state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case a.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ ofGrothendieck C J}
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (toGrothendieck C K) X
⊢ S ∈ GrothendieckTopology.sieves J X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | induction hS with
| of X S hS => apply hJ; assumption
| top => apply J.top_mem
| transitive X R S _ _ H1 H2 => exact J.transitive H1 _ H2 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ ofGrothendieck C J}
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (toGrothendieck C K) X
⊢ S ∈ GrothendieckTopology.sieves J X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | induction hS with
| of X S hS => apply hJ; assumption
| top => apply J.top_mem
| transitive X R S _ _ H1 H2 => exact J.transitive H1 _ H2 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a.of
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ ofGrothendieck C J}
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ Sieve.generate S ∈ GrothendieckTopology.sieves J X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | of X S hS => apply hJ; assumption | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a.of
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ ofGrothendieck C J}
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ Sieve.generate S ∈ GrothendieckTopology.sieves J X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply hJ | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a.of.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ ofGrothendieck C J}
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ S ∈ covering K X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | assumption | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a.top
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ ofGrothendieck C J}
X : C
S : Sieve X
X✝ : C
⊢ ⊤ ∈ GrothendieckTopology.sieves J X✝ | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | top => apply J.top_mem | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a.top
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ ofGrothendieck C J}
X : C
S : Sieve X
X✝ : C
⊢ ⊤ ∈ GrothendieckTopology.sieves J X✝ | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply J.top_mem | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a.transitive
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ ofGrothendieck C J}
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : R ∈ GrothendieckTopology.sieves J X... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | transitive X R S _ _ H1 H2 => exact J.transitive H1 _ H2 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a.transitive
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ ofGrothendieck C J}
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : R ∈ GrothendieckTopology.sieves J X... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact J.transitive H1 _ H2 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
⊢ sInf {J | K ≤ ofGrothendieck C J} ≤ toGrothendieck C K | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply sInf_le | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
⊢ toGrothendieck C K ∈ {J | K ≤ ofGrothendieck C J} | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro X S hS | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ S ∈ covering (ofGrothendieck C (toGrothendieck C K)) X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply saturate.of _ _ hS | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.32430
inst✝ : Category.{?u.32434, ?u.32430} C
x y : Coverage C
⊢ ∀ ⦃X Y : C⦄ (f : Y ⟶ X),
∀ S ∈ (fun B => covering x B ∪ covering y B) X,
∃ T ∈ (fun B => covering x B ∪ covering y B) Y, Presieve.FactorsThruAlong T S f | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rintro X Y f S (hx | hy) | instance : SemilatticeSup (Coverage C) where
sup x y :=
{ covering := fun B ↦ x.covering B ∪ y.covering B
pullback := by
| Mathlib.CategoryTheory.Sites.Coverage.288_0.qkZFgqEgDC2P633 | instance : SemilatticeSup (Coverage C) where
sup x y | Mathlib_CategoryTheory_Sites_Coverage |
case inl
C : Type ?u.32430
inst✝ : Category.{?u.32434, ?u.32430} C
x y : Coverage C
X Y : C
f : Y ⟶ X
S : Presieve X
hx : S ∈ covering x X
⊢ ∃ T ∈ (fun B => covering x B ∪ covering y B) Y, Presieve.FactorsThruAlong T S f | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | obtain ⟨T, hT⟩ := x.pullback f S hx | instance : SemilatticeSup (Coverage C) where
sup x y :=
{ covering := fun B ↦ x.covering B ∪ y.covering B
pullback := by
rintro X Y f S (hx | hy)
· | Mathlib.CategoryTheory.Sites.Coverage.288_0.qkZFgqEgDC2P633 | instance : SemilatticeSup (Coverage C) where
sup x y | Mathlib_CategoryTheory_Sites_Coverage |
case inl.intro
C : Type ?u.32430
inst✝ : Category.{?u.32434, ?u.32430} C
x y : Coverage C
X Y : C
f : Y ⟶ X
S : Presieve X
hx : S ∈ covering x X
T : Presieve Y
hT : T ∈ covering x Y ∧ Presieve.FactorsThruAlong T S f
⊢ ∃ T ∈ (fun B => covering x B ∪ covering y B) Y, Presieve.FactorsThruAlong T S f | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact ⟨T, Or.inl hT.1, hT.2⟩ | instance : SemilatticeSup (Coverage C) where
sup x y :=
{ covering := fun B ↦ x.covering B ∪ y.covering B
pullback := by
rintro X Y f S (hx | hy)
· obtain ⟨T, hT⟩ := x.pullback f S hx
| Mathlib.CategoryTheory.Sites.Coverage.288_0.qkZFgqEgDC2P633 | instance : SemilatticeSup (Coverage C) where
sup x y | Mathlib_CategoryTheory_Sites_Coverage |
case inr
C : Type ?u.32430
inst✝ : Category.{?u.32434, ?u.32430} C
x y : Coverage C
X Y : C
f : Y ⟶ X
S : Presieve X
hy : S ∈ covering y X
⊢ ∃ T ∈ (fun B => covering x B ∪ covering y B) Y, Presieve.FactorsThruAlong T S f | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | obtain ⟨T, hT⟩ := y.pullback f S hy | instance : SemilatticeSup (Coverage C) where
sup x y :=
{ covering := fun B ↦ x.covering B ∪ y.covering B
pullback := by
rintro X Y f S (hx | hy)
· obtain ⟨T, hT⟩ := x.pullback f S hx
exact ⟨T, Or.inl hT.1, hT.2⟩
· | Mathlib.CategoryTheory.Sites.Coverage.288_0.qkZFgqEgDC2P633 | instance : SemilatticeSup (Coverage C) where
sup x y | Mathlib_CategoryTheory_Sites_Coverage |
case inr.intro
C : Type ?u.32430
inst✝ : Category.{?u.32434, ?u.32430} C
x y : Coverage C
X Y : C
f : Y ⟶ X
S : Presieve X
hy : S ∈ covering y X
T : Presieve Y
hT : T ∈ covering y Y ∧ Presieve.FactorsThruAlong T S f
⊢ ∃ T ∈ (fun B => covering x B ∪ covering y B) Y, Presieve.FactorsThruAlong T S f | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact ⟨T, Or.inr hT.1, hT.2⟩ | instance : SemilatticeSup (Coverage C) where
sup x y :=
{ covering := fun B ↦ x.covering B ∪ y.covering B
pullback := by
rintro X Y f S (hx | hy)
· obtain ⟨T, hT⟩ := x.pullback f S hx
exact ⟨T, Or.inl hT.1, hT.2⟩
· obtain ⟨T, hT⟩ := y.pullback f S hy
| Mathlib.CategoryTheory.Sites.Coverage.288_0.qkZFgqEgDC2P633 | instance : SemilatticeSup (Coverage C) where
sup x y | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
⊢ IsSheaf (toGrothendieck C K) P ↔ ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | constructor | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mp
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
⊢ IsSheaf (toGrothendieck C K) P → ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro H X R hR | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mp
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : IsSheaf (toGrothendieck C K) P
X : C
R : Presieve X
hR : R ∈ covering K X
⊢ IsSheafFor P R | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [Presieve.isSheafFor_iff_generate] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mp
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : IsSheaf (toGrothendieck C K) P
X : C
R : Presieve X
hR : R ∈ covering K X
⊢ IsSheafFor P (Sieve.generate R).arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply H _ <| saturate.of _ _ hR | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
⊢ (∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R) → IsSheaf (toGrothendieck C K) P | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro H X S hS | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (toGrothendieck C K) X
⊢ IsSheafFor P S.arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | suffices ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f).arrows by
simpa using this (f := 𝟙 _) | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (toGrothendieck C K) X
this : ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor P (Sieve.pullback f S).arrows
⊢ IsSheafFor P S.arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simpa using this (f := 𝟙 _) | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (toGrothendieck C K) X
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor P (Sieve.pullback f S).arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | induction hS with
| of X S hS =>
intro Y f
obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
apply Presieve.isSheafFor_of_factorsThru (S := T)
· intro Z g hg
obtain ⟨W, i, e, h1, h2⟩ := hT2 hg
exact ⟨Z, 𝟙 _, g, ⟨W, i, e, h1, h2⟩, by simp⟩
· apply H; assumption
· intro Z ... | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (toGrothendieck C K) X
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor P (Sieve.pullback f S).arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | induction hS with
| of X S hS =>
intro Y f
obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
apply Presieve.isSheafFor_of_factorsThru (S := T)
· intro Z g hg
obtain ⟨W, i, e, h1, h2⟩ := hT2 hg
exact ⟨Z, 𝟙 _, g, ⟨W, i, e, h1, h2⟩, by simp⟩
· apply H; assumption
· intro Z ... | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor P (Sieve.pullback f (Sieve.generate S)).arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | of X S hS =>
intro Y f
obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS
apply Presieve.isSheafFor_of_factorsThru (S := T)
· intro Z g hg
obtain ⟨W, i, e, h1, h2⟩ := hT2 hg
exact ⟨Z, 𝟙 _, g, ⟨W, i, e, h1, h2⟩, by simp⟩
· apply H; assumption
· intro Z g _
obtain ⟨R,... | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor P (Sieve.pullback f (Sieve.generate S)).arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro Y f | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
⊢ IsSheafFor P (Sieve.pullback f (Sieve.generate S)).arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | obtain ⟨T, hT1, hT2⟩ := K.pullback f S hS | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of.intro.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
⊢ IsSheafFor P (Sieve.pu... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply Presieve.isSheafFor_of_factorsThru (S := T) | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of.intro.intro.H
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
⊢ FactorsThru T (Sieve... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro Z g hg | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of.intro.intro.H
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
Z : C
g : Z ⟶ Y
hg : T... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | obtain ⟨W, i, e, h1, h2⟩ := hT2 hg | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of.intro.intro.H.intro.intro.intro.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S ... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact ⟨Z, 𝟙 _, g, ⟨W, i, e, h1, h2⟩, by simp⟩ | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
Z : C
g : Z ⟶ Y
hg : T g
W : C
i : Z ⟶ W
e : W ⟶... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of.intro.intro.hS
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
⊢ IsSheafFor P T | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply H | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of.intro.intro.hS.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
⊢ T ∈ covering K Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | assumption | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of.intro.intro.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
⊢ ∀ ⦃Y_1 : C⦄ ⦃f_1 : Y... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro Z g _ | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of.intro.intro.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
Z : C
g : Z ⟶ Y
a✝ : (... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | obtain ⟨R, hR1, hR2⟩ := K.pullback g _ hT1 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of.intro.intro.h.intro.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
T : Presieve Y
hT1 : T ∈ covering K Y
hT2 : FactorsThruAlong T S f
Z : C
g : ... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine ⟨R, (H _ hR1).isSeparatedFor, hR2⟩ | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.top
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X : C
S : Sieve X
X✝ : C
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X✝), IsSheafFor P (Sieve.pullback f ⊤).arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | top => intros; simpa using Presieve.isSheafFor_top_sieve _ | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.top
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X : C
S : Sieve X
X✝ : C
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X✝), IsSheafFor P (Sieve.pullback f ⊤).arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intros | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.top
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X : C
S : Sieve X
X✝ Y✝ : C
f✝ : Y✝ ⟶ X✝
⊢ IsSheafFor P (Sieve.pullback f✝ ⊤).arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simpa using Presieve.isSheafFor_top_sieve _ | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | transitive X R S _ _ H1 H2 =>
intro Y f
simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
choose H1 H1' using H1
choose H2 H2' using H2
refine ⟨?_, fun x hx => ?_⟩
· intro x t₁ t₂ h₁ h₂
refine (H1 f).ext (fun Z g hg => ?_)
refine (H... | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro Y f | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
H : ∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), IsSheafFor... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at * | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | choose H1 H1' using H1 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | choose H2 H2' using H2 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine ⟨?_, fun x hx => ?_⟩ | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_1
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro x t₁ t₂ h₁ h₂ | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_1
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine (H1 f).ext (fun Z g hg => ?_) | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_1
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine (H2 hg (𝟙 _)).ext (fun ZZ gg hgg => ?_) | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_1
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp only [Sieve.pullback_id, Sieve.pullback_apply] at hgg | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_1
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp only [← types_comp_apply] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_1
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [← P.map_comp, ← op_comp, h₁, h₂] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_1.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simpa only [Sieve.pullback_apply, Category.assoc] using hgg | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | let y : ∀ ⦃Z : C⦄ (g : Z ⟶ Y),
((S.pullback (g ≫ f)).pullback (𝟙 _)).arrows.FamilyOfElements P :=
fun Z g ZZ gg hgg => x (gg ≫ g) (by simpa using hgg) | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : FamilyOfE... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simpa using hgg | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | have hy : ∀ ⦃Z : C⦄ (g : Z ⟶ Y), (y g).Compatible := by
intro Z g Y₁ Y₂ ZZ g₁ g₂ f₁ f₂ h₁ h₂ h
rw [hx]
rw [reassoc_of% h] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : FamilyOfE... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro Z g Y₁ Y₂ ZZ g₁ g₂ f₁ f₂ h₁ h₂ h | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : FamilyOfE... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [hx] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : Fa... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [reassoc_of% h] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | choose z hz using fun ⦃Z : C⦄ ⦃g : Z ⟶ Y⦄ (hg : R.pullback f g) =>
H2' hg (𝟙 _) (y g) (hy g) | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | let q : (R.pullback f).arrows.FamilyOfElements P := fun Z g hg => z hg | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | have hq : q.Compatible := by
intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
apply (H2 h₁ g₁).ext
intro ZZ gg hgg
simp only [← types_comp_apply]
rw [← P.map_comp, ← P.map_comp, ← op_comp, ← op_comp, hz, hz]
· dsimp; congr 1; simp only [Category.assoc, h]
· simpa [reassoc_of% h]... | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : FamilyOfE... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : FamilyOfE... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply (H2 h₁ g₁).ext | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : FamilyOfE... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro ZZ gg hgg | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : FamilyOfE... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp only [← types_comp_apply] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : FamilyOfE... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [← P.map_comp, ← P.map_comp, ← op_comp, ← op_comp, hz, hz] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : FamilyOfE... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | dsimp | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : FamilyOfE... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | congr 1 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case e_f
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : ... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp only [Category.assoc, h] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : Fa... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simpa [reassoc_of% h] using hgg | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P R ∧
∀ (x : Fa... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simpa using hgg | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparatedFor P... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | obtain ⟨t, ht⟩ := H1' f q hq | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparate... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine ⟨t, fun Z g hg => ?_⟩ | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparate... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine (H1 (g ≫ f)).ext (fun ZZ gg hgg => ?_) | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparate... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [← types_comp_apply _ (P.map gg.op), ← P.map_comp, ← op_comp, ht] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparate... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | swap | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSepara... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simpa using hgg | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparate... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine (H2 hgg (𝟙 _)).ext (fun ZZZ ggg hggg => ?_) | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparate... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [← types_comp_apply _ (P.map ggg.op), ← P.map_comp, ← op_comp, hz] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparate... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | swap | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSepara... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simpa using hggg | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparate... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine (H2 hgg ggg).ext (fun ZZZZ gggg _ => ?_) | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparate... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [← types_comp_apply _ (P.map gggg.op), ← P.map_comp, ← op_comp] | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSeparate... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply hx | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive.refine_2.intro.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
P : Cᵒᵖ ⥤ Type w
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ covering K X,
IsSepara... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib.CategoryTheory.Sites.Coverage.313_0.qkZFgqEgDC2P633 | /--
The main theorem of this file: Given a coverage `K` on `C`,
a `Type*`-valued presheaf on `C` is a sheaf for `K` if and only if it is a sheaf for
the associated Grothendieck topology.
-/
theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : P... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K L : Coverage C
P : Cᵒᵖ ⥤ Type w
⊢ IsSheaf (toGrothendieck C (K ⊔ L)) P ↔ IsSheaf (toGrothendieck C K) P ∧ IsSheaf (toGrothendieck C L) P | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine ⟨fun h ↦ ⟨Presieve.isSheaf_of_le _ ((gi C).gc.monotone_l le_sup_left) h,
Presieve.isSheaf_of_le _ ((gi C).gc.monotone_l le_sup_right) h⟩, fun h ↦ ?_⟩ | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib.CategoryTheory.Sites.Coverage.388_0.qkZFgqEgDC2P633 | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K L : Coverage C
P : Cᵒᵖ ⥤ Type w
h : IsSheaf (toGrothendieck C K) P ∧ IsSheaf (toGrothendieck C L) P
⊢ IsSheaf (toGrothendieck C (K ⊔ L)) P | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [isSheaf_coverage, isSheaf_coverage] at h | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib.CategoryTheory.Sites.Coverage.388_0.qkZFgqEgDC2P633 | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K L : Coverage C
P : Cᵒᵖ ⥤ Type w
h : (∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R) ∧ ∀ {X : C}, ∀ R ∈ covering L X, IsSheafFor P R
⊢ IsSheaf (toGrothendieck C (K ⊔ L)) P | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [isSheaf_coverage] | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib.CategoryTheory.Sites.Coverage.388_0.qkZFgqEgDC2P633 | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K L : Coverage C
P : Cᵒᵖ ⥤ Type w
h : (∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R) ∧ ∀ {X : C}, ∀ R ∈ covering L X, IsSheafFor P R
⊢ ∀ {X : C}, ∀ R ∈ covering (K ⊔ L) X, IsSheafFor P R | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro X R hR | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib.CategoryTheory.Sites.Coverage.388_0.qkZFgqEgDC2P633 | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K L : Coverage C
P : Cᵒᵖ ⥤ Type w
h : (∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R) ∧ ∀ {X : C}, ∀ R ∈ covering L X, IsSheafFor P R
X : C
R : Presieve X
hR : R ∈ covering (K ⊔ L) X
⊢ IsSheafFor P R | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | cases' hR with hR hR | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib.CategoryTheory.Sites.Coverage.388_0.qkZFgqEgDC2P633 | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib_CategoryTheory_Sites_Coverage |
case inl
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K L : Coverage C
P : Cᵒᵖ ⥤ Type w
h : (∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R) ∧ ∀ {X : C}, ∀ R ∈ covering L X, IsSheafFor P R
X : C
R : Presieve X
hR : R ∈ covering K X
⊢ IsSheafFor P R | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact h.1 R hR | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib.CategoryTheory.Sites.Coverage.388_0.qkZFgqEgDC2P633 | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib_CategoryTheory_Sites_Coverage |
case inr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K L : Coverage C
P : Cᵒᵖ ⥤ Type w
h : (∀ {X : C}, ∀ R ∈ covering K X, IsSheafFor P R) ∧ ∀ {X : C}, ∀ R ∈ covering L X, IsSheafFor P R
X : C
R : Presieve X
hR : R ∈ covering L X
⊢ IsSheafFor P R | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact h.2 R hR | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib.CategoryTheory.Sites.Coverage.388_0.qkZFgqEgDC2P633 | /--
A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
-/
theorem isSheaf_sup (K L : Coverage C) (P : Cᵒᵖ ⥤ Type w) :
(Presieve.IsSheaf ((K ⊔ L).toGrothendieck C)) P ↔
(Presieve.IsSheaf (K.t... | Mathlib_CategoryTheory_Sites_Coverage |
f : ℕ →. ℕ
hf : Partrec f
⊢ Partrec₂ fun a m =>
Part.map (fun x => x + m) (Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a (n + m))) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | refine'
Partrec.map
((@Partrec₂.unpaired' fun a b : ℕ =>
Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + b))).1
_)
(Primrec.nat_add.comp Primrec.snd <| Primrec.snd.comp Primrec.fst).to_comp.to₂ | theorem rfind' {f} (hf : Nat.Partrec f) :
Nat.Partrec
(Nat.unpaired fun a m =>
(Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + m))).map (· + m)) :=
Partrec₂.unpaired'.2 <| by
| Mathlib.Computability.PartrecCode.50_0.A3c3Aev6SyIRjCJ | theorem rfind' {f} (hf : Nat.Partrec f) :
Nat.Partrec
(Nat.unpaired fun a m =>
(Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + m))).map (· + m)) | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
hf : Partrec f
⊢ Partrec (unpaired fun a b => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a (n + b))) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have : Nat.Partrec (fun a => Nat.rfind (fun n => (fun m => decide (m = 0)) <$>
Nat.unpaired (fun a b => f (Nat.pair (Nat.unpair a).1 (b + (Nat.unpair a).2)))
(Nat.pair a n))) :=
rfind
(Partrec₂.unpaired'.2
((Partrec.nat_iff.2 hf).comp
(Primrec₂.pair.comp (Primrec.fst.... | theorem rfind' {f} (hf : Nat.Partrec f) :
Nat.Partrec
(Nat.unpaired fun a m =>
(Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + m))).map (· + m)) :=
Partrec₂.unpaired'.2 <| by
refine'
Partrec.map
((@Partrec₂.unpaired' fun a b : ℕ =>
Nat.rfind fun n => (fun ... | Mathlib.Computability.PartrecCode.50_0.A3c3Aev6SyIRjCJ | theorem rfind' {f} (hf : Nat.Partrec f) :
Nat.Partrec
(Nat.unpaired fun a m =>
(Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + m))).map (· + m)) | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
hf : Partrec f
this :
Partrec fun a =>
Nat.rfind fun n =>
(fun m => decide (m = 0)) <$> unpaired (fun a b => f (Nat.pair (unpair a).1 (b + (unpair a).2))) (Nat.pair a n)
⊢ Partrec (unpaired fun a b => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a (n + b))) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp at this | theorem rfind' {f} (hf : Nat.Partrec f) :
Nat.Partrec
(Nat.unpaired fun a m =>
(Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + m))).map (· + m)) :=
Partrec₂.unpaired'.2 <| by
refine'
Partrec.map
((@Partrec₂.unpaired' fun a b : ℕ =>
Nat.rfind fun n => (fun ... | Mathlib.Computability.PartrecCode.50_0.A3c3Aev6SyIRjCJ | theorem rfind' {f} (hf : Nat.Partrec f) :
Nat.Partrec
(Nat.unpaired fun a m =>
(Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + m))).map (· + m)) | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
hf : Partrec f
this :
Partrec fun a => Nat.rfind fun n => Part.map (fun m => decide (m = 0)) (f (Nat.pair (unpair a).1 (n + (unpair a).2)))
⊢ Partrec (unpaired fun a b => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a (n + b))) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | exact this | theorem rfind' {f} (hf : Nat.Partrec f) :
Nat.Partrec
(Nat.unpaired fun a m =>
(Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + m))).map (· + m)) :=
Partrec₂.unpaired'.2 <| by
refine'
Partrec.map
((@Partrec₂.unpaired' fun a b : ℕ =>
Nat.rfind fun n => (fun ... | Mathlib.Computability.PartrecCode.50_0.A3c3Aev6SyIRjCJ | theorem rfind' {f} (hf : Nat.Partrec f) :
Nat.Partrec
(Nat.unpaired fun a m =>
(Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + m))).map (· + m)) | Mathlib_Computability_PartrecCode |
x✝ : Code.const 0 = Code.const 0
⊢ 0 = 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp | theorem const_inj : ∀ {n₁ n₂}, Nat.Partrec.Code.const n₁ = Nat.Partrec.Code.const n₂ → n₁ = n₂
| 0, 0, _ => by | Mathlib.Computability.PartrecCode.108_0.A3c3Aev6SyIRjCJ | theorem const_inj : ∀ {n₁ n₂}, Nat.Partrec.Code.const n₁ = Nat.Partrec.Code.const n₂ → n₁ = n₂
| 0, 0, _ => by simp
| n₁ + 1, n₂ + 1, h => by
dsimp [Nat.add_one, Nat.Partrec.Code.const] at h
injection h with h₁ h₂
simp only [const_inj h₂] | Mathlib_Computability_PartrecCode |
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