statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
fork : Fork.{v} (leftRes F U) (rightRes F U) | Fork.ofι _ (w F U) | def | TopCat.Presheaf.SheafConditionEqualizerProducts.fork | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | The restriction map `F.obj U ⟶ Π F.obj (U i)` gives a cone over the equalizer diagram
for the sheaf condition. The sheaf condition asserts this cone is a limit cone. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
fork_pt : (fork F U).pt = F.obj (op (iSup U)) | rfl | theorem | TopCat.Presheaf.SheafConditionEqualizerProducts.fork_pt | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [
"iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
fork_ι : (fork F U).ι = res F U | rfl | theorem | TopCat.Presheaf.SheafConditionEqualizerProducts.fork_ι | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
fork_π_app_walkingParallelPair_zero : (fork F U).π.app WalkingParallelPair.zero = res F U | rfl | theorem | TopCat.Presheaf.SheafConditionEqualizerProducts.fork_π_app_walkingParallelPair_zero | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
fork_π_app_walkingParallelPair_one :
(fork F U).π.app WalkingParallelPair.one = res F U ≫ leftRes F U | rfl | theorem | TopCat.Presheaf.SheafConditionEqualizerProducts.fork_π_app_walkingParallelPair_one | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
piOpens.isoOfIso (α : F ≅ G) : piOpens F U ≅ piOpens.{v'} G U | Pi.mapIso fun _ => α.app _ | def | TopCat.Presheaf.SheafConditionEqualizerProducts.piOpens.isoOfIso | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | Isomorphic presheaves have isomorphic `piOpens` for any cover `U`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
piInters.isoOfIso (α : F ≅ G) : piInters F U ≅ piInters.{v'} G U | Pi.mapIso fun _ => α.app _ | def | TopCat.Presheaf.SheafConditionEqualizerProducts.piInters.isoOfIso | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | Isomorphic presheaves have isomorphic `piInters` for any cover `U`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
diagram.isoOfIso (α : F ≅ G) : diagram F U ≅ diagram.{v'} G U | NatIso.ofComponents (by
rintro ⟨⟩
· exact piOpens.isoOfIso U α
· exact piInters.isoOfIso U α)
(by
rintro ⟨⟩ ⟨⟩ ⟨⟩
· simp
· dsimp
refine Pi.hom_ext _ _ fun b ↦ ?_
simp [leftRes]
· dsimp [diagram]
refine Pi.hom_ext _ _ fun b ↦ ?_
simp [rightRes]
... | def | TopCat.Presheaf.SheafConditionEqualizerProducts.diagram.isoOfIso | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | Isomorphic presheaves have isomorphic sheaf condition diagrams. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
fork.isoOfIso (α : F ≅ G) :
fork F U ≅ (Cone.postcompose (diagram.isoOfIso U α).inv).obj (fork G U) | by
fapply Fork.ext
· apply α.app
· dsimp
refine Pi.hom_ext _ _ fun b ↦ ?_
dsimp only [Fork.ι]
simp [res, diagram.isoOfIso] | def | TopCat.Presheaf.SheafConditionEqualizerProducts.fork.isoOfIso | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | If `F G : Presheaf C X` are isomorphic presheaves,
then the `fork F U`, the canonical cone of the sheaf condition diagram for `F`,
is isomorphic to `fork F G` postcomposed with the corresponding isomorphism between
sheaf condition diagrams. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSheafEqualizerProducts (F : Presheaf.{v', v, u} C X) : Prop | ∀ ⦃ι : Type v'⦄ (U : ι → Opens X), Nonempty (IsLimit (SheafConditionEqualizerProducts.fork F U)) | def | TopCat.Presheaf.IsSheafEqualizerProducts | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | The sheaf condition for a `F : Presheaf C X` requires that the morphism
`F.obj U ⟶ ∏ᶜ F.obj (U i)` (where `U` is some open set which is the union of the `U i`)
is the equalizer of the two morphisms
`∏ᶜ F.obj (U i) ⟶ ∏ᶜ F.obj (U i) ⊓ (U j)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coneEquivFunctorObj (c : Cone ((diagram U).op ⋙ F)) :
Cone (SheafConditionEqualizerProducts.diagram F U) | where
pt := c.pt
π :=
{ app := fun Z =>
WalkingParallelPair.casesOn Z (Pi.lift fun i : ι => c.π.app (op (single i)))
(Pi.lift fun b : ι × ι => c.π.app (op (pair b.1 b.2)))
naturality := fun Y Z f => by
cases Y <;> cases Z <;> cases f
· dsimp
ext
simp
... | def | TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [
"Quiver.Hom.op"
] | Implementation of `SheafConditionPairwiseIntersections.coneEquiv`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coneEquivFunctor :
Limits.Cone ((diagram U).op ⋙ F) ⥤
Limits.Cone (SheafConditionEqualizerProducts.diagram F U) | where
obj c := coneEquivFunctorObj F U c
map {c c'} f :=
{ hom := f.hom
w := fun j => by
cases j <;>
· dsimp
ext
simp } | def | TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | Implementation of `SheafConditionPairwiseIntersections.coneEquiv`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coneEquivInverseObj (c : Limits.Cone (SheafConditionEqualizerProducts.diagram F U)) :
Limits.Cone ((diagram U).op ⋙ F) | where
pt := c.pt
π :=
{ app := by
intro x
induction x with | op x => ?_
rcases x with (⟨i⟩ | ⟨i, j⟩)
· exact c.π.app WalkingParallelPair.zero ≫ Pi.π _ i
· exact c.π.app WalkingParallelPair.one ≫ Pi.π _ (i, j)
naturality := by
intro x y f
induction x ... | def | TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | Implementation of `SheafConditionPairwiseIntersections.coneEquiv`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coneEquivInverse :
Limits.Cone (SheafConditionEqualizerProducts.diagram F U) ⥤
Limits.Cone ((diagram U).op ⋙ F) | where
obj c := coneEquivInverseObj F U c
map {c c'} f :=
{ hom := f.hom
w := by
intro x
induction x with | op x => ?_
rcases x with (⟨i⟩ | ⟨i, j⟩)
· dsimp
dsimp only [Fork.ι]
rw [← f.w WalkingParallelPair.zero, Category.assoc]
· dsimp
r... | def | TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | Implementation of `SheafConditionPairwiseIntersections.coneEquiv`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coneEquivUnitIsoApp (c : Cone ((diagram U).op ⋙ F)) :
(𝟭 (Cone ((diagram U).op ⋙ F))).obj c ≅
(coneEquivFunctor F U ⋙ coneEquivInverse F U).obj c | where
hom :=
{ hom := 𝟙 _
w := fun j => by
induction j with | op j => ?_
rcases j with ⟨⟩ <;>
· dsimp [coneEquivInverse]
simp only [Limits.Fan.mk_π_app, Category.id_comp, Limits.limit.lift_π] }
inv :=
{ hom := 𝟙 _
w := fun j => by
induction j with | op... | def | TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | Implementation of `SheafConditionPairwiseIntersections.coneEquiv`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coneEquivUnitIso :
𝟭 (Limits.Cone ((diagram U).op ⋙ F)) ≅ coneEquivFunctor F U ⋙ coneEquivInverse F U | NatIso.ofComponents (coneEquivUnitIsoApp F U) | def | TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | Implementation of `SheafConditionPairwiseIntersections.coneEquiv`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coneEquivCounitIso :
coneEquivInverse F U ⋙ coneEquivFunctor F U ≅
𝟭 (Limits.Cone (SheafConditionEqualizerProducts.diagram F U)) | NatIso.ofComponents
(fun c =>
{ hom :=
{ hom := 𝟙 _
w := by
rintro ⟨_ | _⟩
· dsimp
ext
simp
· dsimp
ext
simp }
inv :=
{ hom := 𝟙 _
w := by
... | def | TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | Implementation of `SheafConditionPairwiseIntersections.coneEquiv`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coneEquiv :
Limits.Cone ((diagram U).op ⋙ F) ≌
Limits.Cone (SheafConditionEqualizerProducts.diagram F U) | where
functor := coneEquivFunctor F U
inverse := coneEquivInverse F U
unitIso := coneEquivUnitIso F U
counitIso := coneEquivCounitIso F U | def | TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [] | Cones over `diagram U ⋙ F` are the same as a cones over the usual sheaf condition equalizer diagram. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLimitMapConeOfIsLimitSheafConditionFork
(P : IsLimit (SheafConditionEqualizerProducts.fork F U)) : IsLimit (F.mapCone (cocone U).op) | IsLimit.ofIsoLimit ((IsLimit.ofConeEquiv (coneEquiv F U).symm).symm P)
{ hom :=
{ hom := 𝟙 _
w := by
intro x
induction x with | op x => ?_
rcases x with ⟨⟩
· simp
rfl
· dsimp [coneEquivInverse, SheafConditionEqualizerProduc... | def | TopCat.Presheaf.SheafConditionPairwiseIntersections.isLimitMapConeOfIsLimitSheafConditionFork | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [
"symm"
] | If `SheafConditionEqualizerProducts.fork` is an equalizer,
then `F.mapCone (cone U)` is a limit cone. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLimitSheafConditionForkOfIsLimitMapCone (Q : IsLimit (F.mapCone (cocone U).op)) :
IsLimit (SheafConditionEqualizerProducts.fork F U) | IsLimit.ofIsoLimit ((IsLimit.ofConeEquiv (coneEquiv F U)).symm Q)
{ hom :=
{ hom := 𝟙 _
w := by
rintro ⟨⟩
· simp
rfl
· dsimp
ext
dsimp [coneEquivInverse, SheafConditionEqualizerProducts.res,
SheafConditi... | def | TopCat.Presheaf.SheafConditionPairwiseIntersections.isLimitSheafConditionForkOfIsLimitMapCone | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [
"symm"
] | If `F.mapCone (cone U)` is a limit cone,
then `SheafConditionEqualizerProducts.fork` is an equalizer. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isSheaf_iff_isSheafEqualizerProducts (F : Presheaf C X) :
F.IsSheaf ↔ F.IsSheafEqualizerProducts | (isSheaf_iff_isSheafPairwiseIntersections F).trans <|
Iff.intro (fun h _ U => ⟨isLimitSheafConditionForkOfIsLimitMapCone F U (h U).some⟩) fun h _ U =>
⟨isLimitMapConeOfIsLimitSheafConditionFork F U (h U).some⟩ | theorem | TopCat.Presheaf.isSheaf_iff_isSheafEqualizerProducts | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | [] | [
"trans"
] | The sheaf condition in terms of an equalizer diagram is equivalent
to the default sheaf condition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
OpensLeCover : Type w | ObjectProperty.FullSubcategory fun V : Opens X ↦ ∃ i, V ≤ U i
deriving Category | def | TopCat.Presheaf.SheafCondition.OpensLeCover | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [] | The category of open sets contained in some element of the cover. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
index (V : OpensLeCover U) : ι | V.property.choose | def | TopCat.Presheaf.SheafCondition.OpensLeCover.index | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [] | An arbitrarily chosen index such that `V ≤ U i`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
homToIndex (V : OpensLeCover U) : V.obj ⟶ U (index V) | V.property.choose_spec.hom | def | TopCat.Presheaf.SheafCondition.OpensLeCover.homToIndex | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [] | The morphism from `V` to `U i` for some `i`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
opensLeCoverCocone : Cocone (ObjectProperty.ι _ : OpensLeCover U ⥤ Opens X) | where
pt := iSup U
ι := { app := fun V : OpensLeCover U => V.homToIndex ≫ Opens.leSupr U _ } | def | TopCat.Presheaf.SheafCondition.opensLeCoverCocone | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [
"iSup"
] | `iSup U` as a cocone over the opens sets contained in some element of the cover.
(In fact this is a colimit cocone.) | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSheafOpensLeCover : Prop | ∀ ⦃ι : Type w⦄ (U : ι → Opens X), Nonempty (IsLimit (F.mapCone (opensLeCoverCocone U).op)) | def | TopCat.Presheaf.IsSheafOpensLeCover | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [] | An equivalent formulation of the sheaf condition
(which we prove equivalent to the usual one below as
`isSheaf_iff_isSheafOpensLeCover`).
A presheaf is a sheaf if `F` sends the cone `(opensLeCoverCocone U).op` to a limit cone.
(Recall `opensLeCoverCocone U`, has cone point `iSup U`,
mapping down to any `V` which is co... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
generateEquivalenceOpensLe_functor' :
(ObjectProperty.FullSubcategory
fun f : Over Y => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom) ⥤
OpensLeCover U | where
obj f :=
⟨f.1.left,
let ⟨_, h, _, ⟨i, hY⟩, _⟩ := f.2
⟨i, hY ▸ h.le⟩⟩
map g := ObjectProperty.homMk g.hom.left | def | TopCat.Presheaf.generateEquivalenceOpensLe_functor' | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [] | Given a family of opens `U` and an open `Y` equal to the union of opens in `U`, we may
take the presieve on `Y` associated to `U` and the sieve generated by it, and form the
full subcategory (subposet) of opens contained in `Y` (`over Y`) consisting of arrows
in the sieve. This full subcategory is equivalent to `OpensL... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
generateEquivalenceOpensLe_inverse' (hY : Y = iSup U) :
OpensLeCover U ⥤
(ObjectProperty.FullSubcategory fun f : Over Y =>
(Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom) | where
obj := fun V => ⟨⟨V.obj, ⟨⟨⟩⟩, homOfLE <| hY ▸ (V.2.choose_spec.trans (le_iSup U (V.2.choose)))⟩,
⟨U V.2.choose, V.2.choose_spec.hom, homOfLE <| hY ▸ le_iSup U V.2.choose,
⟨V.2.choose, rfl⟩, rfl⟩⟩
map g := ObjectProperty.homMk (Over.homMk g.hom) | def | TopCat.Presheaf.generateEquivalenceOpensLe_inverse' | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [
"iSup",
"le_iSup"
] | Given a family of opens `U` and an open `Y` equal to the union of opens in `U`, we may
take the presieve on `Y` associated to `U` and the sieve generated by it, and form the
full subcategory (subposet) of opens contained in `Y` (`over Y`) consisting of arrows
in the sieve. This full subcategory is equivalent to `OpensL... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
generateEquivalenceOpensLe (hY : Y = iSup U) :
(ObjectProperty.FullSubcategory
fun f : Over Y => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom) ≌
OpensLeCover U | where
functor := generateEquivalenceOpensLe_functor' _
inverse := generateEquivalenceOpensLe_inverse' _ hY
unitIso := eqToIso <| CategoryTheory.Functor.ext (by cat_disch)
counitIso := eqToIso <| CategoryTheory.Functor.ext (by cat_disch) | def | TopCat.Presheaf.generateEquivalenceOpensLe | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [
"CategoryTheory.Functor.ext",
"iSup"
] | Given a family of opens `U` and an open `Y` equal to the union of opens in `U`, we may
take the presieve on `Y` associated to `U` and the sieve generated by it, and form the
full subcategory (subposet) of opens contained in `Y` (`over Y`) consisting of arrows
in the sieve. This full subcategory is equivalent to `OpensL... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
whiskerIsoMapGenerateCocone (hY : Y = iSup U) :
(F.mapCone (opensLeCoverCocone U).op).whisker (generateEquivalenceOpensLe U hY).op.functor ≅
F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op | where
hom :=
{ hom := F.map (eqToHom (congr_arg op hY.symm))
w := fun j => by
dsimp
rw [← F.map_comp]
congr 1 }
inv :=
{ hom := F.map (eqToHom (congr_arg op hY))
w := fun j => by
dsimp
rw [← F.map_comp]
congr 1 }
hom_inv_id := by
ext
simp... | def | TopCat.Presheaf.whiskerIsoMapGenerateCocone | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [
"iSup"
] | Given a family of opens `opensLeCoverCocone U` is essentially the natural cocone
associated to the sieve generated by the presieve associated to `U` with indexing
category changed using the above equivalence. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLimitOpensLeEquivGenerate₁ (hY : Y = iSup U) :
IsLimit (F.mapCone (opensLeCoverCocone U).op) ≃
IsLimit (F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op) | (IsLimit.whiskerEquivalenceEquiv (generateEquivalenceOpensLe U hY).op).trans
(IsLimit.equivIsoLimit (whiskerIsoMapGenerateCocone F U hY)) | def | TopCat.Presheaf.isLimitOpensLeEquivGenerate₁ | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [
"iSup",
"trans"
] | Given a presheaf `F` on the topological space `X` and a family of opens `U` of `X`,
the natural cone associated to `F` and `U` used in the definition of
`F.IsSheafOpensLeCover` is a limit cone iff the natural cone associated to `F`
and the sieve generated by the presieve associated to `U` is a limit cone. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLimitOpensLeEquivGenerate₂ (R : Presieve Y)
(hR : Sieve.generate R ∈ Opens.grothendieckTopology X Y) :
IsLimit (F.mapCone (opensLeCoverCocone (coveringOfPresieve Y R)).op) ≃
IsLimit (F.mapCone (Sieve.generate R).arrows.cocone.op) | by
convert!
isLimitOpensLeEquivGenerate₁ F (coveringOfPresieve Y R)
(coveringOfPresieve.iSup_eq_of_mem_grothendieck Y R hR).symm using 1
rw [covering_presieve_eq_self R] | def | TopCat.Presheaf.isLimitOpensLeEquivGenerate₂ | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [
"Opens.grothendieckTopology",
"symm"
] | Given a presheaf `F` on the topological space `X` and a presieve `R` whose generated sieve
is covering for the associated Grothendieck topology (equivalently, the presieve is covering
for the associated pretopology), the natural cone associated to `F` and the family of opens
associated to `R` is a limit cone iff the na... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSheaf.isSheafOpensLeCover (h : F.IsSheaf) :
Nonempty (IsLimit (F.mapCone (opensLeCoverCocone U).op)) | by
rw [(isLimitOpensLeEquivGenerate₁ F U rfl).nonempty_congr]
apply (Presheaf.isSheaf_iff_isLimit _ _).mp h
apply presieveOfCovering.mem_grothendieckTopology | theorem | TopCat.Presheaf.IsSheaf.isSheafOpensLeCover | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isSheaf_iff_isSheafOpensLeCover : F.IsSheaf ↔ F.IsSheafOpensLeCover | by
refine ⟨fun h _ ↦ h.isSheafOpensLeCover,
fun h ↦ (Presheaf.isSheaf_iff_isLimit _ _).mpr fun Y S ↦ ?_⟩
rw [← Sieve.generate_sieve S]
intro hS
rw [← (isLimitOpensLeEquivGenerate₂ F S.1 hS).nonempty_congr]
apply h | theorem | TopCat.Presheaf.isSheaf_iff_isSheafOpensLeCover | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | [] | [] | A presheaf `(opens X)ᵒᵖ ⥤ C` on a topological space `X` is a sheaf on the site `opens X` iff
it satisfies the `IsSheafOpensLeCover` sheaf condition. The latter is not the
official definition of sheaves on spaces, but has the advantage that it does not
require `has_products C`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSheafPairwiseIntersections (F : Presheaf C X) : Prop | ∀ ⦃ι : Type w⦄ (U : ι → Opens X), Nonempty (IsLimit (F.mapCone (Pairwise.cocone U).op)) | def | TopCat.Presheaf.IsSheafPairwiseIntersections | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [] | An alternative formulation of the sheaf condition
(which we prove equivalent to the usual one below as
`isSheaf_iff_isSheafPairwiseIntersections`).
A presheaf is a sheaf if `F` sends the cone `(Pairwise.cocone U).op` to a limit cone.
(Recall `Pairwise.cocone U` has cone point `iSup U`, mapping down to the `U i` and th... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSheafPreservesLimitPairwiseIntersections (F : Presheaf C X) : Prop | ∀ ⦃ι : Type w⦄ (U : ι → Opens X), PreservesLimit (Pairwise.diagram U).op F | def | TopCat.Presheaf.IsSheafPreservesLimitPairwiseIntersections | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [] | An alternative formulation of the sheaf condition
(which we prove equivalent to the usual one below as
`isSheaf_iff_isSheafPreservesLimitPairwiseIntersections`).
A presheaf is a sheaf if `F` preserves the limit of `Pairwise.diagram U`.
(Recall `Pairwise.diagram U` is the diagram consisting of the pairwise intersection... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pairwiseToOpensLeCoverObj : Pairwise ι → OpensLeCover U | | single i => ⟨U i, ⟨i, le_rfl⟩⟩
| Pairwise.pair i j => ⟨U i ⊓ U j, ⟨i, inf_le_left⟩⟩ | def | TopCat.Presheaf.SheafCondition.pairwiseToOpensLeCoverObj | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"Pairwise"
] | Implementation detail:
the object level of `pairwiseToOpensLeCover : Pairwise ι ⥤ OpensLeCover U` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pairwiseToOpensLeCoverMap :
∀ {V W : Pairwise ι}, (V ⟶ W) → (pairwiseToOpensLeCoverObj U V ⟶ pairwiseToOpensLeCoverObj U W) | | _, _, id_single _ => 𝟙 _
| _, _, id_pair _ _ => 𝟙 _
| _, _, left _ _ => ObjectProperty.homMk (homOfLE inf_le_left)
| _, _, right _ _ => ObjectProperty.homMk (homOfLE inf_le_right) | def | TopCat.Presheaf.SheafCondition.pairwiseToOpensLeCoverMap | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"Pairwise"
] | Implementation detail:
the morphism level of `pairwiseToOpensLeCover : Pairwise ι ⥤ OpensLeCover U` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pairwiseToOpensLeCover : Pairwise ι ⥤ OpensLeCover U | where
obj := pairwiseToOpensLeCoverObj U
map {_ _} i := pairwiseToOpensLeCoverMap U i | def | TopCat.Presheaf.SheafCondition.pairwiseToOpensLeCover | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"Pairwise"
] | The category of single and double intersections of the `U i` maps into the category
of open sets below some `U i`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pairwiseDiagramIso :
Pairwise.diagram U ≅ pairwiseToOpensLeCover U ⋙ ObjectProperty.ι _ | where
hom := { app := by rintro (i | ⟨i, j⟩) <;> exact 𝟙 _ }
inv := { app := by rintro (i | ⟨i, j⟩) <;> exact 𝟙 _ } | def | TopCat.Presheaf.SheafCondition.pairwiseDiagramIso | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [] | The diagram in `Opens X` indexed by pairwise intersections from `U` is isomorphic
(in fact, equal) to the diagram factored through `OpensLeCover U`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pairwiseCoconeIso :
(Pairwise.cocone U).op ≅
(Cone.postcomposeEquivalence (NatIso.op (pairwiseDiagramIso U :) :)).functor.obj
((opensLeCoverCocone U).op.whisker (pairwiseToOpensLeCover U).op) | Cone.ext (Iso.refl _) (by cat_disch) | def | TopCat.Presheaf.SheafCondition.pairwiseCoconeIso | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [] | The cocone `Pairwise.cocone U` with cocone point `iSup U` over `Pairwise.diagram U` is isomorphic
to the cocone `opensLeCoverCocone U` (with the same cocone point)
after appropriate whiskering and postcomposition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isLimitOpensLeCoverEquivPairwise :
IsLimit (F.mapCone (opensLeCoverCocone U).op) ≃ IsLimit (F.mapCone (Pairwise.cocone U).op) | calc
IsLimit (F.mapCone (opensLeCoverCocone U).op) ≃
IsLimit ((F.mapCone (opensLeCoverCocone U).op).whisker (pairwiseToOpensLeCover U).op) :=
(Functor.Initial.isLimitWhiskerEquiv (pairwiseToOpensLeCover U).op _).symm
_ ≃ IsLimit (F.mapCone ((opensLeCoverCocone U).op.whisker (pairwiseToOpensLeCover... | def | TopCat.Presheaf.isLimitOpensLeCoverEquivPairwise | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"symm"
] | The diagram over all `{ V : Opens X // ∃ i, V ≤ U i }` is a limit iff the diagram
over `U i` and `U i ⊓ U j` is a limit. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isSheafOpensLeCover_iff_isSheafPairwiseIntersections :
F.IsSheafOpensLeCover ↔ F.IsSheafPairwiseIntersections | forall₂_congr fun _ U ↦ (F.isLimitOpensLeCoverEquivPairwise U).nonempty_congr | theorem | TopCat.Presheaf.isSheafOpensLeCover_iff_isSheafPairwiseIntersections | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [] | The sheaf condition
in terms of a limit diagram over all `{ V : Opens X // ∃ i, V ≤ U i }`
is equivalent to the reformulation
in terms of a limit diagram over `U i` and `U i ⊓ U j`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSheaf.isSheafPairwiseIntersections (h : F.IsSheaf) :
Nonempty (IsLimit (F.mapCone (Pairwise.cocone U).op)) | (h.isSheafOpensLeCover U).map (F.isLimitOpensLeCoverEquivPairwise _) | theorem | TopCat.Presheaf.IsSheaf.isSheafPairwiseIntersections | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isSheaf_iff_isSheafPairwiseIntersections : F.IsSheaf ↔ F.IsSheafPairwiseIntersections | by
rw [isSheaf_iff_isSheafOpensLeCover, isSheafOpensLeCover_iff_isSheafPairwiseIntersections] | theorem | TopCat.Presheaf.isSheaf_iff_isSheafPairwiseIntersections | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [] | The sheaf condition in terms of an equalizer diagram is equivalent
to the reformulation in terms of a limit diagram over `U i` and `U i ⊓ U j`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSheaf.isSheafPreservesLimitPairwiseIntersections (h : F.IsSheaf) :
PreservesLimit (Pairwise.diagram U).op F | preservesLimit_of_preserves_limit_cone (Pairwise.coconeIsColimit U).op
(h.isSheafPairwiseIntersections U).some | theorem | TopCat.Presheaf.IsSheaf.isSheafPreservesLimitPairwiseIntersections | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isSheaf_iff_isSheafPreservesLimitPairwiseIntersections :
F.IsSheaf ↔ F.IsSheafPreservesLimitPairwiseIntersections | by
refine ⟨fun h U ↦ h.isSheafPreservesLimitPairwiseIntersections,
fun h ↦ F.isSheaf_iff_isSheafPairwiseIntersections.mpr fun ι U ↦ ?_⟩
haveI := h U
exact ⟨isLimitOfPreserves _ (Pairwise.coconeIsColimit U).op⟩ | theorem | TopCat.Presheaf.isSheaf_iff_isSheafPreservesLimitPairwiseIntersections | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [] | The sheaf condition in terms of an equalizer diagram is equivalent
to the reformulation in terms of the presheaf preserving the limit of the diagram
consisting of the `U i` and `U i ⊓ U j`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
interUnionPullbackCone :
PullbackCone (F.1.map (homOfLE inf_le_left : U ⊓ V ⟶ _).op)
(F.1.map (homOfLE inf_le_right).op) | PullbackCone.mk (F.1.map (homOfLE le_sup_left).op) (F.1.map (homOfLE le_sup_right).op) <| by
rw [← F.1.map_comp, ← F.1.map_comp]
congr 1 | def | TopCat.Sheaf.interUnionPullbackCone | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"le_sup_left",
"le_sup_right"
] | For a sheaf `F`, `F(U ⊔ V)` is the pullback of `F(U) ⟶ F(U ⊓ V)` and `F(V) ⟶ F(U ⊓ V)`.
This is the pullback cone. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
interUnionPullbackCone_pt : (interUnionPullbackCone F U V).pt = F.1.obj (op <| U ⊔ V) | rfl | theorem | TopCat.Sheaf.interUnionPullbackCone_pt | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interUnionPullbackCone_fst :
(interUnionPullbackCone F U V).fst = F.1.map (homOfLE le_sup_left).op | rfl | theorem | TopCat.Sheaf.interUnionPullbackCone_fst | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interUnionPullbackCone_snd :
(interUnionPullbackCone F U V).snd = F.1.map (homOfLE le_sup_right).op | rfl | theorem | TopCat.Sheaf.interUnionPullbackCone_snd | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"le_sup_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interUnionPullbackConeLift : s.pt ⟶ F.1.obj (op (U ⊔ V)) | by
let ι : ULift.{w} WalkingPair → Opens X := fun j => WalkingPair.casesOn j.down U V
have hι : U ⊔ V = iSup ι := by
ext
rw [Opens.coe_iSup, Set.mem_iUnion]
constructor
· rintro (h | h)
exacts [⟨⟨WalkingPair.left⟩, h⟩, ⟨⟨WalkingPair.right⟩, h⟩]
· rintro ⟨⟨_ | _⟩, h⟩
exacts [Or.inl h,... | def | TopCat.Sheaf.interUnionPullbackConeLift | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"Set.mem_iUnion",
"iSup"
] | (Implementation).
Every cone over `F(U) ⟶ F(U ⊓ V)` and `F(V) ⟶ F(U ⊓ V)` factors through `F(U ⊔ V)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
interUnionPullbackConeLift_left :
interUnionPullbackConeLift F U V s ≫ F.1.map (homOfLE le_sup_left).op = s.fst | by
rw [interUnionPullbackConeLift, Category.assoc, ← F.1.map_comp]
exact
(F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <|
op <| Pairwise.single <| ULift.up WalkingPair.left | theorem | TopCat.Sheaf.interUnionPullbackConeLift_left | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interUnionPullbackConeLift_right :
interUnionPullbackConeLift F U V s ≫ F.1.map (homOfLE le_sup_right).op = s.snd | by
rw [interUnionPullbackConeLift, Category.assoc, ← F.1.map_comp]
exact
(F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <|
op <| Pairwise.single <| ULift.up WalkingPair.right | theorem | TopCat.Sheaf.interUnionPullbackConeLift_right | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"le_sup_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLimitPullbackCone : IsLimit (interUnionPullbackCone F U V) | by
let ι : ULift.{w} WalkingPair → Opens X := fun ⟨j⟩ => WalkingPair.casesOn j U V
have hι : U ⊔ V = iSup ι := by
ext
rw [Opens.coe_iSup, Set.mem_iUnion]
constructor
· rintro (h | h)
exacts [⟨⟨WalkingPair.left⟩, h⟩, ⟨⟨WalkingPair.right⟩, h⟩]
· rintro ⟨⟨_ | _⟩, h⟩
exacts [Or.inl h, Or... | def | TopCat.Sheaf.isLimitPullbackCone | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"Set.mem_iUnion",
"iSup"
] | For a sheaf `F`, `F(U ⊔ V)` is the pullback of `F(U) ⟶ F(U ⊓ V)` and `F(V) ⟶ F(U ⊓ V)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isProductOfDisjoint (h : U ⊓ V = ⊥) :
IsLimit
(BinaryFan.mk (F.1.map (homOfLE le_sup_left : _ ⟶ U ⊔ V).op)
(F.1.map (homOfLE le_sup_right : _ ⟶ U ⊔ V).op)) | isProductOfIsTerminalIsPullback _ _ _ _ (F.isTerminalOfEqEmpty h) (isLimitPullbackCone F U V) | def | TopCat.Sheaf.isProductOfDisjoint | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | [] | [
"isProductOfIsTerminalIsPullback",
"le_sup_left",
"le_sup_right"
] | If `U, V` are disjoint, then `F(U ⊔ V) = F(U) × F(V)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coveringOfPresieve (U : Opens X) (R : Presieve U) : (Σ V, { f : V ⟶ U // R f }) → Opens X | fun f => f.1 | def | TopCat.Presheaf.coveringOfPresieve | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [] | Given a presieve `R` on `U`, we obtain a covering family of open sets in `X`, by taking as index
type the type of dependent pairs `(V, f)`, where `f : V ⟶ U` is in `R`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coveringOfPresieve_apply (U : Opens X) (R : Presieve U) (f : Σ V, { f : V ⟶ U // R f }) :
coveringOfPresieve U R f = f.1 | rfl | theorem | TopCat.Presheaf.coveringOfPresieve_apply | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iSup_eq_of_mem_grothendieck (hR : Sieve.generate R ∈ Opens.grothendieckTopology X U) :
iSup (coveringOfPresieve U R) = U | by
apply le_antisymm
· refine iSup_le ?_
intro f
exact f.2.1.le
intro x hxU
rw [Opens.mem_iSup]
obtain ⟨V, iVU, ⟨W, iVW, iWU, hiWU, -⟩, hxV⟩ := hR x hxU
exact ⟨⟨W, ⟨iWU, hiWU⟩⟩, iVW.le hxV⟩ | theorem | TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieck | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Opens.grothendieckTopology",
"iSup",
"iSup_le",
"le_antisymm"
] | If `R` is a presieve in the Grothendieck topology on `Opens X`, the covering family associated
to `R` really is _covering_, i.e. the union of all open sets equals `U`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
presieveOfCoveringAux {ι : Type v} (U : ι → Opens X) (Y : Opens X) : Presieve Y | fun V _ => ∃ i, V = U i | def | TopCat.Presheaf.presieveOfCoveringAux | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [] | Given a family of opens `U : ι → Opens X` and any open `Y : Opens X`, we obtain a presieve
on `Y` by declaring that a morphism `f : V ⟶ Y` is a member of the presieve if and only if
there exists an index `i : ι` such that `V = U i`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
presieveOfCovering {ι : Type v} (U : ι → Opens X) : Presieve (iSup U) | presieveOfCoveringAux U (iSup U) | def | TopCat.Presheaf.presieveOfCovering | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"iSup"
] | Take `Y` to be `iSup U` and obtain a presieve over `iSup U`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
covering_presieve_eq_self {Y : Opens X} (R : Presieve Y) :
presieveOfCoveringAux (coveringOfPresieve Y R) Y = R | by
funext Z
ext f
exact ⟨fun ⟨⟨_, f', h⟩, rfl⟩ => by rwa [Subsingleton.elim f f'], fun h => ⟨⟨Z, f, h⟩, rfl⟩⟩ | theorem | TopCat.Presheaf.covering_presieve_eq_self | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [] | Given a presieve `R` on `Y`, if we take its associated family of opens via `coveringOfPresieve`
(which may not cover `Y` if `R` is not covering), and take the presieve on `Y` associated to the
family of opens via `presieveOfCoveringAux`, then we get back the original presieve `R`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_grothendieckTopology :
Sieve.generate (presieveOfCovering U) ∈ Opens.grothendieckTopology X (iSup U) | by
intro x hx
obtain ⟨i, hxi⟩ := Opens.mem_iSup.mp hx
exact ⟨U i, Opens.leSupr U i, ⟨U i, 𝟙 _, Opens.leSupr U i, ⟨i, rfl⟩, Category.id_comp _⟩, hxi⟩ | theorem | TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Opens.grothendieckTopology",
"iSup"
] | The sieve generated by `presieveOfCovering U` is a member of the Grothendieck topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
homOfIndex (i : ι) : Σ V, { f : V ⟶ iSup U // presieveOfCovering U f } | ⟨U i, Opens.leSupr U i, i, rfl⟩ | def | TopCat.Presheaf.presieveOfCovering.homOfIndex | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"iSup"
] | An index `i : ι` can be turned into a dependent pair `(V, f)`, where `V` is an open set and
`f : V ⟶ iSup U` is a member of `presieveOfCovering U f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
indexOfHom (f : Σ V, { f : V ⟶ iSup U // presieveOfCovering U f }) : ι | f.2.2.choose | def | TopCat.Presheaf.presieveOfCovering.indexOfHom | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"iSup"
] | By using the axiom of choice, a dependent pair `(V, f)` where `f : V ⟶ iSup U` is a member of
`presieveOfCovering U f` can be turned into an index `i : ι`, such that `V = U i`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
indexOfHom_spec (f : Σ V, { f : V ⟶ iSup U // presieveOfCovering U f }) :
f.1 = U (indexOfHom U f) | f.2.2.choose_spec | theorem | TopCat.Presheaf.presieveOfCovering.indexOfHom_spec | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coverDense_iff_isBasis [Category* ι] (B : ι ⥤ Opens X) :
B.IsCoverDense (Opens.grothendieckTopology X) ↔ Opens.IsBasis (Set.range B.obj) | by
rw [Opens.isBasis_iff_nbhd]
constructor
· intro hd U x hx; rcases hd.1 U x hx with ⟨V, f, ⟨i, f₁, f₂, _⟩, hV⟩
exact ⟨B.obj i, ⟨i, rfl⟩, f₁.le hV, f₂.le⟩
intro hb; constructor; intro U x hx; rcases hb hx with ⟨_, ⟨i, rfl⟩, hx, hi⟩
exact ⟨B.obj i, ⟨⟨hi⟩⟩, ⟨⟨i, 𝟙 _, ⟨⟨hi⟩⟩, rfl⟩⟩, hx⟩ | theorem | TopCat.Opens.coverDense_iff_isBasis | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Opens.grothendieckTopology",
"Set.range",
"f₁",
"f₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coverDense_inducedFunctor {B : ι → Opens X} (h : Opens.IsBasis (Set.range B)) :
(inducedFunctor B).IsCoverDense (Opens.grothendieckTopology X) | (coverDense_iff_isBasis _).2 h | theorem | TopCat.Opens.coverDense_inducedFunctor | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Opens.grothendieckTopology",
"Set.range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsOpenEmbedding.compatiblePreserving (hf : IsOpenEmbedding f) :
CompatiblePreserving (Opens.grothendieckTopology Y) hf.functor | by
haveI : Mono f := (TopCat.mono_iff_injective f).mpr hf.injective
apply compatiblePreservingOfDownwardsClosed
intro U V i
refine ⟨(Opens.map f).obj V, eqToIso <| Opens.ext <| Set.image_preimage_eq_of_subset fun x h ↦ ?_⟩
obtain ⟨_, _, rfl⟩ := i.le h
exact ⟨_, rfl⟩ | theorem | Topology.IsOpenEmbedding.compatiblePreserving | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Opens.grothendieckTopology",
"Set.image_preimage_eq_of_subset",
"TopCat.mono_iff_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpenMap.coverPreserving (hf : IsOpenMap f) :
CoverPreserving (Opens.grothendieckTopology X) (Opens.grothendieckTopology Y) hf.functor | by
constructor
rintro U S hU _ ⟨x, hx, rfl⟩
obtain ⟨V, i, hV, hxV⟩ := hU x hx
exact ⟨_, hf.functor.map i, ⟨_, i, 𝟙 _, hV, rfl⟩, Set.mem_image_of_mem f hxV⟩ | theorem | IsOpenMap.coverPreserving | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"IsOpenMap",
"Opens.grothendieckTopology",
"Set.mem_image_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsOpenEmbedding.functor_isContinuous (h : IsOpenEmbedding f) :
h.functor.IsContinuous (Opens.grothendieckTopology X)
(Opens.grothendieckTopology Y) | by
apply Functor.isContinuous_of_coverPreserving
· exact h.compatiblePreserving
· exact h.isOpenMap.coverPreserving | lemma | Topology.IsOpenEmbedding.functor_isContinuous | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Opens.grothendieckTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TopCat.Presheaf.isSheaf_of_isOpenEmbedding (h : IsOpenEmbedding f) (hF : F.IsSheaf) :
IsSheaf (h.functor.op ⋙ F) | by
have := h.functor_isContinuous
exact Functor.op_comp_isSheaf _ _ _ ⟨_, hF⟩ | theorem | TopCat.Presheaf.isSheaf_of_isOpenEmbedding | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compatiblePreserving_opens_map :
CompatiblePreserving (Opens.grothendieckTopology X) (Opens.map f) | compatiblePreservingOfFlat _ _ | theorem | compatiblePreserving_opens_map | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Opens.grothendieckTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coverPreserving_opens_map : CoverPreserving (Opens.grothendieckTopology Y)
(Opens.grothendieckTopology X) (Opens.map f) | by
constructor
intro U S hS x hx
obtain ⟨V, i, hi, hxV⟩ := hS (f x) hx
exact ⟨_, (Opens.map f).map i, ⟨_, _, 𝟙 _, hi, Subsingleton.elim _ _⟩, hxV⟩ | theorem | coverPreserving_opens_map | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Opens.grothendieckTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isTerminalOfEmpty (F : Sheaf C X) : Limits.IsTerminal (F.obj.obj (op ⊥)) | F.isTerminalOfBotCover ⊥ (fun _ h => h.elim) | def | TopCat.Sheaf.isTerminalOfEmpty | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [] | The empty component of a sheaf is terminal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isTerminalOfEqEmpty (F : X.Sheaf C) {U : Opens X} (h : U = ⊥) :
Limits.IsTerminal (F.obj.obj (op U)) | by
convert! F.isTerminalOfEmpty | def | TopCat.Sheaf.isTerminalOfEqEmpty | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [] | A variant of `isTerminalOfEmpty` that is easier to `apply`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
restrictHomEquivHom (h : Opens.IsBasis (Set.range B)) :
((inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.1) ≃ (F ⟶ F'.1) | @Functor.IsCoverDense.restrictHomEquivHom _ _ _ _ _ _ _ _
(Opens.coverDense_inducedFunctor h) _ F F' | def | TopCat.Sheaf.restrictHomEquivHom | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Set.range"
] | If a family `B` of open sets forms a basis of the topology on `X`, and if `F'`
is a sheaf on `X`, then a homomorphism between a presheaf `F` on `X` and `F'`
is equivalent to a homomorphism between their restrictions to the indexing type
`ι` of `B`, with the induced category structure on `ι`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
extend_hom_app (h : Opens.IsBasis (Set.range B))
(α : (inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.1) (i : ι) :
(restrictHomEquivHom F F' h α).app (op (B i)) = α.app (op i) | by
nth_rw 2 [← (restrictHomEquivHom F F' h).left_inv α]
rfl | theorem | TopCat.Sheaf.extend_hom_app | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Set.range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hom_ext (h : Opens.IsBasis (Set.range B))
{α β : F ⟶ F'.1} (he : ∀ i, α.app (op (B i)) = β.app (op (B i))) : α = β | by
apply (restrictHomEquivHom F F' h).symm.injective
ext i
exact he i.unop | theorem | TopCat.Sheaf.hom_ext | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Set.range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isIso_iff_isIso_basis {F G : Sheaf C X} (h : Opens.IsBasis (Set.range B))
{φ : F ⟶ G} (hi : ∀ i, IsIso (φ.hom.app (op (B i)))) :
IsIso φ | by
have : (inducedFunctor B).IsCoverDense (Opens.grothendieckTopology X) :=
Opens.coverDense_inducedFunctor h
refine Functor.IsCoverDense.iso_of_restrict_iso (G := inducedFunctor B) _ ?_
rw [NatTrans.isIso_iff_isIso_app]
exact fun _ ↦ hi _ | theorem | TopCat.Sheaf.isIso_iff_isIso_basis | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | [] | [
"Opens.grothendieckTopology",
"Set.range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompatible (sf : ∀ i : ι, ToType (F.obj (op (U i)))) : Prop | ∀ i j : ι, F.map (infLELeft (U i) (U j)).op (sf i) = F.map (infLERight (U i) (U j)).op (sf j) | def | TopCat.Presheaf.IsCompatible | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [] | A family of sections `sf` is compatible, if the restrictions of `sf i` and `sf j` to `U i ⊓ U j`
agree, for all `i` and `j` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsGluing (sf : ∀ i : ι, ToType (F.obj (op (U i)))) (s : ToType (F.obj (op (iSup U)))) : Prop | ∀ i : ι, F.map (Opens.leSupr U i).op s = sf i | def | TopCat.Presheaf.IsGluing | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"iSup"
] | A section `s` is a gluing for a family of sections `sf` if it restricts to `sf i` on `U i`,
for all `i` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSheafUniqueGluing : Prop | ∀ ⦃ι : Type x⦄ (U : ι → Opens X) (sf : ∀ i : ι, ToType (F.obj (op (U i)))),
IsCompatible F U sf → ∃! s : ToType (F.obj (op (iSup U))), IsGluing F U sf s | def | TopCat.Presheaf.IsSheafUniqueGluing | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"iSup"
] | The sheaf condition in terms of unique gluings. A presheaf `F : Presheaf C X` satisfies this sheaf
condition if and only if, for every compatible family of sections `sf : Π i : ι, F.obj (op (U i))`,
there exists a unique gluing `s : F.obj (op (iSup U))`.
We prove this to be equivalent to the usual one below in
`TopCat... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
objPairwiseOfFamily (sf : ∀ i, F.obj (op (U i))) :
∀ i, ((Pairwise.diagram U).op ⋙ F).obj i | | ⟨Pairwise.single i⟩ => sf i
| ⟨Pairwise.pair i j⟩ => F.map (infLELeft (U i) (U j)).op (sf i) | def | TopCat.Presheaf.objPairwiseOfFamily | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [] | Given sections over a family of open sets, extend it to include
sections over pairwise intersections of the open sets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompatible.sectionPairwise {sf} (h : IsCompatible F U sf) :
((Pairwise.diagram U).op ⋙ F).sections | by
refine ⟨objPairwiseOfFamily sf, ?_⟩
let G := (Pairwise.diagram U).op ⋙ F
rintro (i | ⟨i, j⟩) (i' | ⟨i', j'⟩) (_ | _ | _ | _)
· exact ConcreteCategory.congr_hom (G.map_id <| op <| Pairwise.single i) _
· rfl
· exact (h i' i).symm
· exact ConcreteCategory.congr_hom (G.map_id <| op <| Pairwise.pair i j) _ | def | TopCat.Presheaf.IsCompatible.sectionPairwise | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"symm"
] | Given a compatible family of sections over open sets, extend it to a
section of the functor `(Pairwise.diagram U).op ⋙ F`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isGluing_iff_pairwise {sf s} : IsGluing F U sf s ↔
∀ i, (F.mapCone (Pairwise.cocone U).op).π.app i s = objPairwiseOfFamily sf i | by
refine ⟨fun h ↦ ?_, fun h i ↦ h (op <| Pairwise.single i)⟩
rintro (i | ⟨i, j⟩)
· exact h i
· rw [← (F.mapCone (Pairwise.cocone U).op).w (op <| Pairwise.Hom.left i j)]
exact congr_arg _ (h i) | theorem | TopCat.Presheaf.isGluing_iff_pairwise | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsSheaf.isSheafUniqueGluing_types (h : F.IsSheaf) (sf : ∀ i : ι, F.obj (op (U i)))
(cpt : IsCompatible F U sf) : ∃! s : F.obj (op (iSup U)), IsGluing F U sf s | by
simp_rw [isGluing_iff_pairwise]
exact (Types.isLimit_iff _).mp (h.isSheafPairwiseIntersections U) _ cpt.sectionPairwise.prop | theorem | TopCat.Presheaf.IsSheaf.isSheafUniqueGluing_types | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isSheaf_iff_isSheafUniqueGluing_types : F.IsSheaf ↔ F.IsSheafUniqueGluing | by
simp_rw [isSheaf_iff_isSheafPairwiseIntersections, IsSheafPairwiseIntersections,
Types.isLimit_iff, IsSheafUniqueGluing, isGluing_iff_pairwise]
refine forall₂_congr fun ι U ↦ ⟨fun h sf cpt ↦ ?_, fun h s hs ↦ ?_⟩
· exact h _ cpt.sectionPairwise.prop
· specialize h (fun i ↦ s <| op <| Pairwise.single i) fu... | theorem | TopCat.Presheaf.isSheaf_iff_isSheafUniqueGluing_types | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"symm",
"trans"
] | For type-valued presheaves, the sheaf condition in terms of unique gluings is equivalent to the
usual sheaf condition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isSheaf_of_isSheafUniqueGluing_types (Fsh : F.IsSheafUniqueGluing) : F.IsSheaf | (isSheaf_iff_isSheafUniqueGluing_types F).mpr Fsh | theorem | TopCat.Presheaf.isSheaf_of_isSheafUniqueGluing_types | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [] | The usual sheaf condition can be obtained from the sheaf condition
in terms of unique gluings. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSheaf.isSheafUniqueGluing (h : F.IsSheaf) {ι : Type*} (U : ι → Opens X)
(sf : ∀ i : ι, ToType (F.obj (op (U i))))
(cpt : IsCompatible F U sf) : ∃! s : ToType (F.obj (op (iSup U))), IsGluing F U sf s | ((isSheaf_iff_isSheaf_comp' (forget C) F).mp h).isSheafUniqueGluing_types sf cpt | theorem | TopCat.Presheaf.IsSheaf.isSheafUniqueGluing | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isSheaf_iff_isSheafUniqueGluing : F.IsSheaf ↔ F.IsSheafUniqueGluing | Iff.trans (isSheaf_iff_isSheaf_comp' (forget C) F)
(isSheaf_iff_isSheafUniqueGluing_types (F ⋙ forget C)) | theorem | TopCat.Presheaf.isSheaf_iff_isSheafUniqueGluing | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [] | For presheaves valued in a concrete category, whose forgetful functor reflects isomorphisms and
preserves limits, the sheaf condition in terms of unique gluings is equivalent to the usual one. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
existsUnique_gluing (sf : ∀ i : ι, ToType (F.1.obj (op (U i))))
(h : IsCompatible F.1 U sf) :
∃! s : ToType (F.1.obj (op (iSup U))), IsGluing F.1 U sf s | IsSheaf.isSheafUniqueGluing F.property U sf h | theorem | TopCat.Sheaf.existsUnique_gluing | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"iSup"
] | A more convenient way of obtaining a unique gluing of sections for a sheaf. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
existsUnique_gluing' (V : Opens X) (iUV : ∀ i : ι, U i ⟶ V) (hcover : V ≤ iSup U)
(sf : ∀ i : ι, ToType (F.1.obj (op (U i)))) (h : IsCompatible F.1 U sf) :
∃! s : ToType (F.1.obj (op V)), ∀ i : ι, F.1.map (iUV i).op s = sf i | by
have V_eq_supr_U : V = iSup U := le_antisymm hcover (iSup_le fun i => (iUV i).le)
obtain ⟨gl, gl_spec, gl_uniq⟩ := F.existsUnique_gluing U sf h
refine ⟨F.1.map (eqToHom V_eq_supr_U).op gl, ?_, ?_⟩
· intro i
rw [← ConcreteCategory.comp_apply, ← F.1.map_comp]
exact gl_spec i
· intro gl' gl'_spec
... | theorem | TopCat.Sheaf.existsUnique_gluing' | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"iSup",
"iSup_le",
"le_antisymm"
] | In this version of the lemma, the inclusion homs `iUV` can be specified directly by the user,
which can be more convenient in practice. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eq_of_locally_eq (s t : ToType (F.1.obj (op (iSup U))))
(h : ∀ i, F.1.map (Opens.leSupr U i).op s = F.1.map (Opens.leSupr U i).op t) : s = t | by
let sf : ∀ i : ι, ToType (F.1.obj (op (U i))) := fun i => F.1.map (Opens.leSupr U i).op s
have sf_compatible : IsCompatible _ U sf := by
intro i j
simp_rw [sf, ← ConcreteCategory.comp_apply, ← F.1.map_comp]
rfl
obtain ⟨gl, -, gl_uniq⟩ := F.existsUnique_gluing U sf sf_compatible
trans gl
· apply... | theorem | TopCat.Sheaf.eq_of_locally_eq | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"iSup",
"symm",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_of_locally_eq' (V : Opens X) (iUV : ∀ i : ι, U i ⟶ V) (hcover : V ≤ iSup U)
(s t : ToType (F.1.obj (op V))) (h : ∀ i, F.1.map (iUV i).op s = F.1.map (iUV i).op t) :
s = t | by
have V_eq_supr_U : V = iSup U := le_antisymm hcover (iSup_le fun i => (iUV i).le)
suffices F.1.map (eqToHom V_eq_supr_U.symm).op s = F.1.map (eqToHom V_eq_supr_U.symm).op t by
convert! congr_arg (F.1.map (eqToHom V_eq_supr_U).op) this <;>
rw [← ConcreteCategory.comp_apply, ← F.1.map_comp, eqToHom_op, eqT... | theorem | TopCat.Sheaf.eq_of_locally_eq' | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"iSup",
"iSup_le",
"le_antisymm"
] | In this version of the lemma, the inclusion homs `iUV` can be specified directly by the user,
which can be more convenient in practice. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eq_of_locally_eq₂ {U₁ U₂ V : Opens X} (i₁ : U₁ ⟶ V) (i₂ : U₂ ⟶ V) (hcover : V ≤ U₁ ⊔ U₂)
(s t : ToType (F.1.obj (op V))) (h₁ : F.1.map i₁.op s = F.1.map i₁.op t)
(h₂ : F.1.map i₂.op s = F.1.map i₂.op t) : s = t | by
classical
fapply F.eq_of_locally_eq' fun t : Bool => if t then U₁ else U₂
· exact fun i => if h : i then eqToHom (if_pos h) ≫ i₁ else eqToHom (if_neg h) ≫ i₂
· refine le_trans hcover ?_
rw [sup_le_iff]
constructor
· exact le_iSup (fun t : Bool => if t then U₁ else U₂) true
· exa... | theorem | TopCat.Sheaf.eq_of_locally_eq₂ | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"le_iSup",
"le_trans",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_app_of_locally_eq {V : Opens X} {G : Sheaf C X} {f : F ⟶ G}
{s : ToType (F.1.obj (op (iSup U)))} {t : ToType (G.1.obj (op V))}
{sf : ∀ i : ι, ToType (F.1.obj (op (U i)))} (h : IsGluing F.1 U sf s) (hV : ∀ i : ι, U i ≤ V)
(ht : ∀ i : ι, f.1.app (op (U i)) (sf i) = G.1.map (homOfLE (hV i)).op t) :
f.ho... | by
refine eq_of_locally_eq G U _ _ (fun _ ↦ ?_)
rw [← NatTrans.naturality_apply, h, ht, ← ConcreteCategory.comp_apply, ← Functor.map_comp]
rfl | theorem | TopCat.Sheaf.eq_app_of_locally_eq | Topology.Sheaves.SheafCondition | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | [] | [
"iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SpectralSpace (X : Type*) [TopologicalSpace X] : Prop extends
T0Space X, CompactSpace X, QuasiSober X, QuasiSeparatedSpace X, PrespectralSpace X | class | SpectralSpace | Topology.Spectral | Mathlib/Topology/Spectral/Basic.lean | [] | [
"CompactSpace",
"PrespectralSpace",
"QuasiSeparatedSpace",
"QuasiSober",
"T0Space",
"TopologicalSpace"
] | A topological space is spectral if it is T0, compact, sober, quasi-separated, and its compact open
subsets form an open basis. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsOpenEmbedding.spectralSpace
[SpectralSpace Y] [CompactSpace X] (hf : IsOpenEmbedding f) :
SpectralSpace X | where
__ := hf.quasiSober
__ := hf.prespectralSpace
__ := hf.quasiSeparatedSpace
__ := hf.isEmbedding.t0Space | theorem | Topology.IsOpenEmbedding.spectralSpace | Topology.Spectral | Mathlib/Topology/Spectral/Basic.lean | [] | [
"CompactSpace",
"SpectralSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
constructibleTopologySubbasis (X : Type*) [TopologicalSpace X] : Set (Set X) | { s | IsOpen s ∧ IsCompact s } ∪ { s | IsClosed s ∧ IsCompact sᶜ } | def | constructibleTopologySubbasis | Topology.Spectral | Mathlib/Topology/Spectral/ConstructibleTopology.lean | [] | [
"IsClosed",
"IsCompact",
"IsOpen",
"Set",
"TopologicalSpace"
] | The subbasis of the constructible topology on a topological space `X`: It consists
of the open and compact sets of `X` and their complements. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.