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

Goal state before:
pp: T : Type u
inst✝ : TopologicalSpace T
X Y : Opens T
f : Y ⟶ X
S : Presieve X
hS : S ∈ {R | ∀ x ∈ X, ∃ U f, R f ∧ x ∈ U}
x : T
hx : x ∈ Y
⊢ ∃ U f_1, Presieve.pullbackArrows f S f_1 ∧ x ∈ U

Tactic:
by

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

Goal state before:
pp: T : Type u
inst✝ : TopologicalSpace T
X : Opens T
S : Presieve X
Ti : ⦃Y : Opens T⦄ → (f : Y ⟶ X) → S f → Presieve Y
hS : S ∈ {R | ∀ x ∈ X, ∃ U f, R f ∧ x ∈ U}
hTi : ∀ ⦃Y : Opens T⦄ (f : Y ⟶ X) (H : S f), Ti f H ∈ {R | ∀ x ∈ Y, ∃ U f, R f ∧ x ∈ U}
x : T
hx : x ∈ X
⊢ ∃ U f, S.bind Ti f ∧ x ∈ U

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝⁹ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁸ : Category.{v₂, u₂} D✝
D' : Type u₃
inst✝⁷ : Category.{v₃, u₃} D'
D'' : Type u₄
inst✝⁶ : Category.{v₄, u₄} D''
J✝ : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
F✝ : C✝ ⥤ D✝
S✝ : Sheaf J✝ RingCat
R : Sheaf K✝ RingCat
inst✝⁵ : F✝.IsContinuous J✝ K✝
φ✝¹ : S✝ ⟶ (F✝.sheafPushforwardContinuous RingCat J✝ K✝).obj R
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F G H : C ⥤ D
T : Sheaf J RingCat
S : Sheaf K RingCat
inst✝² : F.IsContinuous J K
inst✝¹ : G.IsContinuous J K
inst✝ : H.IsContinuous J K
φ✝ : T ⟶ (G.sheafPushforwardContinuous RingCat J K).obj S
α : F ⟶ G
β : G ⟶ H
φ : T ⟶ (H.sheafPushforwardContinuous RingCat J K).obj S
⊢ (φ ≫ (sheafPushforwardContinuousNatTrans β RingCat J K).app S) ≫
      (sheafPushforwardContinuousNatTrans α RingCat J K).app S =
    φ ≫ (sheafPushforwardContinuousNatTrans (α ≫ β) RingCat J K).app S

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F G H : C ⥤ D
T : Sheaf J RingCat
S : Sheaf K RingCat
inst✝² : F.IsContinuous J K
inst✝¹ : G.IsContinuous J K
inst✝ : H.IsContinuous J K
α : F ⟶ G
β : G ⟶ H
φ : T ⟶ (H.sheafPushforwardContinuous RingCat J K).obj S
⊢ pushforwardNatTrans φ (α ≫ β) =
    pushforwardNatTrans φ β ≫
      pushforwardNatTrans (φ ≫ (sheafPushforwardContinuousNatTrans β RingCat J K).app S) α ≫ (pushforwardCongr ⋯).hom

Tactic:
by

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

Goal state before:
pp: T : Type u
inst✝ : TopologicalSpace T
⊢ (grothendieckTopology T).toPretopology = pretopology T

Tactic:
by

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

Goal state before:
pp: T : Type u
inst✝ : TopologicalSpace T
⊢ (pretopology T).toGrothendieck = grothendieckTopology T

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝⁹ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁸ : Category.{v₂, u₂} D✝
D' : Type u₃
inst✝⁷ : Category.{v₃, u₃} D'
D'' : Type u₄
inst✝⁶ : Category.{v₄, u₄} D''
J✝ : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
F✝ : C✝ ⥤ D✝
S✝ : Sheaf J✝ RingCat
R : Sheaf K✝ RingCat
inst✝⁵ : F✝.IsContinuous J✝ K✝
φ✝ : S✝ ⟶ (F✝.sheafPushforwardContinuous RingCat J✝ K✝).obj R
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F G H : C ⥤ D
T : Sheaf J RingCat
S : Sheaf K RingCat
inst✝² : F.IsContinuous J K
inst✝¹ : G.IsContinuous J K
inst✝ : H.IsContinuous J K
φ : T ⟶ (G.sheafPushforwardContinuous RingCat J K).obj S
α : F ≅ G
⊢ (φ ≫ (sheafPushforwardContinuousNatTrans α.hom RingCat J K).app S) ≫
      (sheafPushforwardContinuousNatTrans α.inv RingCat J K).app S =
    φ

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝⁹ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁸ : Category.{v₂, u₂} D✝
D' : Type u₃
inst✝⁷ : Category.{v₃, u₃} D'
D'' : Type u₄
inst✝⁶ : Category.{v₄, u₄} D''
J✝ : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
F✝ : C✝ ⥤ D✝
S✝ : Sheaf J✝ RingCat
R : Sheaf K✝ RingCat
inst✝⁵ : F✝.IsContinuous J✝ K✝
φ✝ : S✝ ⟶ (F✝.sheafPushforwardContinuous RingCat J✝ K✝).obj R
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F G H : C ⥤ D
T : Sheaf J RingCat
S : Sheaf K RingCat
inst✝² : F.IsContinuous J K
inst✝¹ : G.IsContinuous J K
inst✝ : H.IsContinuous J K
φ : T ⟶ (G.sheafPushforwardContinuous RingCat J K).obj S
α : F ≅ G
⊢ pushforwardNatTrans φ α.hom ≫
      pushforwardNatTrans (φ ≫ (sheafPushforwardContinuousNatTrans α.hom RingCat J K).app S) α.inv ≫
        (pushforwardCongr ⋯).hom =
    𝟙 (pushforward φ)

Tactic:
by

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

Goal state before:
pp: T : Type u
inst✝ : TopologicalSpace T
ι : Type u_1
U : ι → Opens T
⊢ (grothendieckTopology T).CoversTop U ↔ IsOpenCover U

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝⁹ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁸ : Category.{v₂, u₂} D✝
D' : Type u₃
inst✝⁷ : Category.{v₃, u₃} D'
D'' : Type u₄
inst✝⁶ : Category.{v₄, u₄} D''
J✝ : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
F✝ : C✝ ⥤ D✝
S✝ : Sheaf J✝ RingCat
R : Sheaf K✝ RingCat
inst✝⁵ : F✝.IsContinuous J✝ K✝
φ✝ : S✝ ⟶ (F✝.sheafPushforwardContinuous RingCat J✝ K✝).obj R
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F G H : C ⥤ D
T : Sheaf J RingCat
S : Sheaf K RingCat
inst✝² : F.IsContinuous J K
inst✝¹ : G.IsContinuous J K
inst✝ : H.IsContinuous J K
φ : T ⟶ (G.sheafPushforwardContinuous RingCat J K).obj S
α : F ≅ G
⊢ (pushforwardNatTrans (φ ≫ (sheafPushforwardContinuousNatTrans α.hom RingCat J K).app S) α.inv ≫
        (pushforwardCongr ⋯).hom) ≫
      pushforwardNatTrans φ α.hom =
    𝟙 (pushforward (φ ≫ (sheafPushforwardContinuousNatTrans α.hom RingCat J K).app S))

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝⁸ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁷ : Category.{v₂, u₂} D✝
D' : Type u₃
inst✝⁶ : Category.{v₃, u₃} D'
D'' : Type u₄
inst✝⁵ : Category.{v₄, u₄} D''
J✝ : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
F✝ : C✝ ⥤ D✝
S✝ : Sheaf J✝ RingCat
R✝ : Sheaf K✝ RingCat
inst✝⁴ : F✝.IsContinuous J✝ K✝
φ✝ : S✝ ⟶ (F✝.sheafPushforwardContinuous RingCat J✝ K✝).obj R✝
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F : C ⥤ D
G : D ⥤ C
S : Sheaf J RingCat
R : Sheaf K RingCat
inst✝¹ : F.IsContinuous J K
inst✝ : G.IsContinuous K J
adj : F ⊣ G
φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R
ψ : R ⟶ (G.sheafPushforwardContinuous RingCat K J).obj S
H₁ : whiskerRight (NatTrans.op adj.counit) R.obj = ψ.hom ≫ G.op.whiskerLeft φ.hom
H₂ : φ.hom ≫ F.op.whiskerLeft ψ.hom ≫ whiskerRight (NatTrans.op adj.unit) S.obj = 𝟙 S.obj
this : (G ⋙ F).IsContinuous K K := isContinuous_comp G F K J K
⊢ 𝟙 (((𝟭 D).sheafPushforwardContinuous RingCat K K).obj R) ≫
      (sheafPushforwardContinuousNatTrans adj.counit RingCat K K).app R =
    ψ ≫ (G.sheafPushforwardContinuous RingCat K J).map φ

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝⁸ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁷ : Category.{v₂, u₂} D✝
D' : Type u₃
inst✝⁶ : Category.{v₃, u₃} D'
D'' : Type u₄
inst✝⁵ : Category.{v₄, u₄} D''
J✝ : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
F✝ : C✝ ⥤ D✝
S✝ : Sheaf J✝ RingCat
R✝ : Sheaf K✝ RingCat
inst✝⁴ : F✝.IsContinuous J✝ K✝
φ✝ : S✝ ⟶ (F✝.sheafPushforwardContinuous RingCat J✝ K✝).obj R✝
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F : C ⥤ D
G : D ⥤ C
S : Sheaf J RingCat
R : Sheaf K RingCat
inst✝¹ : F.IsContinuous J K
inst✝ : G.IsContinuous K J
adj : F ⊣ G
φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R
ψ : R ⟶ (G.sheafPushforwardContinuous RingCat K J).obj S
H₁ : whiskerRight (NatTrans.op adj.counit) R.obj = ψ.hom ≫ G.op.whiskerLeft φ.hom
H₂ : φ.hom ≫ F.op.whiskerLeft ψ.hom ≫ whiskerRight (NatTrans.op adj.unit) S.obj = 𝟙 S.obj
this : (F ⋙ G).IsContinuous J J := isContinuous_comp F G J K J
⊢ (φ ≫ (F.sheafPushforwardContinuous RingCat J K).map ψ) ≫
      (sheafPushforwardContinuousNatTrans adj.unit RingCat J J).app S =
    𝟙 S

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Sheaf

Goal state before:
pp: X : TopCat
F : Presheaf (Discrete Unit) X
x✝² : Discrete Unit
U : Opens ↑X
S : Sieve U
x✝¹ : S ∈ (Opens.grothendieckTopology ↑X) U
x : Presieve.FamilyOfElements (F ⋙ coyoneda.obj (op x✝²)) S.arrows
x✝ : x.Compatible
⊢ (fun t ↦ x.IsAmalgamation t) (eqToHom ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Sheaf

Goal state before:
pp: X : TopCat
F : Presheaf (Discrete Unit) X
x✝³ : Discrete Unit
U : Opens ↑X
S : Sieve U
x✝² : S ∈ (Opens.grothendieckTopology ↑X) U
x : Presieve.FamilyOfElements (F ⋙ coyoneda.obj (op x✝³)) S.arrows
x✝¹ : x.Compatible
x✝ : (F ⋙ coyoneda.obj (op x✝³)).obj (op U)
⊢ (fun t ↦ x.IsAmalgamation t) x✝ → x✝ = eqToHom ⋯

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝⁸ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁷ : Category.{v₂, u₂} D✝
D' : Type u₃
inst✝⁶ : Category.{v₃, u₃} D'
D'' : Type u₄
inst✝⁵ : Category.{v₄, u₄} D''
J✝ : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
F✝ : C✝ ⥤ D✝
S✝ : Sheaf J✝ RingCat
R✝ : Sheaf K✝ RingCat
inst✝⁴ : F✝.IsContinuous J✝ K✝
φ✝ : S✝ ⟶ (F✝.sheafPushforwardContinuous RingCat J✝ K✝).obj R✝
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F : C ⥤ D
G : D ⥤ C
S : Sheaf J RingCat
R : Sheaf K RingCat
inst✝¹ : F.IsContinuous J K
inst✝ : G.IsContinuous K J
adj : F ⊣ G
φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R
ψ : R ⟶ (G.sheafPushforwardContinuous RingCat K J).obj S
H₁ : whiskerRight (NatTrans.op adj.counit) R.obj = ψ.hom ≫ G.op.whiskerLeft φ.hom
H₂ : φ.hom ≫ F.op.whiskerLeft ψ.hom ≫ whiskerRight (NatTrans.op adj.unit) S.obj = 𝟙 S.obj
X : SheafOfModules R
⊢ (pushforward φ).map
        (((pushforwardId R).inv ≫
              pushforwardNatTrans (𝟙 (((𝟭 D).sheafPushforwardContinuous RingCat K K).obj R)) adj.counit ≫
                (pushforwardCongr ⋯).hom ≫ (pushforwardComp ψ φ).inv).app
          X) ≫
      ((pushforwardComp φ ψ).hom ≫
            pushforwardNatTrans (φ ≫ (F.sheafPushforwardContinuous RingCat J K).map ψ) adj.unit ≫
              (pushforwardCongr ⋯).hom ≫ (pushforwardId S).hom).app
        ((pushforward φ).obj X) =
    𝟙 ((pushforward φ).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : TopCat
P Q : Presheaf C X
f g : P ⟶ Q
w : ∀ (U : Opens ↑X), f.app (op U) = g.app (op U)
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
X : Type u_1
inst✝ : CompleteLattice X
v : ℕ → X
w x y z : X
e : v 0 = v 1
x✝¹ : v 1 = v 2
h₀ : v 1 ≤ x
x✝ : x ≤ z ⊓ w
h₂ : x ≤ y ⊓ z
⊢ v 0 ≤ y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: X : TopCat
C : Type u
inst✝² : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
F : Presheaf C X
U V W : Opens ↑X
e₁ : U ≤ V
e₂ : V ≤ W
x : ToType (F.obj (op W))
⊢ ((x |_ V) e₂ |_ U) e₁ = (x |_ U) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: X : TopCat
C : Type u
inst✝² : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
F G : Presheaf C X
e : F ⟶ G
U V : Opens ↑X
h : U ≤ V
x : ToType (F.obj (op V))
⊢ (ConcreteCategory.hom (e.app (op U))) ((x |_ U) h) = ((ConcreteCategory.hom (e.app (op V))) x |_ U) h

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: X : TopCat
C : Type u
inst✝² : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
F : Presheaf C X
U : Opens ↑X
x : ToType (F.obj (op U))
⊢ (x |_ U) ⋯ = x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : TopCat
f g : X ⟶ Y
h : f = g
ℱ : Presheaf C X
⊢ (pushforward C f).obj ℱ = (pushforward C g).obj ℱ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C✝ : Type u
inst✝³ : Category.{v, u} C✝
X✝ : TopCat
C : Type u
inst✝² : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
X Y : TopCat
f g : X ⟶ Y
h : f = g
ℱ : Presheaf C X
U : (Opens ↑Y)ᵒᵖ
⊢ (Opens.map f).op.obj U = (Opens.map g).op.obj U

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝⁸ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁷ : Category.{v₂, u₂} D✝
D' : Type u₃
inst✝⁶ : Category.{v₃, u₃} D'
D'' : Type u₄
inst✝⁵ : Category.{v₄, u₄} D''
J✝ : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
F✝ : C✝ ⥤ D✝
S✝ : Sheaf J✝ RingCat
R✝ : Sheaf K✝ RingCat
inst✝⁴ : F✝.IsContinuous J✝ K✝
φ✝ : S✝ ⟶ (F✝.sheafPushforwardContinuous RingCat J✝ K✝).obj R✝
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F : C ⥤ D
G : D ⥤ C
S : Sheaf J RingCat
R : Sheaf K RingCat
inst✝¹ : F.IsContinuous J K
inst✝ : G.IsContinuous K J
adj : F ⊣ G
φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R
ψ : R ⟶ (G.sheafPushforwardContinuous RingCat K J).obj S
H₁ : whiskerRight (NatTrans.op adj.counit) R.obj = ψ.hom ≫ G.op.whiskerLeft φ.hom
H₂ : φ.hom ≫ F.op.whiskerLeft ψ.hom ≫ whiskerRight (NatTrans.op adj.unit) S.obj = 𝟙 S.obj
X : SheafOfModules S
⊢ ((pushforwardId R).inv ≫
            pushforwardNatTrans (𝟙 (((𝟭 D).sheafPushforwardContinuous RingCat K K).obj R)) adj.counit ≫
              (pushforwardCongr ⋯).hom ≫ (pushforwardComp ψ φ).inv).app
        ((pushforward ψ).obj X) ≫
      (pushforward ψ).map
        (((pushforwardComp φ ψ).hom ≫
              pushforwardNatTrans (φ ≫ (F.sheafPushforwardContinuous RingCat J K).map ψ) adj.unit ≫
                (pushforwardCongr ⋯).hom ≫ (pushforwardId S).hom).app
          X) =
    𝟙 ((pushforward ψ).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : TopCat
f g : X ⟶ Y
h : f = g
ℱ : Presheaf C X
U : (Opens ↑Y)ᵒᵖ
⊢ (pushforwardEq h ℱ).hom.app U = ℱ.map (eqToHom ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C✝ : Type u
inst✝³ : Category.{v, u} C✝
X✝ : TopCat
C : Type u
inst✝² : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
X Y : TopCat
H₁ : X ≅ Y
ℱ : Presheaf C X
𝒢 : Presheaf C Y
H₂ : (pushforward C H₁.hom).obj ℱ ⟶ 𝒢
U : (Opens ↑X)ᵒᵖ
⊢ U = (Opens.map H₁.hom).op.obj (op ((Opens.map H₁.inv).obj (unop U)))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : TopCat
H₁ : X ≅ Y
ℱ : Presheaf C X
𝒢 : Presheaf C Y
H₂ : (pushforward C H₁.hom).obj ℱ ⟶ 𝒢
U : (Opens ↑X)ᵒᵖ
⊢ (toPushforwardOfIso H₁ H₂).app U = ℱ.map (eqToHom ⋯) ≫ H₂.app (op ((Opens.map H₁.inv).obj (unop U)))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C✝ : Type u
inst✝³ : Category.{v, u} C✝
X✝ : TopCat
C : Type u
inst✝² : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
X Y : TopCat
H₁ : X ≅ Y
ℱ : Presheaf C Y
𝒢 : Presheaf C X
H₂ : ℱ ⟶ (pushforward C H₁.hom).obj 𝒢
U : (Opens ↑X)ᵒᵖ
⊢ (Opens.map H₁.hom).op.obj (op ((Opens.map H₁.inv).obj (unop U))) = U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : TopCat
H₁ : X ≅ Y
ℱ : Presheaf C Y
𝒢 : Presheaf C X
H₂ : ℱ ⟶ (pushforward C H₁.hom).obj 𝒢
U : (Opens ↑X)ᵒᵖ
⊢ (pushforwardToOfIso H₁ H₂).app U = H₂.app (op ((Opens.map H₁.inv).obj (unop U))) ≫ 𝒢.map (eqToHom ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Sheaf

Goal state before:
pp: X : TopCat
A : Type u_1
inst✝⁵ : Category.{u, u_1} A
FC : A → A → Type u_2
CC : A → Type u
inst✝⁴ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory A FC
inst✝² : HasLimits A
inst✝¹ : PreservesLimits (forget A)
inst✝ : (forget A).ReflectsIsomorphisms
F : Presheaf A X
hF : F.IsSheaf
U : (Opens ↑X)ᵒᵖ
s t : ToType (F.obj U)
hst :
  ∀ x ∈ unop U,
    ∃ V,
      ∃ (hV : V ≤ unop U),
        x ∈ V ∧ (ConcreteCategory.hom (F.map (homOfLE hV).op)) s = (ConcreteCategory.hom (F.map (homOfLE hV).op)) t
⊢ s = t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C✝ : Type u
inst✝⁴ : Category.{v, u} C✝
X✝ : TopCat
C : Type u
inst✝³ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝¹ : ConcreteCategory C FC
inst✝ : HasColimits C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U : Opens ↑X
H : IsOpen (⇑(ConcreteCategory.hom f) '' ↑U)
x : CostructuredArrow (Opens.map f).op (op U) := CostructuredArrow.mk (homOfLE ⋯).op
s : Cone (empty (CostructuredArrow (Opens.map f).op (op U)))
⊢ s.pt ⟶ (asEmptyCone x).pt

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C✝ : Type u
inst✝⁴ : Category.{v, u} C✝
X✝ : TopCat
C : Type u
inst✝³ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝¹ : ConcreteCategory C FC
inst✝ : HasColimits C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U : Opens ↑X
H : IsOpen (⇑(ConcreteCategory.hom f) '' ↑U)
⊢ ((pullback C f).obj ℱ).obj (op U) ≅ ℱ.obj (op { carrier := ⇑(ConcreteCategory.hom f) '' ↑U, is_open' := H })

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U V : Opens ↑X
HU : IsOpen (⇑(ConcreteCategory.hom f) '' ↑U)
HV : IsOpen (⇑(ConcreteCategory.hom f) '' ↑V)
le : U ≤ V
j : CostructuredArrow (Opens.map f).op (op V)
⊢ ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
              (op V)).ι.app
        j ≫
      ((pullback C f).obj ℱ).map (homOfLE le).op =
    ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
            (op U)).ι.app
      ((CostructuredArrow.map (homOfLE le).op).obj j)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U V : Opens ↑X
HU : IsOpen (⇑(ConcreteCategory.hom f) '' ↑U)
HV : IsOpen (⇑(ConcreteCategory.hom f) '' ↑V)
le : U ≤ V
⊢ ((pullback C f).obj ℱ).map (homOfLE le).op ≫ (pullbackObjObjOfImageOpen f ℱ U HU).hom =
    (pullbackObjObjOfImageOpen f ℱ V HV).hom ≫ ℱ.map (IsOpenMap.functorMap HU HV le).op

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Limits.HasColimits C
X Y : TopCat
f : X ⟶ Y
hf : IsOpenMap ⇑(ConcreteCategory.hom f)
ℱ 𝒢 : TopCat.Presheaf C Y
u : ℱ ⟶ 𝒢
U : Opens ↑X
j : CostructuredArrow (Opens.map f).op (op U)
⊢ ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
              (op U)).ι.app
        j ≫
      ((pullback C f).map u).app (op U) =
    ((CostructuredArrow.proj (Opens.map f).op (op U)).whiskerLeft u).app j ≫
      ((LeftExtension.mk ((Opens.map f).op.leftKanExtension 𝒢) ((Opens.map f).op.leftKanExtensionUnit 𝒢)).coconeAt
              (op U)).ι.app
        j

Tactic:
by

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

Goal state before:
pp: C✝ : Type u₁
inst✝¹¹ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝¹⁰ : Category.{v₂, u₂} D✝
D' : Type u₃
inst✝⁹ : Category.{v₃, u₃} D'
D'' : Type u₄
inst✝⁸ : Category.{v₄, u₄} D''
J✝ : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
F✝ : C✝ ⥤ D✝
S✝ : Sheaf J✝ RingCat
R✝ : Sheaf K✝ RingCat
inst✝⁷ : F✝.IsContinuous J✝ K✝
φ✝ : S✝ ⟶ (F✝.sheafPushforwardContinuous RingCat J✝ K✝).obj R✝
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F : C ⥤ D
G : D ⥤ C
S : Sheaf J RingCat
R : Sheaf K RingCat
inst✝⁴ : F.IsContinuous J K
inst✝³ : G.IsContinuous K J
adj : F ⊣ G
φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R
ψ : R ⟶ (G.sheafPushforwardContinuous RingCat K J).obj S
H₁ : whiskerRight (NatTrans.op adj.counit) R.obj = ψ.hom ≫ G.op.whiskerLeft φ.hom
H₂ : φ.hom ≫ F.op.whiskerLeft ψ.hom ≫ whiskerRight (NatTrans.op adj.unit) S.obj = 𝟙 S.obj
inst✝² : F.IsCocontinuous J K
inst✝¹ : IsIso φ
inst✝ : F.IsLeftAdjoint
⊢ (pushforward φ).IsLeftAdjoint

Tactic:
by

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

Goal state before:
pp: C✝¹ : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} C✝¹
D✝ : Type u₂
inst✝⁹ : Category.{v₂, u₂} D✝
D' : Type u₃
inst✝⁸ : Category.{v₃, u₃} D'
D'' : Type u₄
inst✝⁷ : Category.{v₄, u₄} D''
J✝¹ : GrothendieckTopology C✝¹
K✝ : GrothendieckTopology D✝
F✝ : C✝¹ ⥤ D✝
S✝ : Sheaf J✝¹ RingCat
R✝¹ : Sheaf K✝ RingCat
inst✝⁶ : F✝.IsContinuous J✝¹ K✝
φ✝ : S✝ ⟶ (F✝.sheafPushforwardContinuous RingCat J✝¹ K✝).obj R✝¹
C✝ : Type u₁
inst✝⁵ : Category.{v₁, u₁} C✝
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
J✝ : GrothendieckTopology C✝
K : GrothendieckTopology D
F : C✝ ⥤ D
G : D ⥤ C✝
S : Sheaf J✝ RingCat
R✝ : Sheaf K RingCat
inst✝³ : F.IsContinuous J✝ K
inst✝² : G.IsContinuous K J✝
adj : F ⊣ G
φ : S ⟶ (F.sheafPushforwardContinuous RingCat J✝ K).obj R✝
ψ : R✝ ⟶ (G.sheafPushforwardContinuous RingCat K J✝).obj S
H₁ : whiskerRight (NatTrans.op adj.counit) R✝.obj = ψ.hom ≫ G.op.whiskerLeft φ.hom
H₂ : φ.hom ≫ F.op.whiskerLeft ψ.hom ≫ whiskerRight (NatTrans.op adj.unit) S.obj = 𝟙 S.obj
C : Type u'
inst✝¹ : Category.{v', u'} C
inst✝ : HasBinaryProducts C
J : GrothendieckTopology C
R : Sheaf J RingCat
x : C
U V : Cᵒᵖ
f : U ⟶ V
⊢ R.obj.map f ≫ R.obj.map prod.snd.op =
    R.obj.map prod.snd.op ≫ (((Over.star x).sheafPushforwardContinuous RingCat J (J.over x)).obj (R.over x)).obj.map f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Limits.HasColimits C
X Y : TopCat
f : X ⟶ Y
hf : IsOpenMap ⇑(ConcreteCategory.hom f)
ℱ 𝒢 : TopCat.Presheaf C Y
u : ℱ ⟶ 𝒢
⊢ (pullback C f).map u ≫ (hf.pullbackObjIso 𝒢).hom = (hf.pullbackObjIso ℱ).hom ≫ hf.functor.op.whiskerLeft u

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Category.Pairwise

Goal state before:
pp: ι : Type v
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i j : ι
⊢ ∀ (x : single i ⟶ single j), x ∈ if h : i = j then {eqToHom ⋯} else ∅

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Category.Pairwise

Goal state before:
pp: ι : Type v
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i j k : ι
⊢ ∀ (x : single i ⟶ pair j k), x ∈ ∅

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Category.Pairwise

Goal state before:
pp: ι : Type v
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i j k : ι
⊢ ∀ (x : pair i j ⟶ single k),
    x ∈ (if h : i = k then {left i j ≫ eqToHom ⋯} else ∅) ∪ if h : j = k then {right i j ≫ eqToHom ⋯} else ∅

Tactic:
by

Goal state after: