state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
B : C
F : Cᵒᵖ ⥤ Type (max u v)
hF : EqualizerCondition F
X : C
π : X ⟶ B
πsurj : EffectiveEpi π
inst✝¹ : regular (ofArrows (fun x => X) fun x => π)
inst✝ : hasPullbacks (ofArrows (fun x => X) fun x => π)
y : Unit → F.obj (op X)
h : Arrows.PullbackCompatible F... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have : (Presieve.singleton π).hasPullbacks := by rw [← ofArrows_pUnit]; infer_instance | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F := by
obtain ⟨X, π, hS, πsurj⟩ := Presieve.regular.single_epi (R := S)
subst hS
rw [isSheafFor_arrows_iff_pullbacks]
intro y h
| Mathlib.CategoryTheory.Sites.RegularExtensive.243_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
B : C
F : Cᵒᵖ ⥤ Type (max u v)
hF : EqualizerCondition F
X : C
π : X ⟶ B
πsurj : EffectiveEpi π
inst✝¹ : regular (ofArrows (fun x => X) fun x => π)
inst✝ : hasPullbacks (ofArrows (fun x => X) fun x => π)
y : Unit → F.obj (op X)
h : Arrows.PullbackCompatible F (fun x => π) y
⊢ hasPu... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [← ofArrows_pUnit] | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F := by
obtain ⟨X, π, hS, πsurj⟩ := Presieve.regular.single_epi (R := S)
subst hS
rw [isSheafFor_arrows_iff_pullbacks]
intro y h
have : (Presi... | Mathlib.CategoryTheory.Sites.RegularExtensive.243_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
B : C
F : Cᵒᵖ ⥤ Type (max u v)
hF : EqualizerCondition F
X : C
π : X ⟶ B
πsurj : EffectiveEpi π
inst✝¹ : regular (ofArrows (fun x => X) fun x => π)
inst✝ : hasPullbacks (ofArrows (fun x => X) fun x => π)
y : Unit → F.obj (op X)
h : Arrows.PullbackCompatible F (fun x => π) y
⊢ hasPu... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | infer_instance | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F := by
obtain ⟨X, π, hS, πsurj⟩ := Presieve.regular.single_epi (R := S)
subst hS
rw [isSheafFor_arrows_iff_pullbacks]
intro y h
have : (Presi... | Mathlib.CategoryTheory.Sites.RegularExtensive.243_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
B : C
F : Cᵒᵖ ⥤ Type (max u v)
hF : EqualizerCondition F
X : C
π : X ⟶ B
πsurj : EffectiveEpi π
inst✝¹ : regular (ofArrows (fun x => X) fun x => π)
inst✝ : hasPullbacks (ofArrows (fun x => X) fun x => π)
y : Unit → F.obj (op X)
h : Arrows.PullbackCompatible F... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have : HasPullback π π := hasPullbacks.has_pullbacks Presieve.singleton.mk Presieve.singleton.mk | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F := by
obtain ⟨X, π, hS, πsurj⟩ := Presieve.regular.single_epi (R := S)
subst hS
rw [isSheafFor_arrows_iff_pullbacks]
intro y h
have : (Presi... | Mathlib.CategoryTheory.Sites.RegularExtensive.243_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
B : C
F : Cᵒᵖ ⥤ Type (max u v)
hF : EqualizerCondition F
X : C
π : X ⟶ B
πsurj : EffectiveEpi π
inst✝¹ : regular (ofArrows (fun x => X) fun x => π)
inst✝ : hasPullbacks (ofArrows (fun x => X) fun x => π)
y : Unit → F.obj (op X)
h : Arrows.PullbackCompatible F... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | specialize hF X B π | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F := by
obtain ⟨X, π, hS, πsurj⟩ := Presieve.regular.single_epi (R := S)
subst hS
rw [isSheafFor_arrows_iff_pullbacks]
intro y h
have : (Presi... | Mathlib.CategoryTheory.Sites.RegularExtensive.243_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
B : C
F : Cᵒᵖ ⥤ Type (max u v)
X : C
π : X ⟶ B
πsurj : EffectiveEpi π
inst✝¹ : regular (ofArrows (fun x => X) fun x => π)
inst✝ : hasPullbacks (ofArrows (fun x => X) fun x => π)
y : Unit → F.obj (op X)
h : Arrows.PullbackCompatible F (fun x => π) y
this✝ : ha... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [Function.bijective_iff_existsUnique] at hF | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F := by
obtain ⟨X, π, hS, πsurj⟩ := Presieve.regular.single_epi (R := S)
subst hS
rw [isSheafFor_arrows_iff_pullbacks]
intro y h
have : (Presi... | Mathlib.CategoryTheory.Sites.RegularExtensive.243_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
B : C
F : Cᵒᵖ ⥤ Type (max u v)
X : C
π : X ⟶ B
πsurj : EffectiveEpi π
inst✝¹ : regular (ofArrows (fun x => X) fun x => π)
inst✝ : hasPullbacks (ofArrows (fun x => X) fun x => π)
y : Unit → F.obj (op X)
h : Arrows.PullbackCompatible F (fun x => π) y
this✝ : ha... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | obtain ⟨t, ht, ht'⟩ := hF ⟨y (), h () ()⟩ | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F := by
obtain ⟨X, π, hS, πsurj⟩ := Presieve.regular.single_epi (R := S)
subst hS
rw [isSheafFor_arrows_iff_pullbacks]
intro y h
have : (Presi... | Mathlib.CategoryTheory.Sites.RegularExtensive.243_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
B : C
F : Cᵒᵖ ⥤ Type (max u v)
X : C
π : X ⟶ B
πsurj : EffectiveEpi π
inst✝¹ : regular (ofArrows (fun x => X) fun x => π)
inst✝ : hasPullbacks (ofArrows (fun x => X) fun x => π)
y : Unit → F.obj (op X)
h : Arrows.PullbackCompatible F (fun x => π) ... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | refine ⟨t, fun _ ↦ ?_, fun x h ↦ ht' x ?_⟩ | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F := by
obtain ⟨X, π, hS, πsurj⟩ := Presieve.regular.single_epi (R := S)
subst hS
rw [isSheafFor_arrows_iff_pullbacks]
intro y h
have : (Presi... | Mathlib.CategoryTheory.Sites.RegularExtensive.243_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.refine_1
C : Type u
inst✝² : Category.{v, u} C
B : C
F : Cᵒᵖ ⥤ Type (max u v)
X : C
π : X ⟶ B
πsurj : EffectiveEpi π
inst✝¹ : regular (ofArrows (fun x => X) fun x => π)
inst✝ : hasPullbacks (ofArrows (fun x => X) fun x => π)
y : Unit → F.obj (op X)
h : Arrows.PullbackCompatible F (fun... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | simpa [MapToEqualizer] using ht | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F := by
obtain ⟨X, π, hS, πsurj⟩ := Presieve.regular.single_epi (R := S)
subst hS
rw [isSheafFor_arrows_iff_pullbacks]
intro y h
have : (Presi... | Mathlib.CategoryTheory.Sites.RegularExtensive.243_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
B : C
F : Cᵒᵖ ⥤ Type (max u v)
X : C
π : X ⟶ B
πsurj : EffectiveEpi π
inst✝¹ : regular (ofArrows (fun x => X) fun x => π)
inst✝ : hasPullbacks (ofArrows (fun x => X) fun x => π)
y : Unit → F.obj (op X)
h✝ : Arrows.PullbackCompatible F (fu... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | simpa [MapToEqualizer] using h () | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F := by
obtain ⟨X, π, hS, πsurj⟩ := Presieve.regular.single_epi (R := S)
subst hS
rw [isSheafFor_arrows_iff_pullbacks]
intro y h
have : (Presi... | Mathlib.CategoryTheory.Sites.RegularExtensive.243_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheafFor {B : C} {S : Presieve B} [S.regular] [S.hasPullbacks]
{F : Cᵒᵖ ⥤ Type (max u v)}
(hF : EqualizerCondition F) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
hSF : ∀ {B : C} (S : Presieve B) [inst : regular S] [inst : hasPullbacks S], IsSheafFor F S
⊢ EqualizerCondition F | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | intro X B π _ _ | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
hSF : ∀ {B : C} (S : Presieve B) [inst : regular S] [inst : hasPullbacks S], IsSheafFor F S
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
⊢ Function.Bijective (MapToEqualizer F π pullback.fst pullback.snd (_ : pullback.fst ≫ π = pullback... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩ | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
| Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
hSF : ∀ {B : C} (S : Presieve B) [inst : regular S] [inst : hasPullbacks S], IsSheafFor F S
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this : regular (ofArrows (fun x => X) fun x => π)
⊢ Function.Bijective (MapToEqualizer F π pullback... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have : (ofArrows (fun () ↦ X) (fun _ ↦ π)).hasPullbacks := ⟨
fun hf _ hg ↦ (by cases hf; cases hg; infer_instance)⟩ | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
| Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
hSF : ∀ {B : C} (S : Presieve B) [inst : regular S] [inst : hasPullbacks S], IsSheafFor F S
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this : regular (ofArrows (fun x => X) fun x => π)
Y✝ Z✝ : C
f✝ : Y✝ ⟶ B
hf : ofArrows (fun x => X) ... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | cases hf | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
have : (ofArrows (fun () ↦ X) (fun _ ↦... | Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case mk
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
hSF : ∀ {B : C} (S : Presieve B) [inst : regular S] [inst : hasPullbacks S], IsSheafFor F S
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this : regular (ofArrows (fun x => X) fun x => π)
Z✝ : C
x✝ : Z✝ ⟶ B
hg : ofArrows (fun x =... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | cases hg | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
have : (ofArrows (fun () ↦ X) (fun _ ↦... | Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case mk.mk
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
hSF : ∀ {B : C} (S : Presieve B) [inst : regular S] [inst : hasPullbacks S], IsSheafFor F S
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this : regular (ofArrows (fun x => X) fun x => π)
i✝¹ i✝ : Unit
⊢ HasPullback π π | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | infer_instance | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
have : (ofArrows (fun () ↦ X) (fun _ ↦... | Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
hSF : ∀ {B : C} (S : Presieve B) [inst : regular S] [inst : hasPullbacks S], IsSheafFor F S
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this✝ : regular (ofArrows (fun x => X) fun x => π)
this : hasPullbacks (ofArrows (fun x => X) fun x... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | specialize hSF (ofArrows (fun () ↦ X) (fun _ ↦ π)) | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
have : (ofArrows (fun () ↦ X) (fun _ ↦... | Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this✝ : regular (ofArrows (fun x => X) fun x => π)
this : hasPullbacks (ofArrows (fun x => X) fun x => π)
hSF : IsSheafFor F (ofArrows (fun x => X) fun x => π)
⊢ Function.Bijective (MapToEqua... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [isSheafFor_arrows_iff_pullbacks] at hSF | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
have : (ofArrows (fun () ↦ X) (fun _ ↦... | Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this✝ : regular (ofArrows (fun x => X) fun x => π)
this : hasPullbacks (ofArrows (fun x => X) fun x => π)
hSF : ∀ (x : Unit → F.obj (op X)), Arrows.PullbackCompatible F (fun x => π) x → ∃! t,... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [Function.bijective_iff_existsUnique] | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
have : (ofArrows (fun () ↦ X) (fun _ ↦... | Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this✝ : regular (ofArrows (fun x => X) fun x => π)
this : hasPullbacks (ofArrows (fun x => X) fun x => π)
hSF : ∀ (x : Unit → F.obj (op X)), Arrows.PullbackCompatible F (fun x => π) x → ∃! t,... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | intro ⟨x, hx⟩ | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
have : (ofArrows (fun () ↦ X) (fun _ ↦... | Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this✝ : regular (ofArrows (fun x => X) fun x => π)
this : hasPullbacks (ofArrows (fun x => X) fun x => π)
hSF : ∀ (x : Unit → F.obj (op X)), Arrows.PullbackCompatible F (fun x => π) x → ∃! t,... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | obtain ⟨t, ht, ht'⟩ := hSF (fun _ ↦ x) (fun _ _ ↦ hx) | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
have : (ofArrows (fun () ↦ X) (fun _ ↦... | Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this✝ : regular (ofArrows (fun x => X) fun x => π)
this : hasPullbacks (ofArrows (fun x => X) fun x => π)
hSF : ∀ (x : Unit → F.obj (op X)), Arrows.PullbackCompatible F (fun ... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | refine ⟨t, ?_, fun y h ↦ ht' y ?_⟩ | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
have : (ofArrows (fun () ↦ X) (fun _ ↦... | Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.refine_1
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this✝ : regular (ofArrows (fun x => X) fun x => π)
this : hasPullbacks (ofArrows (fun x => X) fun x => π)
hSF : ∀ (x : Unit → F.obj (op X)), Arrows.PullbackCompatibl... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | simpa [MapToEqualizer] using ht () | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
have : (ofArrows (fun () ↦ X) (fun _ ↦... | Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
X B : C
π : X ⟶ B
inst✝¹ : EffectiveEpi π
inst✝ : HasPullback π π
this✝ : regular (ofArrows (fun x => X) fun x => π)
this : hasPullbacks (ofArrows (fun x => X) fun x => π)
hSF : ∀ (x : Unit → F.obj (op X)), Arrows.PullbackCompatibl... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | simpa [MapToEqualizer] using h | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F := by
intro X B π _ _
have : (ofArrows (fun _ ↦ X) (fun _ ↦ π)).regular := ⟨X, π, rfl, inferInstance⟩
have : (ofArrows (fun () ↦ X) (fun _ ↦... | Mathlib.CategoryTheory.Sites.RegularExtensive.259_0.rkSRr0zuqme90Yu | lemma equalizerCondition_of_regular {F : Cᵒᵖ ⥤ Type (max u v)}
(hSF : ∀ {B : C} (S : Presieve B) [S.regular] [S.hasPullbacks], S.IsSheafFor F) :
EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
X : C
S : Presieve X
inst✝¹ : regular S
inst✝ : Projective X
F : Cᵒᵖ ⥤ Type (max u v)
⊢ IsSheafFor F S | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.single_epi (R := S) | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.275_0.rkSRr0zuqme90Yu | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
X : C
inst✝¹ : Projective X
F : Cᵒᵖ ⥤ Type (max u v)
Y : C
f : Y ⟶ X
hf : EffectiveEpi f
inst✝ : regular (ofArrows (fun x => Y) fun x => f)
⊢ IsSheafFor F (ofArrows (fun x => Y) fun x => f) | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [isSheafFor_arrows_iff] | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F := by
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.single_epi (R := S)
| Mathlib.CategoryTheory.Sites.RegularExtensive.275_0.rkSRr0zuqme90Yu | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
X : C
inst✝¹ : Projective X
F : Cᵒᵖ ⥤ Type (max u v)
Y : C
f : Y ⟶ X
hf : EffectiveEpi f
inst✝ : regular (ofArrows (fun x => Y) fun x => f)
⊢ ∀ (x : Unit → F.obj (op Y)), Arrows.Compatible F (fun x => f) x → ∃! t, ∀ (i : Unit), F.map f.op t = x i | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | refine fun x hx ↦ ⟨F.map (Projective.factorThru (𝟙 _) f).op <| x (), fun _ ↦ ?_, fun y h ↦ ?_⟩ | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F := by
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.single_epi (R := S)
rw [isSheafFor_arrows_iff]
| Mathlib.CategoryTheory.Sites.RegularExtensive.275_0.rkSRr0zuqme90Yu | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.refine_1
C : Type u
inst✝² : Category.{v, u} C
X : C
inst✝¹ : Projective X
F : Cᵒᵖ ⥤ Type (max u v)
Y : C
f : Y ⟶ X
hf : EffectiveEpi f
inst✝ : regular (ofArrows (fun x => Y) fun x => f)
x : Unit → F.obj (op Y)
hx : Arrows.Compatible F (fun x => f) x
x✝ : Unit
⊢ F.map f.op (F.map (Projective.fact... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | simpa using (hx () () Y (𝟙 Y) (f ≫ (Projective.factorThru (𝟙 _) f)) (by simp)).symm | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F := by
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.single_epi (R := S)
rw [isSheafFor_arrows_iff]
refine fun x hx ↦ ⟨F.map (Projective.factorThru (𝟙 _) f).op <| x (), fun _ ↦ ?... | Mathlib.CategoryTheory.Sites.RegularExtensive.275_0.rkSRr0zuqme90Yu | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
X : C
inst✝¹ : Projective X
F : Cᵒᵖ ⥤ Type (max u v)
Y : C
f : Y ⟶ X
hf : EffectiveEpi f
inst✝ : regular (ofArrows (fun x => Y) fun x => f)
x : Unit → F.obj (op Y)
hx : Arrows.Compatible F (fun x => f) x
x✝ : Unit
⊢ 𝟙 Y ≫ (fun x => f) () = (f ≫ Projective.factorThru (𝟙 X) f) ≫ (f... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | simp | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F := by
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.single_epi (R := S)
rw [isSheafFor_arrows_iff]
refine fun x hx ↦ ⟨F.map (Projective.factorThru (𝟙 _) f).op <| x (), fun _ ↦ ?... | Mathlib.CategoryTheory.Sites.RegularExtensive.275_0.rkSRr0zuqme90Yu | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
X : C
inst✝¹ : Projective X
F : Cᵒᵖ ⥤ Type (max u v)
Y : C
f : Y ⟶ X
hf : EffectiveEpi f
inst✝ : regular (ofArrows (fun x => Y) fun x => f)
x : Unit → F.obj (op Y)
hx : Arrows.Compatible F (fun x => f) x
y : F.obj (op X)
h : (fun t => ∀ (i : Unit), F... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | simp only [← h (), ← FunctorToTypes.map_comp_apply, ← op_comp, Projective.factorThru_comp,
op_id, FunctorToTypes.map_id_apply] | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F := by
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.single_epi (R := S)
rw [isSheafFor_arrows_iff]
refine fun x hx ↦ ⟨F.map (Projective.factorThru (𝟙 _) f).op <| x (), fun _ ↦ ?... | Mathlib.CategoryTheory.Sites.RegularExtensive.275_0.rkSRr0zuqme90Yu | lemma isSheafFor_regular_of_projective {X : C} (S : Presieve X) [S.regular] [Projective X]
(F : Cᵒᵖ ⥤ Type (max u v)) : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
inst✝¹ : ∀ ⦃X Y : C⦄ (π : X ⟶ Y) [inst : EffectiveEpi π], HasPullback π π
inst✝ : Preregular C
⊢ IsSheaf (Coverage.toGrothendieck C (regularCoverage C)) F ↔ EqualizerCondition F | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [Presieve.isSheaf_coverage] | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.284_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
inst✝¹ : ∀ ⦃X Y : C⦄ (π : X ⟶ Y) [inst : EffectiveEpi π], HasPullback π π
inst✝ : Preregular C
⊢ (∀ {X : C}, ∀ R ∈ Coverage.covering (regularCoverage C) X, IsSheafFor F R) ↔ EqualizerCondition F | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | refine ⟨fun h ↦ equalizerCondition_of_regular fun S ⟨Y, f, hh⟩ _ ↦ h S ⟨Y, f, hh⟩, ?_⟩ | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F := by
rw [Presieve.isSheaf_coverage]
| Mathlib.CategoryTheory.Sites.RegularExtensive.284_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
inst✝¹ : ∀ ⦃X Y : C⦄ (π : X ⟶ Y) [inst : EffectiveEpi π], HasPullback π π
inst✝ : Preregular C
⊢ EqualizerCondition F → ∀ {X : C}, ∀ R ∈ Coverage.covering (regularCoverage C) X, IsSheafFor F R | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rintro h X S ⟨Y, f, rfl, hf⟩ | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F := by
rw [Presieve.isSheaf_coverage]
refine ⟨fun h ↦ equalizerCondition_of_regular fun S ⟨... | Mathlib.CategoryTheory.Sites.RegularExtensive.284_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
inst✝¹ : ∀ ⦃X Y : C⦄ (π : X ⟶ Y) [inst : EffectiveEpi π], HasPullback π π
inst✝ : Preregular C
h : EqualizerCondition F
X Y : C
f : Y ⟶ X
hf : EffectiveEpi f
⊢ IsSheafFor F (ofArrows (fun x => Y) fun x => f) | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | exact @isSheafFor _ _ _ _ ⟨Y, f, rfl, hf⟩ ⟨fun g _ h ↦ by cases g; cases h; infer_instance⟩ _ h | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F := by
rw [Presieve.isSheaf_coverage]
refine ⟨fun h ↦ equalizerCondition_of_regular fun S ⟨... | Mathlib.CategoryTheory.Sites.RegularExtensive.284_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
inst✝¹ : ∀ ⦃X Y : C⦄ (π : X ⟶ Y) [inst : EffectiveEpi π], HasPullback π π
inst✝ : Preregular C
h✝ : EqualizerCondition F
X Y : C
f : Y ⟶ X
hf : EffectiveEpi f
Y✝ Z✝ : C
f✝ : Y✝ ⟶ X
g : ofArrows (fun x => Y) (fun x => f) f✝
x✝ : Z✝ ⟶ X
h : ofArrows (fun x =>... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | cases g | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F := by
rw [Presieve.isSheaf_coverage]
refine ⟨fun h ↦ equalizerCondition_of_regular fun S ⟨... | Mathlib.CategoryTheory.Sites.RegularExtensive.284_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case mk
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
inst✝¹ : ∀ ⦃X Y : C⦄ (π : X ⟶ Y) [inst : EffectiveEpi π], HasPullback π π
inst✝ : Preregular C
h✝ : EqualizerCondition F
X Y : C
f : Y ⟶ X
hf : EffectiveEpi f
Z✝ : C
x✝ : Z✝ ⟶ X
h : ofArrows (fun x => Y) (fun x => f) x✝
i✝ : Unit
⊢ HasPullback f x✝ | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | cases h | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F := by
rw [Presieve.isSheaf_coverage]
refine ⟨fun h ↦ equalizerCondition_of_regular fun S ⟨... | Mathlib.CategoryTheory.Sites.RegularExtensive.284_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case mk.mk
C : Type u
inst✝² : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max u v)
inst✝¹ : ∀ ⦃X Y : C⦄ (π : X ⟶ Y) [inst : EffectiveEpi π], HasPullback π π
inst✝ : Preregular C
h : EqualizerCondition F
X Y : C
f : Y ⟶ X
hf : EffectiveEpi f
i✝¹ i✝ : Unit
⊢ HasPullback f f | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | infer_instance | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F := by
rw [Presieve.isSheaf_coverage]
refine ⟨fun h ↦ equalizerCondition_of_regular fun S ⟨... | Mathlib.CategoryTheory.Sites.RegularExtensive.284_0.rkSRr0zuqme90Yu | lemma EqualizerCondition.isSheaf_iff (F : Cᵒᵖ ⥤ Type (max u v))
[∀ ⦃X Y : C⦄ (π : X ⟶ Y) [EffectiveEpi π], HasPullback π π] [Preregular C] :
Presieve.IsSheaf (regularCoverage C).toGrothendieck F ↔ EqualizerCondition F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W : C
⊢ IsSheaf (Coverage.toGrothendieck C (regularCoverage C)) (yoneda.obj W) | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [isSheaf_coverage] | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W : C
⊢ ∀ {X : C}, ∀ R ∈ Coverage.covering (regularCoverage C) X, IsSheafFor (yoneda.obj W) R | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | intro X S ⟨_, hS⟩ | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
| Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X : C
S : Presieve X
w✝ : C
hS : ∃ f, (S = ofArrows (fun x => w✝) fun x => f) ∧ EffectiveEpi f
⊢ IsSheafFor (yoneda.obj W) S | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have : S.regular := ⟨_, hS⟩ | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
| Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X : C
S : Presieve X
w✝ : C
hS : ∃ f, (S = ofArrows (fun x => w✝) fun x => f) ∧ EffectiveEpi f
this : regular S
⊢ IsSheafFor (yoneda.obj W) S | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.single_epi (R := S) | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
| Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x => Y) fun x => f) = ofArrows (fun x => w✝) fun x => f_1) ∧ EffectiveEpi f_1
this : regular (ofArrows (fun x => Y) fun x => f)
⊢ IsSheafFor (yoneda.obj W) (ofArrows (... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have h_colim := isColimitOfEffectiveEpiStruct f hf.effectiveEpi.some | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.singl... | Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x => Y) fun x => f) = ofArrows (fun x => w✝) fun x => f_1) ∧ EffectiveEpi f_1
this : regular (ofArrows (fun x => Y) fun x => f)
h_colim : IsColimit (cocone (Sieve.gene... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [← Sieve.generateSingleton_eq, ← Presieve.ofArrows_pUnit] at h_colim | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.singl... | Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x => Y) fun x => f) = ofArrows (fun x => w✝) fun x => f_1) ∧ EffectiveEpi f_1
this : regular (ofArrows (fun x => Y) fun x => f)
h_colim : IsColimit (cocone (Sieve.gene... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | intro x hx | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.singl... | Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x => Y) fun x => f) = ofArrows (fun x => w✝) fun x => f_1) ∧ EffectiveEpi f_1
this : regular (ofArrows (fun x => Y) fun x => f)
h_colim : IsColimit (cocone (Sieve.gene... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | let x_ext := Presieve.FamilyOfElements.sieveExtend x | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.singl... | Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x => Y) fun x => f) = ofArrows (fun x => w✝) fun x => f_1) ∧ EffectiveEpi f_1
this : regular (ofArrows (fun x => Y) fun x => f)
h_colim : IsColimit (cocone (Sieve.gene... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have hx_ext := Presieve.FamilyOfElements.Compatible.sieveExtend hx | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.singl... | Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x => Y) fun x => f) = ofArrows (fun x => w✝) fun x => f_1) ∧ EffectiveEpi f_1
this : regular (ofArrows (fun x => Y) fun x => f)
h_colim : IsColimit (cocone (Sieve.gene... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | let S := Sieve.generate (Presieve.ofArrows (fun () ↦ Y) (fun () ↦ f)) | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.singl... | Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x => Y) fun x => f) = ofArrows (fun x => w✝) fun x => f_1) ∧ EffectiveEpi f_1
this : regular (ofArrows (fun x => Y) fun x => f)
h_colim : IsColimit (cocone (Sieve.gene... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | obtain ⟨t, t_amalg, t_uniq⟩ :=
(Sieve.forallYonedaIsSheaf_iff_colimit S).mpr ⟨h_colim⟩ W x_ext hx_ext | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.singl... | Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x => Y) fun x => f) = ofArrows (fun x => w✝) fun x => f_1) ∧ EffectiveEpi f_1
this : regular (ofArrows (fun x => Y) fun x => f)
h_colim : IsColimit (cocone... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | refine ⟨t, ?_, ?_⟩ | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.singl... | Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.refine_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x => Y) fun x => f) = ofArrows (fun x => w✝) fun x => f_1) ∧ EffectiveEpi f_1
this : regular (ofArrows (fun x => Y) fun x => f)
h_colim : IsColimi... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | convert Presieve.isAmalgamation_restrict (Sieve.le_generate
(Presieve.ofArrows (fun () ↦ Y) (fun () ↦ f))) _ _ t_amalg | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.singl... | Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.e.h.e'_6.h.h.h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x => Y) fun x => f) = ofArrows (fun x => w✝) fun x => f_1) ∧ EffectiveEpi f_1
this : regular (ofArrows (fun x => Y) fun x => f)
h_colim : IsColimit (cocone (Sieve.gener... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | exact (Presieve.restrict_extend hx).symm | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.singl... | Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.refine_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x => Y) fun x => f) = ofArrows (fun x => w✝) fun x => f_1) ∧ EffectiveEpi f_1
this : regular (ofArrows (fun x => Y) fun x => f)
h_colim : IsColimi... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | exact fun y hy ↦ t_uniq y <| Presieve.isAmalgamation_sieveExtend x y hy | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) := by
rw [isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.singl... | Mathlib.CategoryTheory.Sites.RegularExtensive.297_0.rkSRr0zuqme90Yu | /-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) :
Presieve.IsSheaf (regularCoverage C).toGrothendieck (yoneda.obj W) | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
X : C
S : Presieve X
inst✝ : Presieve.extensive S
⊢ ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | obtain ⟨_, _, _, _, rfl, ⟨hc⟩⟩ := Presieve.extensive.arrows_nonempty_isColimit (R := S) | instance {X : C} (S : Presieve X) [S.extensive] : S.hasPullbacks where
has_pullbacks := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.339_0.rkSRr0zuqme90Yu | instance {X : C} (S : Presieve X) [S.extensive] : S.hasPullbacks where
has_pullbacks | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
X : C
w✝³ : Type
w✝² : Fintype w✝³
w✝¹ : w✝³ → C
w✝ : (a : w✝³) → w✝¹ a ⟶ X
inst✝ : Presieve.extensive (Presieve.ofArrows w✝¹ w✝)
hc : IsColimit (Cofan.mk X w✝)
⊢ ∀ {Y Z : C} {f : Y ⟶ X}, Presieve.ofArrows w✝¹... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | intro _ _ _ _ _ hg | instance {X : C} (S : Presieve X) [S.extensive] : S.hasPullbacks where
has_pullbacks := by
obtain ⟨_, _, _, _, rfl, ⟨hc⟩⟩ := Presieve.extensive.arrows_nonempty_isColimit (R := S)
| Mathlib.CategoryTheory.Sites.RegularExtensive.339_0.rkSRr0zuqme90Yu | instance {X : C} (S : Presieve X) [S.extensive] : S.hasPullbacks where
has_pullbacks | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
X : C
w✝³ : Type
w✝² : Fintype w✝³
w✝¹ : w✝³ → C
w✝ : (a : w✝³) → w✝¹ a ⟶ X
inst✝ : Presieve.extensive (Presieve.ofArrows w✝¹ w✝)
hc : IsColimit (Cofan.mk X w✝)
Y✝ Z✝ : C
f✝ : Y✝ ⟶ X
x✝ : Presieve.ofArrows w✝¹... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | cases hg | instance {X : C} (S : Presieve X) [S.extensive] : S.hasPullbacks where
has_pullbacks := by
obtain ⟨_, _, _, _, rfl, ⟨hc⟩⟩ := Presieve.extensive.arrows_nonempty_isColimit (R := S)
intro _ _ _ _ _ hg
| Mathlib.CategoryTheory.Sites.RegularExtensive.339_0.rkSRr0zuqme90Yu | instance {X : C} (S : Presieve X) [S.extensive] : S.hasPullbacks where
has_pullbacks | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.intro.mk
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
X : C
w✝³ : Type
w✝² : Fintype w✝³
w✝¹ : w✝³ → C
w✝ : (a : w✝³) → w✝¹ a ⟶ X
inst✝ : Presieve.extensive (Presieve.ofArrows w✝¹ w✝)
hc : IsColimit (Cofan.mk X w✝)
Y✝ : C
f✝ : Y✝ ⟶ X
x✝ : Presieve.ofArrows w✝¹... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | apply FinitaryPreExtensive.hasPullbacks_of_is_coproduct hc | instance {X : C} (S : Presieve X) [S.extensive] : S.hasPullbacks where
has_pullbacks := by
obtain ⟨_, _, _, _, rfl, ⟨hc⟩⟩ := Presieve.extensive.arrows_nonempty_isColimit (R := S)
intro _ _ _ _ _ hg
cases hg
| Mathlib.CategoryTheory.Sites.RegularExtensive.339_0.rkSRr0zuqme90Yu | instance {X : C} (S : Presieve X) [S.extensive] : S.hasPullbacks where
has_pullbacks | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : FinitaryPreExtensive C
X : C
S : Presieve X
inst✝¹ : extensive S
F : Cᵒᵖ ⥤ Type (max u v)
inst✝ : PreservesFiniteProducts F
⊢ IsSheafFor F S | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | obtain ⟨_, _, Z, π, rfl, ⟨hc⟩⟩ := extensive.arrows_nonempty_isColimit (R := S) | /--
A finite product preserving presheaf is a sheaf for the extensive topology on a category which is
`FinitaryPreExtensive`.
-/
theorem isSheafFor_extensive_of_preservesFiniteProducts {X : C} (S : Presieve X) [S.extensive]
(F : Cᵒᵖ ⥤ Type max u v) [PreservesFiniteProducts F] : S.IsSheafFor F := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.352_0.rkSRr0zuqme90Yu | /--
A finite product preserving presheaf is a sheaf for the extensive topology on a category which is
`FinitaryPreExtensive`.
-/
theorem isSheafFor_extensive_of_preservesFiniteProducts {X : C} (S : Presieve X) [S.extensive]
(F : Cᵒᵖ ⥤ Type max u v) [PreservesFiniteProducts F] : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : FinitaryPreExtensive C
X : C
F : Cᵒᵖ ⥤ Type (max u v)
inst✝¹ : PreservesFiniteProducts F
w✝¹ : Type
w✝ : Fintype w✝¹
Z : w✝¹ → C
π : (a : w✝¹) → Z a ⟶ X
inst✝ : extensive (ofArrows Z π)
hc : IsColimit (Cofan.mk X π)
⊢ IsSheafFor F (... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have : (ofArrows Z (Cofan.mk X π).inj).hasPullbacks :=
(inferInstance : (ofArrows Z π).hasPullbacks) | /--
A finite product preserving presheaf is a sheaf for the extensive topology on a category which is
`FinitaryPreExtensive`.
-/
theorem isSheafFor_extensive_of_preservesFiniteProducts {X : C} (S : Presieve X) [S.extensive]
(F : Cᵒᵖ ⥤ Type max u v) [PreservesFiniteProducts F] : S.IsSheafFor F := by
obtain ⟨_, _,... | Mathlib.CategoryTheory.Sites.RegularExtensive.352_0.rkSRr0zuqme90Yu | /--
A finite product preserving presheaf is a sheaf for the extensive topology on a category which is
`FinitaryPreExtensive`.
-/
theorem isSheafFor_extensive_of_preservesFiniteProducts {X : C} (S : Presieve X) [S.extensive]
(F : Cᵒᵖ ⥤ Type max u v) [PreservesFiniteProducts F] : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : FinitaryPreExtensive C
X : C
F : Cᵒᵖ ⥤ Type (max u v)
inst✝¹ : PreservesFiniteProducts F
w✝¹ : Type
w✝ : Fintype w✝¹
Z : w✝¹ → C
π : (a : w✝¹) → Z a ⟶ X
inst✝ : extensive (ofArrows Z π)
hc : IsColimit (Cofan.mk X π)
this : hasPullba... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | exact isSheafFor_of_preservesProduct _ _ hc | /--
A finite product preserving presheaf is a sheaf for the extensive topology on a category which is
`FinitaryPreExtensive`.
-/
theorem isSheafFor_extensive_of_preservesFiniteProducts {X : C} (S : Presieve X) [S.extensive]
(F : Cᵒᵖ ⥤ Type max u v) [PreservesFiniteProducts F] : S.IsSheafFor F := by
obtain ⟨_, _,... | Mathlib.CategoryTheory.Sites.RegularExtensive.352_0.rkSRr0zuqme90Yu | /--
A finite product preserving presheaf is a sheaf for the extensive topology on a category which is
`FinitaryPreExtensive`.
-/
theorem isSheafFor_extensive_of_preservesFiniteProducts {X : C} (S : Presieve X) [S.extensive]
(F : Cᵒᵖ ⥤ Type max u v) [PreservesFiniteProducts F] : S.IsSheafFor F | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
⊢ IsSheaf (Coverage.toGrothendieck C (extensiveCoverage C)) F ↔ Nonempty (PreservesFiniteProducts F) | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | refine ⟨fun hF ↦ ⟨⟨fun α _ ↦ ⟨fun {K} ↦ ?_⟩⟩⟩, fun hF ↦ ?_⟩ | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : IsSheaf (Coverage.toGrothendieck C (extensiveCoverage C)) F
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
⊢ PreservesLimit K F | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [Presieve.isSheaf_coverage] at hF | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
⊢ PreservesLimit K F | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | let Z : α → C := fun i ↦ unop (K.obj ⟨i⟩) | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
⊢ Preserves... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have : (Presieve.ofArrows Z (Cofan.mk (∐ Z) (Sigma.ι Z)).inj).hasPullbacks :=
(inferInstance : (Presieve.ofArrows Z (Sigma.ι Z)).hasPullbacks) | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
this : hasP... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have : ∀ (i : α), Mono (Cofan.inj (Cofan.mk (∐ Z) (Sigma.ι Z)) i) :=
(inferInstance : ∀ (i : α), Mono (Sigma.ι Z i)) | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
this✝ : has... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | let i : K ≅ Discrete.functor (fun i ↦ op (Z i)) := Discrete.natIsoFunctor | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
this✝ : has... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | let _ : PreservesLimit (Discrete.functor (fun i ↦ op (Z i))) F :=
Presieve.preservesProductOfIsSheafFor F ?_ initialIsInitial _ (coproductIsCoproduct Z)
(FinitaryExtensive.isPullback_initial_to_sigma_ι Z)
(hF (Presieve.ofArrows Z (fun i ↦ Sigma.ι Z i)) ?_) | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1.refine_3
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝¹ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
t... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | exact preservesLimitOfIsoDiagram F i.symm | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1.refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
th... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | apply hF | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1.refine_1.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | refine ⟨Empty, inferInstance, Empty.elim, IsEmpty.elim inferInstance, rfl, ⟨default,?_, ?_⟩⟩ | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1.refine_1.a.refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | ext b | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1.refine_1.a.refine_1.h
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as :=... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | cases b | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1.refine_1.a.refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | simp only [eq_iff_true_of_subsingleton] | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1.refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
th... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | refine ⟨α, inferInstance, Z, (fun i ↦ Sigma.ι Z i), rfl, ?_⟩ | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1.refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
th... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | suffices Sigma.desc (fun i ↦ Sigma.ι Z i) = 𝟙 _ by rw [this]; infer_instance | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
this✝¹ : hasPullbacks (of... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [this] | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
this✝¹ : hasPullbacks (of... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | infer_instance | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1.refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
th... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | ext | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1.refine_2.h
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R
α : Type
x✝ : Fintype α
K : Discrete α ⥤ Cᵒᵖ
Z : α → C := fun i => (K.obj { as := i }).unop
... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | simp | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : Nonempty (PreservesFiniteProducts F)
⊢ IsSheaf (Coverage.toGrothendieck C (extensiveCoverage C)) F | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | let _ := hF.some | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : Nonempty (PreservesFiniteProducts F)
x✝ : PreservesFiniteProducts F := Nonempty.some hF
⊢ IsSheaf (Coverage.toGrothendieck C (extensiveCoverage C)) F | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [Presieve.isSheaf_coverage] | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : Nonempty (PreservesFiniteProducts F)
x✝ : PreservesFiniteProducts F := Nonempty.some hF
⊢ ∀ {X : C}, ∀ R ∈ Coverage.covering (extensiveCoverage C) X, IsSheafFor F R | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | intro X R ⟨Y, α, Z, π, hR, hi⟩ | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : Nonempty (PreservesFiniteProducts F)
x✝ : PreservesFiniteProducts F := Nonempty.some hF
X : C
R : Presieve X
Y : Type
α : Fintype Y
Z : Y → C
π : (a : Y) → Z a ⟶ X
hR : R = ofArr... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have : IsIso (Sigma.desc (Cofan.inj (Cofan.mk X π))) := hi | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : Nonempty (PreservesFiniteProducts F)
x✝ : PreservesFiniteProducts F := Nonempty.some hF
X : C
R : Presieve X
Y : Type
α : Fintype Y
Z : Y → C
π : (a : Y) → Z a ⟶ X
hR : R = ofArr... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have : R.extensive := ⟨Y, α, Z, π, hR, ⟨Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X π)⟩⟩ | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type (max u v)
hF : Nonempty (PreservesFiniteProducts F)
x✝ : PreservesFiniteProducts F := Nonempty.some hF
X : C
R : Presieve X
Y : Type
α : Fintype Y
Z : Y → C
π : (a : Y) → Z a ⟶ X
hR : R = ofArr... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | exact isSheafFor_extensive_of_preservesFiniteProducts R F | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) := by
refine ⟨fu... | Mathlib.CategoryTheory.Sites.RegularExtensive.366_0.rkSRr0zuqme90Yu | /--
A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
-/
theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
Nonempty (PreservesFiniteProducts F) | Mathlib_CategoryTheory_Sites_RegularExtensive |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
p : FractionalIdeal S P
s : Set P
hs : s = ↑p
⊢ IsFractional S (Submodule.copy (↑p) s hs) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | convert p.isFractional | /-- Copy of a `FractionalIdeal` with a new underlying set equal to the old one.
Useful to fix definitional equalities. -/
protected def copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : FractionalIdeal S P :=
⟨Submodule.copy p s hs, by
| Mathlib.RingTheory.FractionalIdeal.153_0.90B1BH8AtSmfl9S | /-- Copy of a `FractionalIdeal` with a new underlying set equal to the old one.
Useful to fix definitional equalities. -/
protected def copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : FractionalIdeal S P | Mathlib_RingTheory_FractionalIdeal |
case h.e'_7
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
p : FractionalIdeal S P
s : Set P
hs : s = ↑p
⊢ Submodule.copy (↑p) s hs = ↑p | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | ext | /-- Copy of a `FractionalIdeal` with a new underlying set equal to the old one.
Useful to fix definitional equalities. -/
protected def copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : FractionalIdeal S P :=
⟨Submodule.copy p s hs, by
convert p.isFractional
| Mathlib.RingTheory.FractionalIdeal.153_0.90B1BH8AtSmfl9S | /-- Copy of a `FractionalIdeal` with a new underlying set equal to the old one.
Useful to fix definitional equalities. -/
protected def copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : FractionalIdeal S P | Mathlib_RingTheory_FractionalIdeal |
case h.e'_7.h
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
p : FractionalIdeal S P
s : Set P
hs : s = ↑p
x✝ : P
⊢ x✝ ∈ Submodule.copy (↑p) s hs ↔ x✝ ∈ ↑p | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp only [hs] | /-- Copy of a `FractionalIdeal` with a new underlying set equal to the old one.
Useful to fix definitional equalities. -/
protected def copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : FractionalIdeal S P :=
⟨Submodule.copy p s hs, by
convert p.isFractional
ext
| Mathlib.RingTheory.FractionalIdeal.153_0.90B1BH8AtSmfl9S | /-- Copy of a `FractionalIdeal` with a new underlying set equal to the old one.
Useful to fix definitional equalities. -/
protected def copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : FractionalIdeal S P | Mathlib_RingTheory_FractionalIdeal |
case h.e'_7.h
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
p : FractionalIdeal S P
s : Set P
hs : s = ↑p
x✝ : P
⊢ x✝ ∈ Submodule.copy ↑p ↑p (_ : ↑p = ↑↑p) ↔ x✝ ∈ ↑p | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rfl | /-- Copy of a `FractionalIdeal` with a new underlying set equal to the old one.
Useful to fix definitional equalities. -/
protected def copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : FractionalIdeal S P :=
⟨Submodule.copy p s hs, by
convert p.isFractional
ext
simp only [hs]
| Mathlib.RingTheory.FractionalIdeal.153_0.90B1BH8AtSmfl9S | /-- Copy of a `FractionalIdeal` with a new underlying set equal to the old one.
Useful to fix definitional equalities. -/
protected def copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : FractionalIdeal S P | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
h : I ≤ 1
⊢ IsFractional S I | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | use 1, S.one_mem | theorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I := by
| Mathlib.RingTheory.FractionalIdeal.206_0.90B1BH8AtSmfl9S | theorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I | Mathlib_RingTheory_FractionalIdeal |
case right
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
h : I ≤ 1
⊢ ∀ b ∈ I, IsInteger R (1 • b) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro b hb | theorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I := by
use 1, S.one_mem
| Mathlib.RingTheory.FractionalIdeal.206_0.90B1BH8AtSmfl9S | theorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I | Mathlib_RingTheory_FractionalIdeal |
case right
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
h : I ≤ 1
b : P
hb : b ∈ I
⊢ IsInteger R (1 • b) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | rw [one_smul] | theorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I := by
use 1, S.one_mem
intro b hb
| Mathlib.RingTheory.FractionalIdeal.206_0.90B1BH8AtSmfl9S | theorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I | Mathlib_RingTheory_FractionalIdeal |
case right
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
h : I ≤ 1
b : P
hb : b ∈ I
⊢ IsInteger R b | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | obtain ⟨b', b'_mem, rfl⟩ := h hb | theorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I := by
use 1, S.one_mem
intro b hb
rw [one_smul]
| Mathlib.RingTheory.FractionalIdeal.206_0.90B1BH8AtSmfl9S | theorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I | Mathlib_RingTheory_FractionalIdeal |
case right.intro.refl
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
h : I ≤ 1
b' : R
hb : (Algebra.linearMap R P) b' ∈ I
⊢ IsInteger R ((Algebra.linearMap R P) b') | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact Set.mem_range_self b' | theorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I := by
use 1, S.one_mem
intro b hb
rw [one_smul]
obtain ⟨b', b'_mem, rfl⟩ := h hb
| Mathlib.RingTheory.FractionalIdeal.206_0.90B1BH8AtSmfl9S | theorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
J : FractionalIdeal S P
hIJ : I ≤ ↑J
⊢ IsFractional S I | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | obtain ⟨a, a_mem, ha⟩ := J.isFractional | theorem isFractional_of_le {I : Submodule R P} {J : FractionalIdeal S P} (hIJ : I ≤ J) :
IsFractional S I := by
| Mathlib.RingTheory.FractionalIdeal.214_0.90B1BH8AtSmfl9S | theorem isFractional_of_le {I : Submodule R P} {J : FractionalIdeal S P} (hIJ : I ≤ J) :
IsFractional S I | Mathlib_RingTheory_FractionalIdeal |
case intro.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
J : FractionalIdeal S P
hIJ : I ≤ ↑J
a : R
a_mem : a ∈ S
ha : ∀ b ∈ ↑J, IsInteger R (a • b)
⊢ IsFractional S I | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | use a, a_mem | theorem isFractional_of_le {I : Submodule R P} {J : FractionalIdeal S P} (hIJ : I ≤ J) :
IsFractional S I := by
obtain ⟨a, a_mem, ha⟩ := J.isFractional
| Mathlib.RingTheory.FractionalIdeal.214_0.90B1BH8AtSmfl9S | theorem isFractional_of_le {I : Submodule R P} {J : FractionalIdeal S P} (hIJ : I ≤ J) :
IsFractional S I | Mathlib_RingTheory_FractionalIdeal |
case right
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
J : FractionalIdeal S P
hIJ : I ≤ ↑J
a : R
a_mem : a ∈ S
ha : ∀ b ∈ ↑J, IsInteger R (a • b)
⊢ ∀ b ∈ I, IsInteger R (a • b) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | intro b b_mem | theorem isFractional_of_le {I : Submodule R P} {J : FractionalIdeal S P} (hIJ : I ≤ J) :
IsFractional S I := by
obtain ⟨a, a_mem, ha⟩ := J.isFractional
use a, a_mem
| Mathlib.RingTheory.FractionalIdeal.214_0.90B1BH8AtSmfl9S | theorem isFractional_of_le {I : Submodule R P} {J : FractionalIdeal S P} (hIJ : I ≤ J) :
IsFractional S I | Mathlib_RingTheory_FractionalIdeal |
case right
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
J : FractionalIdeal S P
hIJ : I ≤ ↑J
a : R
a_mem : a ∈ S
ha : ∀ b ∈ ↑J, IsInteger R (a • b)
b : P
b_mem : b ∈ I
⊢ IsInteger R (a • b) | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | exact ha b (hIJ b_mem) | theorem isFractional_of_le {I : Submodule R P} {J : FractionalIdeal S P} (hIJ : I ≤ J) :
IsFractional S I := by
obtain ⟨a, a_mem, ha⟩ := J.isFractional
use a, a_mem
intro b b_mem
| Mathlib.RingTheory.FractionalIdeal.214_0.90B1BH8AtSmfl9S | theorem isFractional_of_le {I : Submodule R P} {J : FractionalIdeal S P} (hIJ : I ≤ J) :
IsFractional S I | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Ideal R
⊢ coeSubmodule P I ≤ 1 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simpa using coeSubmodule_mono P (le_top : I ≤ ⊤) | /-- Map an ideal `I` to a fractional ideal by forgetting `I` is integral.
This is the function that implements the coercion `Ideal R → FractionalIdeal S P`. -/
@[coe]
def coeIdeal (I : Ideal R) : FractionalIdeal S P :=
⟨coeSubmodule P I,
isFractional_of_le_one _ <| by | Mathlib.RingTheory.FractionalIdeal.222_0.90B1BH8AtSmfl9S | /-- Map an ideal `I` to a fractional ideal by forgetting `I` is integral.
This is the function that implements the coercion `Ideal R → FractionalIdeal S P`. -/
@[coe]
def coeIdeal (I : Ideal R) : FractionalIdeal S P | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
x : P
x✝ : x ∈ 0
x' : R
x'_mem_zero : x' ∈ ↑0
x'_eq_x : (Algebra.linearMap R P) x' = x
⊢ x = 0 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | have x'_eq_zero : x' = 0 := x'_mem_zero | @[simp]
theorem mem_zero_iff {x : P} : x ∈ (0 : FractionalIdeal S P) ↔ x = 0 :=
⟨fun ⟨x', x'_mem_zero, x'_eq_x⟩ => by
| Mathlib.RingTheory.FractionalIdeal.275_0.90B1BH8AtSmfl9S | @[simp]
theorem mem_zero_iff {x : P} : x ∈ (0 : FractionalIdeal S P) ↔ x = 0 | Mathlib_RingTheory_FractionalIdeal |
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
x : P
x✝ : x ∈ 0
x' : R
x'_mem_zero : x' ∈ ↑0
x'_eq_x : (Algebra.linearMap R P) x' = x
x'_eq_zero : x' = 0
⊢ x = 0 | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integer
import Mathlib.Ri... | simp [x'_eq_x.symm, x'_eq_zero] | @[simp]
theorem mem_zero_iff {x : P} : x ∈ (0 : FractionalIdeal S P) ↔ x = 0 :=
⟨fun ⟨x', x'_mem_zero, x'_eq_x⟩ => by
have x'_eq_zero : x' = 0 := x'_mem_zero
| Mathlib.RingTheory.FractionalIdeal.275_0.90B1BH8AtSmfl9S | @[simp]
theorem mem_zero_iff {x : P} : x ∈ (0 : FractionalIdeal S P) ↔ x = 0 | Mathlib_RingTheory_FractionalIdeal |
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