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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology C
inst✝ : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ 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
hf' : Sieve.ofArrows X' f' ∈ J S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Y : C
q : Y ⟶ X i
j : ι'
a : Y ⟶ X' j
fac : a ≫ f' j = q ≫ f i
⊢ Sieve.generate (sieve f f' i).arrows ∈ (J.over (X i)) (Over.mk (𝟙 (X i)))

Tactic:
simpa using

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology C
inst✝ : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ 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
hf' : Sieve.ofArrows X' f' ∈ J S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Y : C
q : Y ⟶ X i
j : ι'
a : Y ⟶ X' j
fac : a ≫ f' j = q ≫ f i
⊢ sieve f f' i ∈ (J.over (X i)) (Over.mk (𝟙 (X i)))
Module: Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : Precoverage C
inst✝² : J.toGrothendieck.Subcanonical
inst✝¹ : Limits.HasPullbacks C
inst✝ : J.IsStableUnderBaseChange
P X Y Z : C
fst : P ⟶ X
snd : P ⟶ Y
f : X ⟶ Z
g : Y ⟶ Z
𝒰 : J.ZeroHypercover X
H : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰.f i)) (pullback.fst fst (𝒰.f i) ≫ snd) (𝒰.f i ≫ f) g
h : fst ≫ f = snd ≫ g
i : (𝒰.pullbackCoverOfLeft f g).I₀
m : pullback (𝒰.f i ≫ f) g ⟶ pullback f g := pullback.map (𝒰.f i ≫ f) g f g (𝒰.f i) (𝟙 Y) (𝟙 Z) ⋯ ⋯
this : IsPullback (pullback.fst (𝒰.f i ≫ f) g) m (𝒰.f i) (pullback.fst f g)
⊢ IsPullback (pullback.fst fst (𝒰.f i))
    (pullback.lift (pullback.snd fst (𝒰.f i)) (pullback.fst fst (𝒰.f i) ≫ snd) ⋯ ≫ pullback.fst (𝒰.f i ≫ f) g)
    (pullback.lift fst snd h ≫ pullback.fst f g) (𝒰.f i)

Tactic:
simpa using

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
J : Precoverage C
inst✝² : J.toGrothendieck.Subcanonical
inst✝¹ : Limits.HasPullbacks C
inst✝ : J.IsStableUnderBaseChange
P X Y Z : C
fst : P ⟶ X
snd : P ⟶ Y
f : X ⟶ Z
g : Y ⟶ Z
𝒰 : J.ZeroHypercover X
H : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰.f i)) (pullback.fst fst (𝒰.f i) ≫ snd) (𝒰.f i ≫ f) g
h : fst ≫ f = snd ≫ g
i : (𝒰.pullbackCoverOfLeft f g).I₀
m : pullback (𝒰.f i ≫ f) g ⟶ pullback f g := pullback.map (𝒰.f i ≫ f) g f g (𝒰.f i) (𝟙 Y) (𝟙 Z) ⋯ ⋯
this : IsPullback (pullback.fst (𝒰.f i ≫ f) g) m (𝒰.f i) (pullback.fst f g)
⊢ IsPullback (pullback.fst fst (𝒰.f i)) (pullback.snd fst (𝒰.f i)) fst (𝒰.f i)
Module: Mathlib.CategoryTheory.Sites.Descent.Precoverage

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology C
inst✝ : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ 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
hf' : Sieve.ofArrows X' f' ∈ J S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Y : C
q : Y ⟶ X i
j : ι'
a : Y ⟶ X' j
fac : a ≫ f' j = q ≫ f i
s : (F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i)))) := ⋯.amalgamate (familyOfElements w φ i) ⋯
⊢ Sieve.generate (sieve f f' i).arrows ∈ (J.over (X i)) (Over.mk (𝟙 (X i)))

Tactic:
simpa using

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology C
inst✝ : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ 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
hf' : Sieve.ofArrows X' f' ∈ J S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Y : C
q : Y ⟶ X i
j : ι'
a : Y ⟶ X' j
fac : a ≫ f' j = q ≫ f i
s : (F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i)))) := ⋯.amalgamate (familyOfElements w φ i) ⋯
⊢ sieve f f' i ∈ (J.over (X i)) (Over.mk (𝟙 (X i)))
Module: Mathlib.CategoryTheory.Sites.Descent.Precoverage

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology C
inst✝ : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ 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
hf' : Sieve.ofArrows X' f' ∈ J S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Y : C
q : Y ⟶ X i
j : ι'
a : Y ⟶ X' j
fac : a ≫ f' j = q ≫ f i
s : (F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i)))) := ⋯.amalgamate (familyOfElements w φ i) ⋯
hs : (familyOfElements w φ i).IsAmalgamation s
⊢ (F.map q.op.toLoc).toFunctor.map (hom w hf' φ i) = mor w φ q a fac

Tactic:
simpa using

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
J : GrothendieckTopology C
inst✝ : F.IsPrestack J
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ 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
hf' : Sieve.ofArrows X' f' ∈ J S
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Y : C
q : Y ⟶ X i
j : ι'
a : Y ⟶ X' j
fac : a ≫ f' j = q ≫ f i
s : (F.presheafHom (D₁.obj i) (D₂.obj i)).obj (op (Over.mk (𝟙 (X i)))) := ⋯.amalgamate (familyOfElements w φ i) ⋯
hs : (familyOfElements w φ i).IsAmalgamation s
⊢ (F.map q.op.toLoc).toFunctor.map ((F.mapId { as := op (X i) }).inv.toNatTrans.app (D₁.obj i)) ≫
      (F.map q.op.toLoc).toFunctor.map (⋯.amalgamate (familyOfElements w φ i) ⋯) ≫
        (F.map q.op.toLoc).toFunctor.map ((F.mapId { as := op (X i) }).hom.toNatTrans.app (D₂.obj i)) =
    mor w φ q a fac
Module: Mathlib.CategoryTheory.Sites.SheafHom

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
φ : (Over.forget (unop (op X))).op ⋙ F ⟶ (Over.forget (unop (op X))).op ⋙ G
Y : Over (unop (op X))
⊢ ((ConcreteCategory.hom (↾(Over.map (𝟙 (op X)).unop).op.whiskerLeft)).toFun φ).app (op Y) =
    ((ConcreteCategory.hom (𝟙 ((Over.forget (unop (op X))).op ⋙ F ⟶ (Over.forget (unop (op X))).op ⋙ G))).toFun φ).app
      (op Y)

Tactic:
simpa using

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
φ : (Over.forget (unop (op X))).op ⋙ F ⟶ (Over.forget (unop (op X))).op ⋙ G
Y : Over (unop (op X))
⊢ φ.app (op ((Over.map (𝟙 X)).obj Y)) = φ.app (op Y)
Module: Mathlib.CategoryTheory.Sites.SheafHom

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X Y Z : C
f : Y ⟶ X
g : Z ⟶ Y
φ : (Over.forget (unop (op X))).op ⋙ F ⟶ (Over.forget (unop (op X))).op ⋙ G
W : Over (unop (op Z))
⊢ ((ConcreteCategory.hom (↾(Over.map (op f ≫ op g).unop).op.whiskerLeft)).toFun φ).app (op W) =
    ((ConcreteCategory.hom
              (↾(Over.map (Quiver.Hom.unop (op f))).op.whiskerLeft ≫
                ↾(Over.map (Quiver.Hom.unop (op g))).op.whiskerLeft)).toFun
          φ).app
      (op W)

Tactic:
simpa using

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X Y Z : C
f : Y ⟶ X
g : Z ⟶ Y
φ : (Over.forget (unop (op X))).op ⋙ F ⟶ (Over.forget (unop (op X))).op ⋙ G
W : Over (unop (op Z))
⊢ φ.app (op ((Over.map (g ≫ f)).obj W)) = φ.app (op ((Over.map f).obj ((Over.map g).obj W)))
Module: Mathlib.CategoryTheory.Sites.MayerVietorisSquare

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
sq : Square C
inst✝¹ : Mono sq.f₂₄
inst✝ : Mono sq.f₃₄
h₁ : sq.IsPullback
h₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄
this : Mono sq.f₁₃
F : Sheaf J (Type v)
s : PullbackCone (sq.op.map F.obj).f₂₄ (sq.op.map F.obj).f₃₄
W : C
a b : W ⟶ pairFunction sq.X₂ sq.X₃ WalkingPair.left
fac : a ≫ WalkingPair.casesOn WalkingPair.left sq.f₂₄ sq.f₃₄ = b ≫ WalkingPair.casesOn WalkingPair.left sq.f₂₄ sq.f₃₄
⊢ a = b

Tactic:
simpa using

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
sq : Square C
inst✝¹ : Mono sq.f₂₄
inst✝ : Mono sq.f₃₄
h₁ : sq.IsPullback
h₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄
this : Mono sq.f₁₃
F : Sheaf J (Type v)
s : PullbackCone (sq.op.map F.obj).f₂₄ (sq.op.map F.obj).f₃₄
W : C
a b : W ⟶ pairFunction sq.X₂ sq.X₃ WalkingPair.left
fac : a ≫ WalkingPair.casesOn WalkingPair.left sq.f₂₄ sq.f₃₄ = b ≫ WalkingPair.casesOn WalkingPair.left sq.f₂₄ sq.f₃₄
⊢ a ≫ sq.f₂₄ = b ≫ sq.f₂₄
Module: Mathlib.CategoryTheory.Sites.MayerVietorisSquare

Goal state before:
pp: case left.right
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
sq : Square C
inst✝¹ : Mono sq.f₂₄
inst✝ : Mono sq.f₃₄
h₁ : sq.IsPullback
h₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄
this : Mono sq.f₁₃
F : Sheaf J (Type v)
s : PullbackCone (sq.op.map F.obj).f₂₄ (sq.op.map F.obj).f₃₄
W : C
φ : W ⟶ sq.pullbackCone.pt
fac :
  (φ ≫ sq.pullbackCone.fst) ≫ WalkingPair.casesOn WalkingPair.left sq.f₂₄ sq.f₃₄ =
    (φ ≫ sq.pullbackCone.snd) ≫ WalkingPair.casesOn WalkingPair.right sq.f₂₄ sq.f₃₄
⊢ WalkingPair.casesOn WalkingPair.left s.fst s.snd ≫ F.obj.map (φ ≫ sq.pullbackCone.fst).op =
    WalkingPair.casesOn WalkingPair.right s.fst s.snd ≫ F.obj.map (φ ≫ sq.pullbackCone.snd).op

Tactic:
simpa using

Goal state after:
case left.right
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
sq : Square C
inst✝¹ : Mono sq.f₂₄
inst✝ : Mono sq.f₃₄
h₁ : sq.IsPullback
h₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄
this : Mono sq.f₁₃
F : Sheaf J (Type v)
s : PullbackCone (sq.op.map F.obj).f₂₄ (sq.op.map F.obj).f₃₄
W : C
φ : W ⟶ sq.pullbackCone.pt
fac :
  (φ ≫ sq.pullbackCone.fst) ≫ WalkingPair.casesOn WalkingPair.left sq.f₂₄ sq.f₃₄ =
    (φ ≫ sq.pullbackCone.snd) ≫ WalkingPair.casesOn WalkingPair.right sq.f₂₄ sq.f₃₄
⊢ s.fst ≫ F.obj.map sq.f₁₂.op ≫ F.obj.map φ.op = s.snd ≫ F.obj.map sq.f₁₃.op ≫ F.obj.map φ.op
Module: Mathlib.CategoryTheory.Sites.MayerVietorisSquare

Goal state before:
pp: case right.left
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
sq : Square C
inst✝¹ : Mono sq.f₂₄
inst✝ : Mono sq.f₃₄
h₁ : sq.IsPullback
h₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄
this : Mono sq.f₁₃
F : Sheaf J (Type v)
s : PullbackCone (sq.op.map F.obj).f₂₄ (sq.op.map F.obj).f₃₄
W : C
φ : W ⟶ sq.pullbackCone.pt
fac :
  (φ ≫ sq.pullbackCone.snd) ≫ WalkingPair.casesOn WalkingPair.right sq.f₂₄ sq.f₃₄ =
    (φ ≫ sq.pullbackCone.fst) ≫ WalkingPair.casesOn WalkingPair.left sq.f₂₄ sq.f₃₄
⊢ WalkingPair.casesOn WalkingPair.right s.fst s.snd ≫ F.obj.map (φ ≫ sq.pullbackCone.snd).op =
    WalkingPair.casesOn WalkingPair.left s.fst s.snd ≫ F.obj.map (φ ≫ sq.pullbackCone.fst).op

Tactic:
simpa using

Goal state after:
case right.left
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
sq : Square C
inst✝¹ : Mono sq.f₂₄
inst✝ : Mono sq.f₃₄
h₁ : sq.IsPullback
h₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄
this : Mono sq.f₁₃
F : Sheaf J (Type v)
s : PullbackCone (sq.op.map F.obj).f₂₄ (sq.op.map F.obj).f₃₄
W : C
φ : W ⟶ sq.pullbackCone.pt
fac :
  (φ ≫ sq.pullbackCone.snd) ≫ WalkingPair.casesOn WalkingPair.right sq.f₂₄ sq.f₃₄ =
    (φ ≫ sq.pullbackCone.fst) ≫ WalkingPair.casesOn WalkingPair.left sq.f₂₄ sq.f₃₄
⊢ s.snd ≫ F.obj.map sq.f₁₃.op ≫ F.obj.map φ.op = s.fst ≫ F.obj.map sq.f₁₂.op ≫ F.obj.map φ.op
Module: Mathlib.CategoryTheory.Sites.MayerVietorisSquare

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
sq : Square C
inst✝¹ : Mono sq.f₂₄
inst✝ : Mono sq.f₃₄
h₁ : sq.IsPullback
h₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄
this : Mono sq.f₁₃
F : Sheaf J (Type v)
s : PullbackCone (sq.op.map F.obj).f₂₄ (sq.op.map F.obj).f₃₄
W : C
a b : W ⟶ pairFunction sq.X₂ sq.X₃ WalkingPair.right
fac : a ≫ WalkingPair.casesOn WalkingPair.right sq.f₂₄ sq.f₃₄ = b ≫ WalkingPair.casesOn WalkingPair.right sq.f₂₄ sq.f₃₄
⊢ a = b

Tactic:
simpa using

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
sq : Square C
inst✝¹ : Mono sq.f₂₄
inst✝ : Mono sq.f₃₄
h₁ : sq.IsPullback
h₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄
this : Mono sq.f₁₃
F : Sheaf J (Type v)
s : PullbackCone (sq.op.map F.obj).f₂₄ (sq.op.map F.obj).f₃₄
W : C
a b : W ⟶ pairFunction sq.X₂ sq.X₃ WalkingPair.right
fac : a ≫ WalkingPair.casesOn WalkingPair.right sq.f₂₄ sq.f₃₄ = b ≫ WalkingPair.casesOn WalkingPair.right sq.f₂₄ sq.f₃₄
⊢ a ≫ sq.f₃₄ = b ≫ sq.f₃₄
Module: Mathlib.CategoryTheory.Sites.Descent.Precoverage

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
inst✝³ : HasPullbacks C
J : Precoverage C
inst✝² : J.HasIsos
inst✝¹ : J.IsStableUnderBaseChange
inst✝ : J.IsStableUnderComposition
hF : ∀ (S : C), ∀ R ∈ J.coverings S, F.IsPrestackFor R
S : C
M N : ↑(F.obj { as := op S })
X : Over S
R : Presieve X
hR : Presieve.map (Over.forget S) R ∈ J.coverings X.left
R' : Sieve X.left
hR' : (Sieve.overEquiv X).symm R' = Sieve.generate R
⊢ F.IsPrestackFor ((Sieve.overEquiv X) (Sieve.generate R)).arrows

Tactic:
simpa using

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
inst✝³ : HasPullbacks C
J : Precoverage C
inst✝² : J.HasIsos
inst✝¹ : J.IsStableUnderBaseChange
inst✝ : J.IsStableUnderComposition
hF : ∀ (S : C), ∀ R ∈ J.coverings S, F.IsPrestackFor R
S : C
M N : ↑(F.obj { as := op S })
X : Over S
R : Presieve X
hR : Presieve.map (Over.forget S) R ∈ J.coverings X.left
R' : Sieve X.left
hR' : (Sieve.overEquiv X).symm R' = Sieve.generate R
⊢ F.IsPrestackFor (Presieve.map (Over.forget S) R)
Module: Mathlib.CategoryTheory.Sites.NonabelianCohomology.H1

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
G : Cᵒᵖ ⥤ GrpCat
I : Type w'
U : I → C
γ : OneCocycle G U
i : I
T : C
a : T ⟶ U i
⊢ γ.ev i i a a = 1

Tactic:
simpa using

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
G : Cᵒᵖ ⥤ GrpCat
I : Type w'
U : I → C
γ : OneCocycle G U
i : I
T : C
a : T ⟶ U i
⊢ γ.ev i i a a = 1
Module: Mathlib.CategoryTheory.Sites.SheafHom

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y₁ Y₂ : Over X
φ : Y₂ ⟶ Y₁
Z : Over Y₂.left
hZ : (Sieve.pullback Y₂.hom S).arrows Z.hom
⊢ S.arrows ((Z.hom ≫ Over.Hom.left φ) ≫ Y₁.hom)

Tactic:
simpa using

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y₁ Y₂ : Over X
φ : Y₂ ⟶ Y₁
Z : Over Y₂.left
hZ : (Sieve.pullback Y₂.hom S).arrows Z.hom
⊢ S.arrows (Z.hom ≫ Y₂.hom)
Module: Mathlib.CategoryTheory.Sites.SheafHom

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y : C
g : Y ⟶ X
hg : S.arrows g
⊢ S.arrows (𝟙 Y ≫ g)

Tactic:
simpa using

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y : C
g : Y ⟶ X
hg : S.arrows g
⊢ S.arrows g
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
inst✝ : IsCofiltered N
U : N
X : C
f₁ f₂ : p.obj U ⟶ X
hf : fiberMk f₁ = fiberMk f₂
V : Nᵒᵖ
g : op U ⟶ V
hg :
  (hom ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).map g)) (shrinkYonedaObjObjEquiv.symm f₁) =
    (hom ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).map g)) (shrinkYonedaObjObjEquiv.symm f₂)
⊢ p.map g.unop ≫ f₁ = p.map g.unop ≫ f₂

Tactic:
simpa using

Goal state after:
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
inst✝ : IsCofiltered N
U : N
X : C
f₁ f₂ : p.obj U ⟶ X
hf : fiberMk f₁ = fiberMk f₂
V : Nᵒᵖ
g : op U ⟶ V
hg :
  (hom ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).map g)) (shrinkYonedaObjObjEquiv.symm f₁) =
    (hom ((p.op ⋙ shrinkYoneda.{w, v, u}.obj X).map g)) (shrinkYonedaObjObjEquiv.symm f₂)
⊢ p.map g.unop ≫ f₁ = p.map g.unop ≫ f₂
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:
simpa using

Goal state after:
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
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:
simpa using

Goal state after:
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))
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:
simpa using

Goal state after:
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 = ⊤
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)
this : b ≫ a = 𝟙 X
⊢ S ∈ ⊥ X

Tactic:
simpa using

Goal state after:
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)
this : b ≫ a = 𝟙 X
⊢ S.arrows (𝟙 X)
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
⊢ fiberMk φ₁ = fiberMk φ₂

Tactic:
simpa using

Goal state after:
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
⊢ fiberMk φ₁ = fiberMk φ₂
Module: Mathlib.CategoryTheory.Sites.RegularEpi

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{u_4, u_2} D
J : GrothendieckTopology C
inst✝⁴ : HasPullbacks D
inst✝³ : HasPushouts D
inst✝² : IsRegularEpiCategory D
h : ∀ {F G : Sheaf J D} (f : F ⟶ G) [Epi f], ∃ I p i, Epi p ∧ Mono i ∧ p ≫ i = f.hom
inst✝¹ : HasSheafify J D
inst✝ : Balanced (Sheaf J D)
F G : Sheaf J D
f : F ⟶ G
x✝ : Epi f
I : Cᵒᵖ ⥤ D
p : F.obj ⟶ I
i : I ⟶ G.obj
hp : Epi p
hi : Mono i
hpi : p ≫ i = f.hom
⊢ (presheafToSheaf J D).map f.hom =
    (sheafificationAdjunction J D).counit.app F ≫ f ≫ inv ((sheafificationAdjunction J D).counit.app G)

Tactic:
simpa using

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{u_4, u_2} D
J : GrothendieckTopology C
inst✝⁴ : HasPullbacks D
inst✝³ : HasPushouts D
inst✝² : IsRegularEpiCategory D
h : ∀ {F G : Sheaf J D} (f : F ⟶ G) [Epi f], ∃ I p i, Epi p ∧ Mono i ∧ p ≫ i = f.hom
inst✝¹ : HasSheafify J D
inst✝ : Balanced (Sheaf J D)
F G : Sheaf J D
f : F ⟶ G
x✝ : Epi f
I : Cᵒᵖ ⥤ D
p : F.obj ⟶ I
i : I ⟶ G.obj
hp : Epi p
hi : Mono i
hpi : p ≫ i = f.hom
⊢ (presheafToSheaf J D).map f.hom ≫ (sheafificationAdjunction J D).counit.app G =
    (sheafificationAdjunction J D).counit.app F ≫ f
Module: Mathlib.CategoryTheory.Subfunctor.Finite

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
ι : Type w'
X : ι → Cᵒᵖ
x : (i : ι) → F.obj (X i)
h : PresheafIsGeneratedBy F x
F' : Cᵒᵖ ⥤ Type w
f : F ⟶ F'
⊢ (Subfunctor.range f).IsGeneratedBy fun i ↦ (ConcreteCategory.hom (f.app (X i))) (x i)

Tactic:
simpa using

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
ι : Type w'
X : ι → Cᵒᵖ
x : (i : ι) → F.obj (X i)
h : PresheafIsGeneratedBy F x
F' : Cᵒᵖ ⥤ Type w
f : F ⟶ F'
⊢ (⊤.image f).IsGeneratedBy fun i ↦ (ConcreteCategory.hom (f.app (X i))) (x i)
Module: Mathlib.CategoryTheory.Subfunctor.Finite

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
ι : Type w'
X : ι → Cᵒᵖ
x : (i : ι) → F.obj (X i)
h : PresheafIsGeneratedBy F x
F' : Cᵒᵖ ⥤ Type w
f : F ⟶ F'
inst✝ : Epi f
⊢ PresheafIsGeneratedBy F' fun i ↦ (ConcreteCategory.hom (f.app (X i))) (x i)

Tactic:
simpa using

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
ι : Type w'
X : ι → Cᵒᵖ
x : (i : ι) → F.obj (X i)
h : PresheafIsGeneratedBy F x
F' : Cᵒᵖ ⥤ Type w
f : F ⟶ F'
inst✝ : Epi f
⊢ PresheafIsGeneratedBy F' fun i ↦ (ConcreteCategory.hom (f.app (X i))) (x i)