ C
inst✝³ : InitiallySmall N
J : GrothendieckTopology C
inst✝² : IsCofiltered N
hp : ∀ ⦃X : C⦄, ∀ R ∈ J X, ∀ ⦃U : N⦄ (f : p.obj U ⟶ X), ∃ Y g, ∃ (_ : R.arrows g), ∃ V q a, a ≫ g = p.map q ≫ f
A : Type u''
inst✝¹ : Category.{v'', u''} A
inst✝ : HasColimitsOfSize.{w, w, v'', u''} A
V U : N
f : V ⟶ U
P : Cᵒᵖ ⥤ A
⊢ P.map (p.map f).op ≫ toPresheafFiberOfIsCofiltered p hp V P = toPresheafFiberOfIsCofiltered p hp U P

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
h : K.Generates J
F : Cᵒᵖ ⥤ Type w
H : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
X : C
S : Sieve X
hS : S ∈ J X
x : Presieve.FamilyOfElements F S.arrows
t₁ t₂ : F.obj (Opposite.op X)
ht₁ : x.IsAmalgamation t₁
ht₂ : x.IsAmalgamation t₂
𝒮 : (Z : C) → Set (F.obj (Opposite.op Z)) :=
  fun Z ↦
    ((Set.range fun g ↦ x ↑g ⋯) ∪ Set.range fun g ↦ (ConcreteCategory.hom (F.map g.op)) t₁) ∪
      Set.range fun g ↦ (ConcreteCategory.hom (F.map g.op)) t₂
Q : Subfunctor F := K.subsheafify 𝒮
this : ∀ (Z : C), _root_.Small.{max u v, w} (Q.toFunctor.obj (Opposite.op Z))
hQ : Presieve.IsSheaf J Q.toFunctor
x' : Presieve.FamilyOfElements Q.toFunctor S.arrows := fun Z g hg ↦ ⟨x g hg, ⋯⟩
⊢ (fun g ↦ (ConcreteCategory.hom (F.map g.op)) t₁) (𝟙 (Opposite.unop (Opposite.op X))) = t₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : LocallySmall.{w, v, u} C
N : Type u'
inst✝⁴ : Category.{v', u'} N
p : N ⥤ C
inst✝³ : InitiallySmall N
J : GrothendieckTopology C
inst✝² : IsCofiltered N
hp : ∀ ⦃X : C⦄, ∀ R ∈ J X, ∀ ⦃U : N⦄ (f : p.obj U ⟶ X), ∃ Y g, ∃ (_ : R.arrows g), ∃ V q a, a ≫ g = p.map q ≫ f
A : Type u''
inst✝¹ : Category.{v'', u''} A
inst✝ : HasColimitsOfSize.{w, w, v'', u''} A
P Q : Cᵒᵖ ⥤ A
g : P ⟶ Q
U : N
⊢ toPresheafFiberOfIsCofiltered p hp U P ≫ (ofIsCofiltered p hp).presheafFiber.map g =
    g.app (op (p.obj U)) ≫ toPresheafFiberOfIsCofiltered p hp U Q

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
h : K.Generates J
F : Cᵒᵖ ⥤ Type w
H : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
X : C
S : Sieve X
hS : S ∈ J X
x : Presieve.FamilyOfElements F S.arrows
t₁ t₂ : F.obj (Opposite.op X)
ht₁ : x.IsAmalgamation t₁
ht₂ : x.IsAmalgamation t₂
𝒮 : (Z : C) → Set (F.obj (Opposite.op Z)) :=
  fun Z ↦
    ((Set.range fun g ↦ x ↑g ⋯) ∪ Set.range fun g ↦ (ConcreteCategory.hom (F.map g.op)) t₁) ∪
      Set.range fun g ↦ (ConcreteCategory.hom (F.map g.op)) t₂
Q : Subfunctor F := K.subsheafify 𝒮
this : ∀ (Z : C), _root_.Small.{max u v, w} (Q.toFunctor.obj (Opposite.op Z))
hQ : Presieve.IsSheaf J Q.toFunctor
x' : Presieve.FamilyOfElements Q.toFunctor S.arrows := fun Z g hg ↦ ⟨x g hg, ⋯⟩
ht₁' : t₁ ∈ Q.obj (Opposite.op X)
⊢ (fun g ↦ (ConcreteCategory.hom (F.map g.op)) t₂) (𝟙 (Opposite.unop (Opposite.op X))) = t₂

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : J.Subcanonical
K : Type u_1
inst✝¹ : Category.{v_1, u_1} K
F : K ⥤ C
inst✝ : ∀ (X : Sheaf J (Type v)), PreservesLimit F.op X.obj
c : Cocone F
hc : IsColimit c
⊢ IsColimit (J.yoneda.mapCocone c)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : J.Subcanonical
K : Type u_1
inst✝¹ : Category.{v_1, u_1} K
F : K ⥤ C
inst✝ : ∀ (X : Sheaf J (Type (max v v'))), PreservesLimit F.op X.obj
c : Cocone F
hc : IsColimit c
⊢ IsColimit ((GrothendieckTopology.uliftYoneda.{v', v, u} J).mapCocone c)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
F : Cᵒᵖ ⥤ Type w
hF : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
𝒮 : (Z : C) → Set (F.obj (Opposite.op Z))
h : ∀ (Z : C), Small.{max u v, w} ↑(𝒮 Z)
X : C
ι : C → Type (max u v) := fun X ↦ Shrink.{max u v, w} ↑(𝒮 X)
t : (X : C) → ι X → F.obj (Opposite.op X) := fun X i ↦ ↑((equivShrink ↑(𝒮 X)).symm i)
x✝ : F.obj (Opposite.op X)
Z✝ : C
R✝ : Presieve Z✝
hR : R✝ ∈ K.coverings Z✝
y✝ : Presieve.FamilyOfElements F R✝
hy : y✝.Compatible
hmem : ∀ ⦃W : C⦄ (r : W ⟶ Z✝) (hr : R✝ r), K.SubsheafClosure 𝒮 W (y✝ r hr)
t✝ : F.obj (Opposite.op Z✝)
ht : y✝.IsAmalgamation t✝
x : ⦃W : C⦄ → (r : W ⟶ Z✝) → R✝ r → Witness K ι W
hx : ∀ ⦃W : C⦄ (r : W ⟶ Z✝) (hr : R✝ r), Witness.eval hF ι t (x r hr) = some (y✝ r hr)
⊢ Witness.eval hF ι t (Witness.amalgamate hR x) = some t✝

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
inst✝¹ : J.Subcanonical
K : Type u_1
inst✝ : Category.{v_1, u_1} K
⊢ ∀ {K_1 : K ⥤ C}, PreservesLimit K_1 J.yoneda

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
inst✝¹ : J.Subcanonical
K : Type u_1
inst✝ : Category.{v_1, u_1} K
⊢ ∀ {K_1 : K ⥤ C}, PreservesLimit K_1 (GrothendieckTopology.uliftYoneda.{v', v, u} J)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
F : Cᵒᵖ ⥤ Type w
hF : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
𝒮 : (Z : C) → Set (F.obj (Opposite.op Z))
h : ∀ (Z : C), Small.{max u v, w} ↑(𝒮 Z)
X : C
ι : C → Type (max u v) := fun X ↦ Shrink.{max u v, w} ↑(𝒮 X)
t : (X : C) → ι X → F.obj (Opposite.op X) := fun X i ↦ ↑((equivShrink ↑(𝒮 X)).symm i)
x : F.obj (Opposite.op X)
hx : K.SubsheafClosure 𝒮 X x
⊢ ∃ i, Witness.eval hF ι t i = some x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
h : K.Generates J
F : Cᵒᵖ ⥤ Type w
H : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
⊢ Presieve.IsSheaf J F

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
H : K.Generates J
F : Cᵒᵖ ⥤ Type w
⊢ Presieve.IsSheaf J F ↔ ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R

Tactic:
by

Goal state after: