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 |
|---|---|---|---|---|---|---|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y : C
f : Y ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a => pullback f (π a)
⊢ ∃ T ∈ (fun B => {S | ∃ α x X π, S = Presieve.ofArrows 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.... | let π' : (a : α) → Z' a ⟶ Y := fun a ↦ pullback.fst | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y : C
f : Y ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a => pullback f (π a)
π' : (a : α) → Z' a ⟶ Y := fun a => pullback.fst
⊢ ∃ T ∈ (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 ⟨@Presieve.ofArrows C _ _ α Z' π', ⟨?_, ?_⟩⟩ | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y : C
f : Y ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a => pullback f (π a)
π' : (a : α) → Z' a ⟶ Y := fun a => pullback.fst... | /-
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.... | constructor | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_1.h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y : C
f : Y ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a => pullback f (π a)
π' : (a : α) → Z' a ⟶ Y := fun a => pullback.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 ⟨hα, Z', π', ⟨by simp only, FinitaryPreExtensive.sigma_desc_iso (fun x => π x) f h_iso⟩⟩ | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y : C
f : Y ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a => pullback f (π a)
π' : (a : α) → Z' a ⟶ Y := fun a => pullback.fst
⊢ Presieve.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.... | simp only | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y : C
f : Y ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a => pullback f (π a)
π' : (a : α) → Z' a ⟶ Y := fun a => pullback.fst... | /-
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 W g hg | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y : C
f : Y ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a => pullback f (π a)
π' : (a : α) → Z' a ⟶ Y := fun a => pullback.fst... | /-
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.... | rcases hg with ⟨a⟩ | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_2.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y✝ : C
f : Y✝ ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a => pullback f (π a)
π' : (a : α) → Z' a ⟶ Y✝ := fun a => pullba... | /-
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 ⟨Z a, pullback.snd, π a, ?_, by rw [CategoryTheory.Limits.pullback.condition]⟩ | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y✝ : C
f : Y✝ ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a => pullback f (π a)
π' : (a : α) → Z' a ⟶ Y✝ := fun a => pullback.fst
Y : C
a : ... | /-
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 [CategoryTheory.Limits.pullback.condition] | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_2.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y✝ : C
f : Y✝ ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a => pullback f (π a)
π' : (a : α) → Z' a ⟶ Y✝ := fun a => pullba... | /-
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 [hS] | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case refine_2.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y✝ : C
f : Y✝ ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a => pullback f (π a)
π' : (a : α) → Z' a ⟶ Y✝ := fun a => pullba... | /-
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.ofArrows.mk a | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
α : Type
inst✝ : Fintype α
B : C
X : α → C
π : (a : α) → X a ⟶ B
⊢ EffectiveEpi (Sigma.desc π) ↔ EffectiveEpiFamily 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.... | exact ⟨fun h ↦ ⟨⟨@effectiveEpiFamilyStructOfEffectiveEpiDesc _ _ _ _ X π _ h _ _ (fun g ↦
(FinitaryPreExtensive.sigma_desc_iso (fun a ↦ Sigma.ι X a) g inferInstance).epi_of_iso)⟩⟩,
fun _ ↦ inferInstance⟩ | theorem effectiveEpi_desc_iff_effectiveEpiFamily [FinitaryPreExtensive C] {α : Type} [Fintype α]
{B : C} (X : α → C) (π : (a : α) → X a ⟶ B) :
EffectiveEpi (Sigma.desc π) ↔ EffectiveEpiFamily X π := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.127_0.rkSRr0zuqme90Yu | theorem effectiveEpi_desc_iff_effectiveEpiFamily [FinitaryPreExtensive C] {α : Type} [Fintype α]
{B : C} (X : α → C) (π : (a : α) → X a ⟶ B) :
EffectiveEpi (Sigma.desc π) ↔ EffectiveEpiFamily X π | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
⊢ ∃ β x X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ i ι, ∀ (b : β), ι b ≫ π₁ (i b) = π₂ b ≫ 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 ⟨α, inferInstance, ?_⟩ | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
⊢ ∃ X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ i ι, ∀ (b : α), ι b ≫ π₁ (i b) = π₂ b ≫ 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.... | obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁) | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
| Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.desc π₁ = 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.... | let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a) | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
| Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.desc π₁ = 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.... | let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigma.ι X₁ a) ≫ g | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
| Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.desc π₁ = 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.... | let π' := fun a ↦ pullback.fst (f := g') (g := Sigma.ι X₁ a) | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.desc π₁ = 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.... | have _ := FinitaryPreExtensive.sigma_desc_iso (fun a ↦ Sigma.ι X₁ a) g' inferInstance | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝¹ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.desc π₁ = ... | /-
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 ⟨X₂, π₂, ?_, ?_⟩ | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝¹ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.d... | /-
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 : (Sigma.desc π' ≫ g) = Sigma.desc π₂ := by ext; simp | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝¹ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.desc π₁ = g ≫ f
X₂ : α → C := fun a => ... | /-
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 | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝¹ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.desc π₁ = g ≫ f
X₂ : α → C := 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.... | simp | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝¹ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.d... | /-
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 [← effectiveEpi_desc_iff_effectiveEpiFamily, ← this] | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝¹ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.d... | /-
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 | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝¹ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.d... | /-
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 ⟨id, fun b ↦ pullback.snd, fun b ↦ ?_⟩ | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝¹ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.d... | /-
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 [id_eq, Category.assoc, ← hg] | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝¹ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.d... | /-
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 [← Category.assoc, pullback.condition] | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝¹ : Fintype α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ fun b => X₁ b
hg : g' ≫ Sigma.d... | /-
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 | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h := by
refine ⟨α, inferInstance, ?_⟩
obtain ⟨Y, g, _, g', hg⟩ := Preregular.exists_fac f (Sigma.desc π₁)
let X₂ := fun a ↦ pullback g' (Sigma.ι X₁ a)
let π₂ := fun a ↦ pullback.fst (f := g') (g := Sigm... | Mathlib.CategoryTheory.Sites.RegularExtensive.134_0.rkSRr0zuqme90Yu | instance [FinitaryPreExtensive C] [Preregular C] : Precoherent C where
pullback {B₁ B₂} f α _ X₁ π₁ h | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
⊢ Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C) = coherentTopology C | /-
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 S | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
⊢ S ∈ GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)) B ↔
S ∈ GrothendieckTopology.sieves (coherentTopology C) B | /-
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 ↦ ?_, fun h ↦ ?_⟩ | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
| Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
h : S ∈ GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)) B
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology C) B | /-
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.... | induction h with
| of Y T hT =>
apply Coverage.saturate.of
simp only [Coverage.sup_covering, Set.mem_union] at hT
exact Or.elim hT
(fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
(fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩)
... | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
h : S ∈ GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)) B
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology C) B | /-
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.... | induction h with
| of Y T hT =>
apply Coverage.saturate.of
simp only [Coverage.sup_covering, Set.mem_union] at hT
exact Or.elim hT
(fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
(fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩)
... | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1.of
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
hT : T ∈ Coverage.covering (extensiveCoverage C ⊔ regularCoverage C) Y
⊢ Sieve.generate T ∈ GrothendieckTopology.sieves (coherentTopology C) Y | /-
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.... | | of Y T hT =>
apply Coverage.saturate.of
simp only [Coverage.sup_covering, Set.mem_union] at hT
exact Or.elim hT
(fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
(fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩) | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1.of
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
hT : T ∈ Coverage.covering (extensiveCoverage C ⊔ regularCoverage C) Y
⊢ Sieve.generate T ∈ GrothendieckTopology.sieves (coherentTopology C) Y | /-
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 Coverage.saturate.of | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1.of.hS
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
hT : T ∈ Coverage.covering (extensiveCoverage C ⊔ regularCoverage C) Y
⊢ T ∈ Coverage.covering (coherentCoverage C) Y | /-
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 [Coverage.sup_covering, Set.mem_union] at hT | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1.of.hS
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
hT : T ∈ Coverage.covering (extensiveCoverage C) Y ∨ T ∈ Coverage.covering (regularCoverage C) Y
⊢ T ∈ Coverage.covering (coherentCoverage C) Y | /-
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 Or.elim hT
(fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
(fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance, fun _ ↦ Z, fun _ ↦ f, ⟨h, inferInstance⟩⟩) | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1.top
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
X✝ : C
⊢ ⊤ ∈ GrothendieckTopology.sieves (coherentTopology C) 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.... | | top => apply Coverage.saturate.top | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1.top
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
X✝ : C
⊢ ⊤ ∈ GrothendieckTopology.sieves (coherentTopology C) 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.... | apply Coverage.saturate.top | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1.transitive
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (extensiveCoverage C ⊔ regularCoverage C) Y T
a✝ :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Coverage.saturate (extensiveCoverage C... | /-
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.... | | transitive Y T => apply Coverage.saturate.transitive Y T<;> [assumption; assumption] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1.transitive
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (extensiveCoverage C ⊔ regularCoverage C) Y T
a✝ :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Coverage.saturate (extensiveCoverage C... | /-
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 Coverage.saturate.transitive Y T<;> [assumption; assumption] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1.transitive
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (extensiveCoverage C ⊔ regularCoverage C) Y T
a✝ :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Coverage.saturate (extensiveCoverage C... | /-
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 Coverage.saturate.transitive Y T | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1.transitive.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (extensiveCoverage C ⊔ regularCoverage C) Y T
a✝ :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Coverage.saturate (extensiveCoverage... | /-
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.... | assumption | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_1.transitive.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (extensiveCoverage C ⊔ regularCoverage C) Y T
a✝ :
∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄,
T.arrows f → Coverage.saturate (extensiveCoverage... | /-
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.... | assumption | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
h : S ∈ GrothendieckTopology.sieves (coherentTopology C) B
⊢ S ∈ GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)) B | /-
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.... | induction h with
| of Y T hT =>
obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
let φ := fun (i : I) ↦ Sigma.ι X i
let F := Sigma.desc f
let Z := Sieve.generate T
let Xs := (∐ fun (i : I) => X i)
let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
ap... | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
h : S ∈ GrothendieckTopology.sieves (coherentTopology C) B
⊢ S ∈ GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)) B | /-
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.... | induction h with
| of Y T hT =>
obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
let φ := fun (i : I) ↦ Sigma.ι X i
let F := Sigma.desc f
let Z := Sieve.generate T
let Xs := (∐ fun (i : I) => X i)
let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
ap... | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
hT : T ∈ Coverage.covering (coherentCoverage C) Y
⊢ Sieve.generate T ∈ GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)... | /-
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.... | | of Y T hT =>
obtain ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT
let φ := fun (i : I) ↦ Sigma.ι X i
let F := Sigma.desc f
let Z := Sieve.generate T
let Xs := (∐ fun (i : I) => X i)
let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F))
apply Coverage.saturate... | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
hT : T ∈ Coverage.covering (coherentCoverage C) Y
⊢ Sieve.generate T ∈ GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)... | /-
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 ⟨I, hI, X, f, ⟨h, hT⟩⟩ := hT | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
⊢ Sieve.generate T ∈ Gr... | /-
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 φ := fun (i : I) ↦ Sigma.ι X i | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ 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.... | let F := Sigma.desc f | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ 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.... | let Z := Sieve.generate T | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ 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.... | let Xs := (∐ fun (i : I) => X i) | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ 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.... | let Zf := Sieve.generate (Presieve.ofArrows (fun (_ : Unit) ↦ Xs) (fun (_ : Unit) ↦ F)) | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ 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.... | apply Coverage.saturate.transitive Y Zf | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → 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.... | apply Coverage.saturate.of | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.hS
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → 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.... | simp only [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
Set.mem_setOf_eq] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.hS
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → 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.... | exact Or.inr ⟨Xs, F, ⟨rfl, inferInstance⟩⟩ | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → 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.... | intro R g hZfg | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → 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.... | dsimp at hZfg | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → 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.... | rw [Presieve.ofArrows_pUnit] at hZfg | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → 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.... | obtain ⟨W, ψ, σ, ⟨hW, hW'⟩⟩ := hZfg | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily 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.... | induction hW | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamil... | /-
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 [← hW', Sieve.pullback_comp Z] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamil... | /-
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 Sieve.pullback ψ ((Sieve.pullback F) Z) ∈ GrothendieckTopology.sieves
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck R by assumption | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := ... | /-
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.... | assumption | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamil... | /-
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 GrothendieckTopology.pullback_stable' | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFam... | /-
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 Coverage.saturate ((extensiveCoverage C) ⊔ (regularCoverage C)) Xs
(Z.pullback F) by assumption | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := ... | /-
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.... | assumption | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFam... | /-
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 : Sieve.generate (Presieve.ofArrows X φ) ≤ Z.pullback F | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFam... | /-
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 Coverage.saturate_of_superset _ this | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFam... | /-
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 Coverage.saturate.of | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a.hS
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpi... | /-
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 [Coverage.sup_covering, extensiveCoverage, regularCoverage, Set.mem_union,
Set.mem_setOf_eq] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a.hS
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpi... | /-
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 Or.inl ⟨I, hI, X, φ, ⟨rfl, ?_⟩⟩ | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a.hS
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpi... | /-
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 φ = 𝟙 _ by rw [this]; infer_instance | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := ... | /-
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] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => X b) ⟶ Y := ... | /-
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 | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a.hS
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpi... | /-
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 | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.of.intro.intro.intro.intro.intro.a.intro.intro.intro.intro.mk.a.hS.h
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveE... | /-
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 [colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app, Category.comp_id] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case this
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => 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.... | intro Q q hq | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case this
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => 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.... | simp only [Sieve.pullback_apply, Sieve.generate_apply] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case this
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => 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.... | simp only [Sieve.generate_apply] at hq | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case this
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι X i
F : (∐ fun b => 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.... | obtain ⟨E, e, r, hq⟩ := hq | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case this.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Sigma.ι 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' ⟨E, e, r ≫ F, ⟨_, _⟩⟩ | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case this.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Si... | /-
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 [h] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case this.intro.intro.intro.refine'_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Si... | /-
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.... | induction hq.1 | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case this.intro.intro.intro.refine'_1.mk
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun 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 [colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case this.intro.intro.intro.refine'_1.mk
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun 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.... | exact Presieve.ofArrows.mk _ | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case this.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Si... | /-
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 [← hq.2] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case this.intro.intro.intro.refine'_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T : Presieve Y
I : Type
hI : Fintype I
X : I → C
f : (a : I) → X a ⟶ Y
h : T = Presieve.ofArrows X f
hT : EffectiveEpiFamily X f
φ : (i : I) → X i ⟶ ∐ X := fun i => Si... | /-
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 [Category.assoc] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.top
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
X✝ : C
⊢ ⊤ ∈ GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)) 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.... | | top => apply Coverage.saturate.top | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.top
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
X✝ : C
⊢ ⊤ ∈ GrothendieckTopology.sieves (Coverage.toGrothendieck C (extensiveCoverage C ⊔ regularCoverage C)) 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.... | apply Coverage.saturate.top | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.transitive
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y T
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback 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.... | | transitive Y T => apply Coverage.saturate.transitive Y T<;> [assumption; assumption] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.transitive
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y T
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback 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.... | apply Coverage.saturate.transitive Y T<;> [assumption; assumption] | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.transitive
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y T
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback 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.... | apply Coverage.saturate.transitive Y T | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.transitive.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y T
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback 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.... | assumption | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
case h.h.h.refine_2.transitive.a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T S✝ : Sieve Y
a✝¹ : Coverage.saturate (coherentCoverage C) Y T
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → Coverage.saturate (coherentCoverage C) Y_1 (Sieve.pullback 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.... | assumption | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_... | Mathlib.CategoryTheory.Sites.RegularExtensive.151_0.rkSRr0zuqme90Yu | /-- The union of the extensive and regular coverages generates the coherent topology on `C`. -/
lemma extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max u v)
W X B : C
f : X ⟶ B
g₁ g₂ : W ⟶ X
w : g₁ ≫ f = g₂ ≫ f
t : P.obj (op B)
⊢ P.map f.op t ∈ {x | P.map g₁.op x = P.map g₂.op 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.... | simp only [Set.mem_setOf_eq, ← FunctorToTypes.map_comp_apply, ← op_comp, w] | /--
The map to the explicit equalizer used in the sheaf condition.
-/
def MapToEqualizer (P : Cᵒᵖ ⥤ Type (max u v)) {W X B : C} (f : X ⟶ B)
(g₁ g₂ : W ⟶ X) (w : g₁ ≫ f = g₂ ≫ f) :
P.obj (op B) → { x : P.obj (op X) | P.map g₁.op x = P.map g₂.op x } := fun t ↦
⟨P.map f.op t, by | Mathlib.CategoryTheory.Sites.RegularExtensive.227_0.rkSRr0zuqme90Yu | /--
The map to the explicit equalizer used in the sheaf condition.
-/
def MapToEqualizer (P : Cᵒᵖ ⥤ Type (max u v)) {W X B : C} (f : X ⟶ B)
(g₁ g₂ : W ⟶ X) (w : g₁ ≫ f = g₂ ≫ f) :
P.obj (op B) → { x : P.obj (op X) | P.map g₁.op x = P.map g₂.op x } | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
B : C
S : Presieve B
inst✝¹ : regular S
inst✝ : hasPullbacks S
F : Cᵒᵖ ⥤ Type (max u v)
hF : EqualizerCondition 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 ⟨X, π, hS, πsurj⟩ := Presieve.regular.single_epi (R := S) | 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
| 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
S : Presieve B
inst✝¹ : regular S
inst✝ : hasPullbacks S
F : Cᵒᵖ ⥤ Type (max u v)
hF : EqualizerCondition F
X : C
π : X ⟶ B
hS : S = ofArrows (fun x => X) fun x => π
πsurj : EffectiveEpi π
⊢ 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.... | subst hS | 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)
| 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)
⊢ IsSheafFor F (ofArrows (fun x => X) 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.... | rw [isSheafFor_arrows_iff_pullbacks] | 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
| 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)
⊢ ∀ (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.... | intro 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]
| 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 |
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