statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
section_ext (F : sheaf C X) (U : opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.presheaf.germ x t) :
s = t | begin
-- We use `germ_eq` and the axiom of choice, to pick for every point `x` a neighbourhood
-- `V x`, such that the restrictions of `s` and `t` to `V x` coincide.
choose V m i₁ i₂ heq using λ x : U, F.presheaf.germ_eq x.1 x.2 x.2 s t (h x),
-- Since `F` is a sheaf, we can prove the equality locally, if we ca... | lemma | Top.presheaf.section_ext | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [] | Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
app_injective_of_stalk_functor_map_injective {F : sheaf C X} {G : presheaf C X}
(f : F.1 ⟶ G) (U : opens X) (h : ∀ x : U, function.injective ((stalk_functor C x.val).map f)) :
function.injective (f.app (op U)) | λ s t hst, section_ext F _ _ _ $ λ x, h x $ by
rw [stalk_functor_map_germ_apply, stalk_functor_map_germ_apply, hst] | lemma | Top.presheaf.app_injective_of_stalk_functor_map_injective | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_injective_iff_stalk_functor_map_injective {F : sheaf C X}
{G : presheaf C X} (f : F.1 ⟶ G) :
(∀ x : X, function.injective ((stalk_functor C x).map f)) ↔
(∀ U : opens X, function.injective (f.app (op U))) | ⟨λ h U, app_injective_of_stalk_functor_map_injective f U (λ x, h x.1),
stalk_functor_map_injective_of_app_injective f⟩ | lemma | Top.presheaf.app_injective_iff_stalk_functor_map_injective | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_functor_preserves_mono (x : X) :
functor.preserves_monomorphisms (sheaf.forget C X ⋙ stalk_functor C x) | ⟨λ 𝓐 𝓑 f m, concrete_category.mono_of_injective _ $
(app_injective_iff_stalk_functor_map_injective f.1).mpr
(λ c, (@@concrete_category.mono_iff_injective_of_preserves_pullback _ _ (f.1.app (op c)) _).mp
((nat_trans.mono_iff_mono_app _ f.1).mp
(@@category_theory.presheaf_mono_of_mono _ _ _ _ _ _ _ ... | instance | Top.presheaf.stalk_functor_preserves_mono | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [
"category_theory.presheaf_mono_of_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_mono_of_mono {F G : sheaf C X} (f : F ⟶ G) [mono f] :
Π x, mono $ (stalk_functor C x).map f.1 | λ x, by convert functor.map_mono (sheaf.forget.{v} C X ⋙ stalk_functor C x) f | lemma | Top.presheaf.stalk_mono_of_mono | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_stalk_mono {F G : sheaf C X} (f : F ⟶ G)
[Π x, mono $ (stalk_functor C x).map f.1] : mono f | (Sheaf.hom.mono_iff_presheaf_mono _ _ _).mpr $ (nat_trans.mono_iff_mono_app _ _).mpr $ λ U,
(concrete_category.mono_iff_injective_of_preserves_pullback _).mpr $
app_injective_of_stalk_functor_map_injective f.1 U.unop $ λ ⟨x, hx⟩,
(concrete_category.mono_iff_injective_of_preserves_pullback _).mp $ infer_instance | lemma | Top.presheaf.mono_of_stalk_mono | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_stalk_mono {F G : sheaf C X} (f : F ⟶ G) :
mono f ↔ ∀ x, mono ((stalk_functor C x).map f.1) | ⟨by { introI m, exact stalk_mono_of_mono _ }, by { introI m, exact mono_of_stalk_mono _ }⟩ | lemma | Top.presheaf.mono_iff_stalk_mono | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_surjective_of_injective_of_locally_surjective {F G : sheaf C X} (f : F ⟶ G)
(U : opens X) (hinj : ∀ x : U, function.injective ((stalk_functor C x.1).map f.1))
(hsurj : ∀ (t) (x : U), ∃ (V : opens X) (m : x.1 ∈ V) (iVU : V ⟶ U) (s : F.1.obj (op V)),
f.1.app (op V) s = G.1.map iVU.op t) :
function.surjectiv... | begin
intro t,
-- We use the axiom of choice to pick around each point `x` an open neighborhood `V` and a
-- preimage under `f` on `V`.
choose V mV iVU sf heq using hsurj t,
-- These neighborhoods clearly cover all of `U`.
have V_cover : U ≤ supr V,
{ intros x hxU,
rw [opens.mem_supr],
exact ⟨⟨x, ... | lemma | Top.presheaf.app_surjective_of_injective_of_locally_surjective | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [
"inf_le_left",
"supr"
] | For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
app_surjective_of_stalk_functor_map_bijective {F G : sheaf C X} (f : F ⟶ G)
(U : opens X) (h : ∀ x : U, function.bijective ((stalk_functor C x.val).map f.1)) :
function.surjective (f.1.app (op U)) | begin
refine app_surjective_of_injective_of_locally_surjective f U (λ x, (h x).1) (λ t x, _),
-- Now we need to prove our initial claim: That we can find preimages of `t` locally.
-- Since `f` is surjective on stalks, we can find a preimage `s₀` of the germ of `t` at `x`
obtain ⟨s₀,hs₀⟩ := (h x).2 (G.presheaf.g... | lemma | Top.presheaf.app_surjective_of_stalk_functor_map_bijective | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_bijective_of_stalk_functor_map_bijective {F G : sheaf C X} (f : F ⟶ G)
(U : opens X) (h : ∀ x : U, function.bijective ((stalk_functor C x.val).map f.1)) :
function.bijective (f.1.app (op U)) | ⟨app_injective_of_stalk_functor_map_injective f.1 U (λ x, (h x).1),
app_surjective_of_stalk_functor_map_bijective f U h⟩ | lemma | Top.presheaf.app_bijective_of_stalk_functor_map_bijective | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_is_iso_of_stalk_functor_map_iso {F G : sheaf C X} (f : F ⟶ G) (U : opens X)
[∀ x : U, is_iso ((stalk_functor C x.val).map f.1)] : is_iso (f.1.app (op U)) | begin
-- Since the forgetful functor of `C` reflects isomorphisms, it suffices to see that the
-- underlying map between types is an isomorphism, i.e. bijective.
suffices : is_iso ((forget C).map (f.1.app (op U))),
{ exactI is_iso_of_reflects_iso (f.1.app (op U)) (forget C) },
rw is_iso_iff_bijective,
apply... | lemma | Top.presheaf.app_is_iso_of_stalk_functor_map_iso | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_stalk_functor_map_iso {F G : sheaf C X} (f : F ⟶ G)
[∀ x : X, is_iso ((stalk_functor C x).map f.1)] : is_iso f | begin
-- Since the inclusion functor from sheaves to presheaves is fully faithful, it suffices to
-- show that `f`, as a morphism between _presheaves_, is an isomorphism.
suffices : is_iso ((sheaf.forget C X).map f),
{ exactI is_iso_of_fully_faithful (sheaf.forget C X) f },
-- We show that all components of `... | lemma | Top.presheaf.is_iso_of_stalk_functor_map_iso | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [
"opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_iff_stalk_functor_map_iso {F G : sheaf C X} (f : F ⟶ G) :
is_iso f ↔ ∀ x : X, is_iso ((stalk_functor C x).map f.1) | begin
split,
{ intros h x, resetI,
exact @functor.map_is_iso _ _ _ _ _ _ (stalk_functor C x) f.1
((sheaf.forget C X).map_is_iso f) },
{ intro h,
exactI is_iso_of_stalk_functor_map_iso f }
end | lemma | Top.presheaf.is_iso_iff_stalk_functor_map_iso | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [] | Let `F` and `G` be sheaves valued in a concrete category, whose forgetful functor reflects
isomorphisms, preserves limits and filtered colimits. Then a morphism `f : F ⟶ G` is an
isomorphism if and only if all of its stalk maps are isomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_open_algebra_map {X : Top} (F : X.presheaf CommRing) {U : opens X} (x : U) :
algebra_map (F.obj $ op U) (F.stalk x) = F.germ x | rfl | lemma | Top.presheaf.stalk_open_algebra_map | topology.sheaves | src/topology/sheaves/stalks.lean | [
"topology.category.Top.open_nhds",
"topology.sheaves.presheaf",
"topology.sheaves.sheaf_condition.unique_gluing",
"category_theory.adjunction.evaluation",
"category_theory.limits.types",
"category_theory.limits.preserves.filtered",
"category_theory.limits.final",
"tactic.elementwise",
"algebra.categ... | [
"CommRing",
"Top",
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_opens : C | ∏ (λ i : ι, F.obj (op (U i))) | def | Top.presheaf.sheaf_condition_equalizer_products.pi_opens | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | The product of the sections of a presheaf over a family of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_inters : C | ∏ (λ p : ι × ι, F.obj (op (U p.1 ⊓ U p.2))) | def | Top.presheaf.sheaf_condition_equalizer_products.pi_inters | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | The product of the sections of a presheaf over the pairwise intersections of
a family of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_res : pi_opens F U ⟶ pi_inters.{v'} F U | pi.lift (λ p : ι × ι, pi.π _ p.1 ≫ F.map (inf_le_left (U p.1) (U p.2)).op) | def | Top.presheaf.sheaf_condition_equalizer_products.left_res | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [
"inf_le_left"
] | The morphism `Π F.obj (U i) ⟶ Π F.obj (U i) ⊓ (U j)` whose components
are given by the restriction maps from `U i` to `U i ⊓ U j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_res : pi_opens F U ⟶ pi_inters.{v'} F U | pi.lift (λ p : ι × ι, pi.π _ p.2 ≫ F.map (inf_le_right (U p.1) (U p.2)).op) | def | Top.presheaf.sheaf_condition_equalizer_products.right_res | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [
"inf_le_right"
] | The morphism `Π F.obj (U i) ⟶ Π F.obj (U i) ⊓ (U j)` whose components
are given by the restriction maps from `U j` to `U i ⊓ U j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
res : F.obj (op (supr U)) ⟶ pi_opens.{v'} F U | pi.lift (λ i : ι, F.map (topological_space.opens.le_supr U i).op) | def | Top.presheaf.sheaf_condition_equalizer_products.res | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [
"supr",
"topological_space.opens.le_supr"
] | The morphism `F.obj U ⟶ Π F.obj (U i)` whose components
are given by the restriction maps from `U j` to `U i ⊓ U j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
res_π (i : ι) : res F U ≫ limit.π _ ⟨i⟩ = F.map (opens.le_supr U i).op | by rw [res, limit.lift_π, fan.mk_π_app] | lemma | Top.presheaf.sheaf_condition_equalizer_products.res_π | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w : res F U ≫ left_res F U = res F U ≫ right_res F U | begin
dsimp [res, left_res, right_res],
ext,
simp only [limit.lift_π, limit.lift_π_assoc, fan.mk_π_app, category.assoc],
rw [←F.map_comp],
rw [←F.map_comp],
congr,
end | lemma | Top.presheaf.sheaf_condition_equalizer_products.w | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram : walking_parallel_pair ⥤ C | parallel_pair (left_res.{v'} F U) (right_res F U) | def | Top.presheaf.sheaf_condition_equalizer_products.diagram | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | The equalizer diagram for the sheaf condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork : fork.{v} (left_res F U) (right_res F U) | fork.of_ι _ (w F U) | def | Top.presheaf.sheaf_condition_equalizer_products.fork | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork_X : (fork F U).X = F.obj (op (supr U)) | rfl | lemma | Top.presheaf.sheaf_condition_equalizer_products.fork_X | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork_ι : (fork F U).ι = res F U | rfl | lemma | Top.presheaf.sheaf_condition_equalizer_products.fork_ι | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork_π_app_walking_parallel_pair_zero :
(fork F U).π.app walking_parallel_pair.zero = res F U | rfl | lemma | Top.presheaf.sheaf_condition_equalizer_products.fork_π_app_walking_parallel_pair_zero | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork_π_app_walking_parallel_pair_one :
(fork F U).π.app walking_parallel_pair.one = res F U ≫ left_res F U | rfl | lemma | Top.presheaf.sheaf_condition_equalizer_products.fork_π_app_walking_parallel_pair_one | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_opens.iso_of_iso (α : F ≅ G) : pi_opens F U ≅ pi_opens.{v'} G U | pi.map_iso (λ X, α.app _) | def | Top.presheaf.sheaf_condition_equalizer_products.pi_opens.iso_of_iso | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | Isomorphic presheaves have isomorphic `pi_opens` for any cover `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_inters.iso_of_iso (α : F ≅ G) : pi_inters F U ≅ pi_inters.{v'} G U | pi.map_iso (λ X, α.app _) | def | Top.presheaf.sheaf_condition_equalizer_products.pi_inters.iso_of_iso | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | Isomorphic presheaves have isomorphic `pi_inters` for any cover `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram.iso_of_iso (α : F ≅ G) : diagram F U ≅ diagram.{v'} G U | nat_iso.of_components
begin rintro ⟨⟩, exact pi_opens.iso_of_iso U α, exact pi_inters.iso_of_iso U α end
begin
rintro ⟨⟩ ⟨⟩ ⟨⟩,
{ simp, },
{ ext, simp [left_res], },
{ ext, simp [right_res], },
{ simp, },
end. | def | Top.presheaf.sheaf_condition_equalizer_products.diagram.iso_of_iso | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | Isomorphic presheaves have isomorphic sheaf condition diagrams. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.iso_of_iso (α : F ≅ G) :
fork F U ≅ (cones.postcompose (diagram.iso_of_iso U α).inv).obj (fork G U) | begin
fapply fork.ext,
{ apply α.app, },
{ ext,
dunfold fork.ι, -- Ugh, `simp` can't unfold abbreviations.
simp [res, diagram.iso_of_iso], }
end | def | Top.presheaf.sheaf_condition_equalizer_products.fork.iso_of_iso | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_equalizer_products (F : presheaf.{v' v u} C X) : Prop | ∀ ⦃ι : Type v'⦄ (U : ι → opens X), nonempty (is_limit (sheaf_condition_equalizer_products.fork F U)) | def | Top.presheaf.is_sheaf_equalizer_products | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | The sheaf condition for a `F : presheaf C X` requires that the morphism
`F.obj U ⟶ ∏ 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
`∏ F.obj (U i) ⟶ ∏ F.obj (U i) ⊓ (U j)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_equiv_functor_obj (c : cone ((diagram U).op ⋙ F)) :
cone (sheaf_condition_equalizer_products.diagram F U) | { X := c.X,
π :=
{ app := λ Z,
walking_parallel_pair.cases_on Z
(pi.lift (λ (i : ι), c.π.app (op (single i))))
(pi.lift (λ (b : ι × ι), c.π.app (op (pair b.1 b.2)))),
naturality' := λ Y Z f,
begin
cases Y; cases Z; cases f,
{ ext i, dsimp,
simp only [limit.lift_π, c... | def | Top.presheaf.sheaf_condition_pairwise_intersections.cone_equiv_functor_obj | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [
"quiver.hom.op"
] | Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_equiv_functor :
limits.cone ((diagram U).op ⋙ F) ⥤
limits.cone (sheaf_condition_equalizer_products.diagram F U) | { obj := λ c, cone_equiv_functor_obj F U c,
map := λ c c' f,
{ hom := f.hom,
w' := λ j, begin
cases j;
{ ext, simp only [limits.fan.mk_π_app, limits.cone_morphism.w,
limits.limit.lift_π, category.assoc, cone_equiv_functor_obj_π_app], },
end }, }. | def | Top.presheaf.sheaf_condition_pairwise_intersections.cone_equiv_functor | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_equiv_inverse_obj
(c : limits.cone (sheaf_condition_equalizer_products.diagram F U)) :
limits.cone ((diagram U).op ⋙ F) | { X := c.X,
π :=
{ app :=
begin
intro x,
induction x using opposite.rec,
rcases x with (⟨i⟩|⟨i,j⟩),
{ exact c.π.app (walking_parallel_pair.zero) ≫ pi.π _ i, },
{ exact c.π.app (walking_parallel_pair.one) ≫ pi.π _ (i, j), }
end,
naturality' :=
begin
intros x y f,
... | def | Top.presheaf.sheaf_condition_pairwise_intersections.cone_equiv_inverse_obj | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [
"opposite.rec"
] | Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_equiv_inverse :
limits.cone (sheaf_condition_equalizer_products.diagram F U) ⥤
limits.cone ((diagram U).op ⋙ F) | { obj := λ c, cone_equiv_inverse_obj F U c,
map := λ c c' f,
{ hom := f.hom,
w' :=
begin
intro x,
induction x using opposite.rec,
rcases x with (⟨i⟩|⟨i,j⟩),
{ dsimp,
dunfold fork.ι,
rw [←(f.w walking_parallel_pair.zero), category.assoc], },
{ dsimp,
rw [... | def | Top.presheaf.sheaf_condition_pairwise_intersections.cone_equiv_inverse | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [
"opposite.rec"
] | Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_equiv_unit_iso_app
(c : cone ((diagram U).op ⋙ F)) :
(𝟭 (cone ((diagram U).op ⋙ F))).obj c ≅
(cone_equiv_functor F U ⋙ cone_equiv_inverse F U).obj c | { hom :=
{ hom := 𝟙 _,
w' := λ j, begin
induction j using opposite.rec, rcases j;
{ dsimp, simp only [limits.fan.mk_π_app, category.id_comp, limits.limit.lift_π], }
end, },
inv :=
{ hom := 𝟙 _,
w' := λ j, begin
induction j using opposite.rec, rcases j;
{ dsimp, simp only [lim... | def | Top.presheaf.sheaf_condition_pairwise_intersections.cone_equiv_unit_iso_app | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [
"opposite.rec"
] | Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_equiv_unit_iso :
𝟭 (limits.cone ((diagram U).op ⋙ F)) ≅
cone_equiv_functor F U ⋙ cone_equiv_inverse F U | nat_iso.of_components (cone_equiv_unit_iso_app F U) (by tidy) | def | Top.presheaf.sheaf_condition_pairwise_intersections.cone_equiv_unit_iso | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_equiv_counit_iso :
cone_equiv_inverse F U ⋙ cone_equiv_functor F U ≅
𝟭 (limits.cone (sheaf_condition_equalizer_products.diagram F U)) | nat_iso.of_components (λ c,
{ hom :=
{ hom := 𝟙 _,
w' :=
begin
rintro ⟨_|_⟩,
{ ext ⟨j⟩, dsimp, simp only [category.id_comp, limits.fan.mk_π_app, limits.limit.lift_π], },
{ ext ⟨i,j⟩, dsimp, simp only [category.id_comp, limits.fan.mk_π_app, limits.limit.lift_π], },
end },
inv :=
{ ho... | def | Top.presheaf.sheaf_condition_pairwise_intersections.cone_equiv_counit_iso | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_equiv :
limits.cone ((diagram U).op ⋙ F) ≌ limits.cone (sheaf_condition_equalizer_products.diagram F U) | { functor := cone_equiv_functor F U,
inverse := cone_equiv_inverse F U,
unit_iso := cone_equiv_unit_iso F U,
counit_iso := cone_equiv_counit_iso F U, } | def | Top.presheaf.sheaf_condition_pairwise_intersections.cone_equiv | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | Cones over `diagram U ⋙ F` are the same as a cones over the usual sheaf condition equalizer diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_map_cone_of_is_limit_sheaf_condition_fork
(P : is_limit (sheaf_condition_equalizer_products.fork F U)) :
is_limit (F.map_cone (cocone U).op) | is_limit.of_iso_limit ((is_limit.of_cone_equiv (cone_equiv F U).symm).symm P)
{ hom :=
{ hom := 𝟙 _,
w' :=
begin
intro x,
induction x using opposite.rec,
rcases x with ⟨⟩,
{ dsimp, simp, refl, },
{ dsimp,
simp only [limit.lift_π, limit.lift_π_assoc, category.id_comp, fan... | def | Top.presheaf.sheaf_condition_pairwise_intersections.is_limit_map_cone_of_is_limit_sheaf_condition_fork | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [
"opposite.rec"
] | If `sheaf_condition_equalizer_products.fork` is an equalizer,
then `F.map_cone (cone U)` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_sheaf_condition_fork_of_is_limit_map_cone
(Q : is_limit (F.map_cone (cocone U).op)) :
is_limit (sheaf_condition_equalizer_products.fork F U) | is_limit.of_iso_limit ((is_limit.of_cone_equiv (cone_equiv F U)).symm Q)
{ hom :=
{ hom := 𝟙 _,
w' :=
begin
rintro ⟨⟩,
{ dsimp, simp, refl, },
{ dsimp, ext ⟨i, j⟩,
simp only [limit.lift_π, limit.lift_π_assoc, category.id_comp, fan.mk_π_app,
category.assoc],
rw ←F.m... | def | Top.presheaf.sheaf_condition_pairwise_intersections.is_limit_sheaf_condition_fork_of_is_limit_map_cone | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | If `F.map_cone (cone U)` is a limit cone,
then `sheaf_condition_equalizer_products.fork` is an equalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_iff_is_sheaf_equalizer_products (F : presheaf C X) :
F.is_sheaf ↔ F.is_sheaf_equalizer_products | (is_sheaf_iff_is_sheaf_pairwise_intersections F).trans $
iff.intro (λ h ι U, ⟨is_limit_sheaf_condition_fork_of_is_limit_map_cone F U (h U).some⟩)
(λ h ι U, ⟨is_limit_map_cone_of_is_limit_sheaf_condition_fork F U (h U).some⟩) | lemma | Top.presheaf.is_sheaf_iff_is_sheaf_equalizer_products | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/equalizer_products.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.products",
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | The sheaf condition in terms of an equalizer diagram is equivalent
to the default sheaf condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
opens_le_cover : Type w | full_subcategory (λ (V : opens X), ∃ i, V ≤ U i) | def | Top.presheaf.sheaf_condition.opens_le_cover | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/opens_le_cover.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [] | The category of open sets contained in some element of the cover. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
index (V : opens_le_cover U) : ι | V.property.some | def | Top.presheaf.sheaf_condition.opens_le_cover.index | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/opens_le_cover.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [] | An arbitrarily chosen index such that `V ≤ U i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_to_index (V : opens_le_cover U) : V.obj ⟶ U (index V) | (V.property.some_spec).hom | def | Top.presheaf.sheaf_condition.opens_le_cover.hom_to_index | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/opens_le_cover.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [] | The morphism from `V` to `U i` for some `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
opens_le_cover_cocone : cocone (full_subcategory_inclusion _ : opens_le_cover U ⥤ opens X) | { X := supr U,
ι := { app := λ V : opens_le_cover U, V.hom_to_index ≫ opens.le_supr U _, } } | def | Top.presheaf.sheaf_condition.opens_le_cover_cocone | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/opens_le_cover.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [
"supr"
] | `supr 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_opens_le_cover : Prop | ∀ ⦃ι : Type w⦄ (U : ι → opens X), nonempty (is_limit (F.map_cone (opens_le_cover_cocone U).op)) | def | Top.presheaf.is_sheaf_opens_le_cover | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/opens_le_cover.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [] | An equivalent formulation of the sheaf condition
(which we prove equivalent to the usual one below as
`is_sheaf_iff_is_sheaf_opens_le_cover`).
A presheaf is a sheaf if `F` sends the cone `(opens_le_cover_cocone U).op` to a limit cone.
(Recall `opens_le_cover_cocone U`, has cone point `supr U`,
mapping down to any `V` ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
generate_equivalence_opens_le :
full_subcategory (λ (f : over Y), (sieve.generate (presieve_of_covering_aux U Y)).arrows f.hom) ≌
opens_le_cover U | { functor :=
{ obj := λ f, ⟨f.1.left, let ⟨_,h,_,⟨i,hY⟩,_⟩ := f.2 in ⟨i, hY ▸ h.le⟩⟩,
map := λ _ _ g, g.left },
inverse :=
{ obj := λ V, ⟨over.mk (hY.substr (let ⟨i,h⟩ := V.2 in h.trans (le_supr U i))).hom,
let ⟨i,h⟩ := V.2 in ⟨U i, h.hom, (hY.substr (le_supr U i)).hom, ⟨i, rfl⟩, rfl⟩⟩,
map := λ _ _... | def | Top.presheaf.generate_equivalence_opens_le | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/opens_le_cover.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [
"category_theory.functor.ext",
"category_theory.functor.hext",
"le_supr"
] | 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 equivalen... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_iso_map_generate_cocone :
(F.map_cone (opens_le_cover_cocone U).op).whisker (generate_equivalence_opens_le U hY).op.functor
≅ F.map_cone (sieve.generate (presieve_of_covering_aux U Y)).arrows.cocone.op | { hom :=
{ hom := F.map (eq_to_hom (congr_arg op hY.symm)),
w' := λ j, by { erw ← F.map_comp, congr } },
inv :=
{ hom := F.map (eq_to_hom (congr_arg op hY)),
w' := λ j, by { erw ← F.map_comp, congr } },
hom_inv_id' := by { ext, simp [eq_to_hom_map], },
inv_hom_id' := by { ext, simp [eq_to_hom_map], } ... | def | Top.presheaf.whisker_iso_map_generate_cocone | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/opens_le_cover.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [] | Given a family of opens `opens_le_cover_cocone 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_opens_le_equiv_generate₁ :
is_limit (F.map_cone (opens_le_cover_cocone U).op) ≃
is_limit (F.map_cone (sieve.generate (presieve_of_covering_aux U Y)).arrows.cocone.op) | (is_limit.whisker_equivalence_equiv (generate_equivalence_opens_le U hY).op).trans
(is_limit.equiv_iso_limit (whisker_iso_map_generate_cocone F U hY)) | def | Top.presheaf.is_limit_opens_le_equiv_generate₁ | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/opens_le_cover.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [] | 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.is_sheaf_opens_le_cover` is a limit cone iff the natural cone associated to `F`
and the sieve generated by the presieve associated to `U` is a limit con... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_opens_le_equiv_generate₂ (R : presieve Y)
(hR : sieve.generate R ∈ opens.grothendieck_topology X Y) :
is_limit (F.map_cone (opens_le_cover_cocone (covering_of_presieve Y R)).op) ≃
is_limit (F.map_cone (sieve.generate R).arrows.cocone.op) | begin
convert is_limit_opens_le_equiv_generate₁ F (covering_of_presieve Y R)
(covering_of_presieve.supr_eq_of_mem_grothendieck Y R hR).symm using 2;
rw covering_presieve_eq_self R,
end | def | Top.presheaf.is_limit_opens_le_equiv_generate₂ | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/opens_le_cover.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [
"opens.grothendieck_topology"
] | 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 con... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_iff_is_sheaf_opens_le_cover :
F.is_sheaf ↔ F.is_sheaf_opens_le_cover | begin
refine (presheaf.is_sheaf_iff_is_limit _ _).trans _,
split,
{ intros h ι U, rw (is_limit_opens_le_equiv_generate₁ F U rfl).nonempty_congr,
apply h, apply presieve_of_covering.mem_grothendieck_topology },
{ intros h Y S, rw ← sieve.generate_sieve S, intro hS,
rw ← (is_limit_opens_le_equiv_generate₂... | lemma | Top.presheaf.is_sheaf_iff_is_sheaf_opens_le_cover | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/opens_le_cover.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [] | A presheaf `(opens X)ᵒᵖ ⥤ C` on a topological space `X` is a sheaf on the site `opens X` iff
it satisfies the `is_sheaf_opens_le_cover` 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_pairwise_intersections (F : presheaf C X) : Prop | ∀ ⦃ι : Type w⦄ (U : ι → opens X), nonempty (is_limit (F.map_cone (pairwise.cocone U).op)) | def | Top.presheaf.is_sheaf_pairwise_intersections | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [] | An alternative formulation of the sheaf condition
(which we prove equivalent to the usual one below as
`is_sheaf_iff_is_sheaf_pairwise_intersections`).
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 `supr U`, mapping down to the `U i` an... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_preserves_limit_pairwise_intersections (F : presheaf C X) : Prop | ∀ ⦃ι : Type w⦄ (U : ι → opens X), nonempty (preserves_limit (pairwise.diagram U).op F) | def | Top.presheaf.is_sheaf_preserves_limit_pairwise_intersections | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [] | An alternative formulation of the sheaf condition
(which we prove equivalent to the usual one below as
`is_sheaf_iff_is_sheaf_preserves_limit_pairwise_intersections`).
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 inters... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pairwise_to_opens_le_cover_obj : pairwise ι → opens_le_cover U | | (single i) := ⟨U i, ⟨i, le_rfl⟩⟩
| (pair i j) := ⟨U i ⊓ U j, ⟨i, inf_le_left⟩⟩ | def | Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_obj | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"pairwise"
] | Implementation detail:
the object level of `pairwise_to_opens_le_cover : pairwise ι ⥤ opens_le_cover U` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pairwise_to_opens_le_cover_map :
Π {V W : pairwise ι},
(V ⟶ W) → (pairwise_to_opens_le_cover_obj U V ⟶ pairwise_to_opens_le_cover_obj U W) | | _ _ (id_single i) := 𝟙 _
| _ _ (id_pair i j) := 𝟙 _
| _ _ (left i j) := hom_of_le inf_le_left
| _ _ (right i j) := hom_of_le inf_le_right | def | Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_map | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"inf_le_left",
"inf_le_right",
"pairwise"
] | Implementation detail:
the morphism level of `pairwise_to_opens_le_cover : pairwise ι ⥤ opens_le_cover U` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pairwise_to_opens_le_cover : pairwise ι ⥤ opens_le_cover U | { obj := pairwise_to_opens_le_cover_obj U,
map := λ V W i, pairwise_to_opens_le_cover_map U i, } | def | Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pairwise_diagram_iso :
pairwise.diagram U ≅
pairwise_to_opens_le_cover U ⋙ full_subcategory_inclusion _ | { hom := { app := begin rintro (i|⟨i,j⟩); exact 𝟙 _, end, },
inv := { app := begin rintro (i|⟨i,j⟩); exact 𝟙 _, end, }, } | def | Top.presheaf.sheaf_condition.pairwise_diagram_iso | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [] | The diagram in `opens X` indexed by pairwise intersections from `U` is isomorphic
(in fact, equal) to the diagram factored through `opens_le_cover U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pairwise_cocone_iso :
(pairwise.cocone U).op ≅
(cones.postcompose_equivalence (nat_iso.op (pairwise_diagram_iso U : _) : _)).functor.obj
((opens_le_cover_cocone U).op.whisker (pairwise_to_opens_le_cover U).op) | cones.ext (iso.refl _) (by tidy) | def | Top.presheaf.sheaf_condition.pairwise_cocone_iso | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [] | The cocone `pairwise.cocone U` with cocone point `supr U` over `pairwise.diagram U` is isomorphic
to the cocone `opens_le_cover_cocone U` (with the same cocone point)
after appropriate whiskering and postcomposition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_opens_le_cover_iff_is_sheaf_pairwise_intersections :
F.is_sheaf_opens_le_cover ↔ F.is_sheaf_pairwise_intersections | forall₂_congr $ λ ι U, equiv.nonempty_congr $
calc is_limit (F.map_cone (opens_le_cover_cocone U).op)
≃ is_limit ((F.map_cone (opens_le_cover_cocone U).op).whisker (pairwise_to_opens_le_cover U).op)
: (functor.initial.is_limit_whisker_equiv (pairwise_to_opens_le_cover U).op _).symm
... ≃ is_limit (F.map_c... | lemma | Top.presheaf.is_sheaf_opens_le_cover_iff_is_sheaf_pairwise_intersections | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"equiv.nonempty_congr",
"forall₂_congr"
] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_iff_is_sheaf_pairwise_intersections :
F.is_sheaf ↔ F.is_sheaf_pairwise_intersections | by rw [is_sheaf_iff_is_sheaf_opens_le_cover,
is_sheaf_opens_le_cover_iff_is_sheaf_pairwise_intersections] | lemma | Top.presheaf.is_sheaf_iff_is_sheaf_pairwise_intersections | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_iff_is_sheaf_preserves_limit_pairwise_intersections :
F.is_sheaf ↔ F.is_sheaf_preserves_limit_pairwise_intersections | begin
rw is_sheaf_iff_is_sheaf_pairwise_intersections,
split,
{ intros h ι U,
exact ⟨preserves_limit_of_preserves_limit_cone (pairwise.cocone_is_colimit U).op (h U).some⟩ },
{ intros h ι U,
haveI := (h U).some,
exact ⟨preserves_limit.preserves (pairwise.cocone_is_colimit U).op⟩ }
end | lemma | Top.presheaf.is_sheaf_iff_is_sheaf_preserves_limit_pairwise_intersections | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inter_union_pullback_cone : pullback_cone
(F.1.map (hom_of_le inf_le_left : U ⊓ V ⟶ _).op) (F.1.map (hom_of_le inf_le_right).op) | pullback_cone.mk (F.1.map (hom_of_le le_sup_left).op) (F.1.map (hom_of_le le_sup_right).op)
(by { rw [← F.1.map_comp, ← F.1.map_comp], congr }) | def | Top.sheaf.inter_union_pullback_cone | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"inf_le_left",
"inf_le_right",
"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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inter_union_pullback_cone_X :
(inter_union_pullback_cone F U V).X = F.1.obj (op $ U ⊔ V) | rfl | lemma | Top.sheaf.inter_union_pullback_cone_X | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_union_pullback_cone_fst :
(inter_union_pullback_cone F U V).fst = F.1.map (hom_of_le le_sup_left).op | rfl | lemma | Top.sheaf.inter_union_pullback_cone_fst | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_union_pullback_cone_snd :
(inter_union_pullback_cone F U V).snd = F.1.map (hom_of_le le_sup_right).op | rfl | lemma | Top.sheaf.inter_union_pullback_cone_snd | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_union_pullback_cone_lift : s.X ⟶ F.1.obj (op (U ⊔ V)) | begin
let ι : ulift.{w} walking_pair → opens X := λ j, walking_pair.cases_on j.down U V,
have hι : U ⊔ V = supr ι,
{ ext,
rw [opens.coe_supr, set.mem_Union],
split,
{ rintros (h|h),
exacts [⟨⟨walking_pair.left⟩, h⟩, ⟨⟨walking_pair.right⟩, h⟩] },
{ rintro ⟨⟨_ | _⟩, h⟩,
exacts [or.inl h,... | def | Top.sheaf.inter_union_pullback_cone_lift | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"inf_comm",
"inf_le_left",
"opposite.rec",
"set.mem_Union",
"supr"
] | (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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inter_union_pullback_cone_lift_left :
inter_union_pullback_cone_lift F U V s ≫ F.1.map (hom_of_le le_sup_left).op = s.fst | begin
erw [category.assoc, ←F.1.map_comp],
exact (F.presheaf.is_sheaf_iff_is_sheaf_pairwise_intersections.mp F.2 _).some.fac _
(op $ pairwise.single (ulift.up walking_pair.left))
end | lemma | Top.sheaf.inter_union_pullback_cone_lift_left | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_union_pullback_cone_lift_right :
inter_union_pullback_cone_lift F U V s ≫ F.1.map (hom_of_le le_sup_right).op = s.snd | begin
erw [category.assoc, ←F.1.map_comp],
exact (F.presheaf.is_sheaf_iff_is_sheaf_pairwise_intersections.mp F.2 _).some.fac _
(op $ pairwise.single (ulift.up walking_pair.right))
end | lemma | Top.sheaf.inter_union_pullback_cone_lift_right | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit_pullback_cone : is_limit (inter_union_pullback_cone F U V) | begin
let ι : ulift.{w} walking_pair → opens X := λ ⟨j⟩, walking_pair.cases_on j U V,
have hι : U ⊔ V = supr ι,
{ ext,
rw [opens.coe_supr, set.mem_Union],
split,
{ rintros (h|h),
exacts [⟨⟨walking_pair.left⟩, h⟩, ⟨⟨walking_pair.right⟩, h⟩] },
{ rintro ⟨⟨_ | _⟩, h⟩,
exacts [or.inl h, or... | def | Top.sheaf.is_limit_pullback_cone | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"opposite.rec",
"set.mem_Union",
"supr"
] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_product_of_disjoint (h : U ⊓ V = ⊥) : is_limit
(binary_fan.mk (F.1.map (hom_of_le le_sup_left : _ ⟶ U ⊔ V).op)
(F.1.map (hom_of_le le_sup_right : _ ⟶ U ⊔ V).op)) | is_product_of_is_terminal_is_pullback _ _ _ _
(F.is_terminal_of_eq_empty h) (is_limit_pullback_cone F U V) | def | Top.sheaf.is_product_of_disjoint | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"is_product_of_is_terminal_is_pullback",
"le_sup_left",
"le_sup_right"
] | If `U, V` are disjoint, then `F(U ⊔ V) = F(U) × F(V)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_sup_iso_prod_eq_locus {X : Top} (F : X.sheaf CommRing)
(U V : opens X) :
F.1.obj (op $ U ⊔ V) ≅ CommRing.of (ring_hom.eq_locus _ _) | (F.is_limit_pullback_cone U V).cone_point_unique_up_to_iso (CommRing.pullback_cone_is_limit _ _) | def | Top.sheaf.obj_sup_iso_prod_eq_locus | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"CommRing",
"CommRing.of",
"CommRing.pullback_cone_is_limit",
"Top",
"ring_hom.eq_locus"
] | `F(U ⊔ V)` is isomorphic to the `eq_locus` of the two maps `F(U) × F(V) ⟶ F(U ⊓ V)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_sup_iso_prod_eq_locus_hom_fst {X : Top} (F : X.sheaf CommRing)
(U V : opens X) (x) :
((F.obj_sup_iso_prod_eq_locus U V).hom x).1.fst = F.1.map (hom_of_le le_sup_left).op x | concrete_category.congr_hom ((F.is_limit_pullback_cone U V).cone_point_unique_up_to_iso_hom_comp
(CommRing.pullback_cone_is_limit _ _) walking_cospan.left) x | lemma | Top.sheaf.obj_sup_iso_prod_eq_locus_hom_fst | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"CommRing",
"CommRing.pullback_cone_is_limit",
"Top",
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj_sup_iso_prod_eq_locus_hom_snd {X : Top} (F : X.sheaf CommRing)
(U V : opens X) (x) :
((F.obj_sup_iso_prod_eq_locus U V).hom x).1.snd = F.1.map (hom_of_le le_sup_right).op x | concrete_category.congr_hom ((F.is_limit_pullback_cone U V).cone_point_unique_up_to_iso_hom_comp
(CommRing.pullback_cone_is_limit _ _) walking_cospan.right) x | lemma | Top.sheaf.obj_sup_iso_prod_eq_locus_hom_snd | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"CommRing",
"CommRing.pullback_cone_is_limit",
"Top",
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj_sup_iso_prod_eq_locus_inv_fst {X : Top} (F : X.sheaf CommRing)
(U V : opens X) (x) :
F.1.map (hom_of_le le_sup_left).op ((F.obj_sup_iso_prod_eq_locus U V).inv x) = x.1.1 | concrete_category.congr_hom ((F.is_limit_pullback_cone U V).cone_point_unique_up_to_iso_inv_comp
(CommRing.pullback_cone_is_limit _ _) walking_cospan.left) x | lemma | Top.sheaf.obj_sup_iso_prod_eq_locus_inv_fst | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"CommRing",
"CommRing.pullback_cone_is_limit",
"Top",
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj_sup_iso_prod_eq_locus_inv_snd {X : Top} (F : X.sheaf CommRing)
(U V : opens X) (x) :
F.1.map (hom_of_le le_sup_right).op ((F.obj_sup_iso_prod_eq_locus U V).inv x) = x.1.2 | concrete_category.congr_hom ((F.is_limit_pullback_cone U V).cone_point_unique_up_to_iso_inv_comp
(CommRing.pullback_cone_is_limit _ _) walking_cospan.right) x | lemma | Top.sheaf.obj_sup_iso_prod_eq_locus_inv_snd | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/pairwise_intersections.lean | [
"topology.sheaves.sheaf_condition.opens_le_cover",
"category_theory.limits.final",
"category_theory.limits.preserves.basic",
"category_theory.category.pairwise",
"category_theory.limits.constructions.binary_products",
"algebra.category.Ring.constructions"
] | [
"CommRing",
"CommRing.pullback_cone_is_limit",
"Top",
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covering_of_presieve (U : opens X) (R : presieve U) : (Σ V, {f : V ⟶ U // R f}) → opens X | λ f, f.1 | def | Top.presheaf.covering_of_presieve | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covering_of_presieve_apply (U : opens X) (R : presieve U) (f : Σ V, {f : V ⟶ U // R f}) :
covering_of_presieve U R f = f.1 | rfl | lemma | Top.presheaf.covering_of_presieve_apply | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_eq_of_mem_grothendieck (hR : sieve.generate R ∈ opens.grothendieck_topology X U) :
supr (covering_of_presieve U R) = U | begin
apply le_antisymm,
{ refine supr_le _,
intro f,
exact f.2.1.le, },
intros x hxU,
rw [opens.mem_supr],
obtain ⟨V, iVU, ⟨W, iVW, iWU, hiWU, -⟩, hxV⟩ := hR x hxU,
exact ⟨⟨W, ⟨iWU, hiWU⟩⟩, iVW.le hxV⟩,
end | lemma | Top.presheaf.covering_of_presieve.supr_eq_of_mem_grothendieck | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"opens.grothendieck_topology",
"supr",
"supr_le"
] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
presieve_of_covering_aux {ι : Type v} (U : ι → opens X) (Y : opens X) : presieve Y | λ V f, ∃ i, V = U i | def | Top.presheaf.presieve_of_covering_aux | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
presieve_of_covering {ι : Type v} (U : ι → opens X) : presieve (supr U) | presieve_of_covering_aux U (supr U) | def | Top.presheaf.presieve_of_covering | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"supr"
] | Take `Y` to be `supr U` and obtain a presieve over `supr U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covering_presieve_eq_self {Y : opens X} (R : presieve Y) :
presieve_of_covering_aux (covering_of_presieve Y R) Y = R | by { ext Z f, exact ⟨λ ⟨⟨_,_,h⟩,rfl⟩, by convert h, λ h, ⟨⟨Z,f,h⟩,rfl⟩⟩ } | lemma | Top.presheaf.covering_presieve_eq_self | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [] | Given a presieve `R` on `Y`, if we take its associated family of opens via
`covering_of_presieve` (which may not cover `Y` if `R` is not covering), and take
the presieve on `Y` associated to the family of opens via `presieve_of_covering_aux`,
then we get back the original presieve `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_grothendieck_topology :
sieve.generate (presieve_of_covering U) ∈ opens.grothendieck_topology X (supr U) | begin
intros x hx,
obtain ⟨i, hxi⟩ := opens.mem_supr.mp hx,
exact ⟨U i, opens.le_supr U i, ⟨U i, 𝟙 _, opens.le_supr U i, ⟨i, rfl⟩, category.id_comp _⟩, hxi⟩,
end | lemma | Top.presheaf.presieve_of_covering.mem_grothendieck_topology | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"opens.grothendieck_topology",
"supr"
] | The sieve generated by `presieve_of_covering U` is a member of the grothendieck topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_of_index (i : ι) : Σ V, {f : V ⟶ supr U // presieve_of_covering U f} | ⟨U i, opens.le_supr U i, i, rfl⟩ | def | Top.presheaf.presieve_of_covering.hom_of_index | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"supr"
] | An index `i : ι` can be turned into a dependent pair `(V, f)`, where `V` is an open set and
`f : V ⟶ supr U` is a member of `presieve_of_covering U f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
index_of_hom (f : Σ V, {f : V ⟶ supr U // presieve_of_covering U f}) : ι | f.2.2.some | def | Top.presheaf.presieve_of_covering.index_of_hom | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"supr"
] | By using the axiom of choice, a dependent pair `(V, f)` where `f : V ⟶ supr U` is a member of
`presieve_of_covering U f` can be turned into an index `i : ι`, such that `V = U i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
index_of_hom_spec (f : Σ V, {f : V ⟶ supr U // presieve_of_covering U f}) :
f.1 = U (index_of_hom U f) | f.2.2.some_spec | lemma | Top.presheaf.presieve_of_covering.index_of_hom_spec | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cover_dense_iff_is_basis [category ι] (B : ι ⥤ opens X) :
cover_dense (opens.grothendieck_topology X) B ↔ opens.is_basis (set.range B.obj) | begin
rw opens.is_basis_iff_nbhd,
split, intros hd U x hx, rcases hd.1 U x hx with ⟨V,f,⟨i,f₁,f₂,hc⟩,hV⟩,
exact ⟨B.obj i, ⟨i,rfl⟩, f₁.le hV, f₂.le⟩,
intro hb, split, intros U x hx, rcases hb hx with ⟨_,⟨i,rfl⟩,hx,hi⟩,
exact ⟨B.obj i, ⟨⟨hi⟩⟩, ⟨⟨i, 𝟙 _, ⟨⟨hi⟩⟩, rfl⟩⟩, hx⟩,
end | lemma | Top.opens.cover_dense_iff_is_basis | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"opens.grothendieck_topology",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cover_dense_induced_functor {B : ι → opens X} (h : opens.is_basis (set.range B)) :
cover_dense (opens.grothendieck_topology X) (induced_functor B) | (cover_dense_iff_is_basis _).2 h | lemma | Top.opens.cover_dense_induced_functor | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"opens.grothendieck_topology",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.compatible_preserving (hf : open_embedding f) :
compatible_preserving (opens.grothendieck_topology Y) hf.is_open_map.functor | begin
haveI : mono f := (Top.mono_iff_injective f).mpr hf.inj,
apply compatible_preserving_of_downwards_closed,
intros U V i,
refine ⟨(opens.map f).obj V, eq_to_iso $ opens.ext $ set.image_preimage_eq_of_subset $ λ x h, _⟩,
obtain ⟨_, _, rfl⟩ := i.le h,
exact ⟨_, rfl⟩
end | lemma | open_embedding.compatible_preserving | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"Top.mono_iff_injective",
"open_embedding",
"opens.grothendieck_topology",
"set.image_preimage_eq_of_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map.cover_preserving (hf : is_open_map f) :
cover_preserving (opens.grothendieck_topology X) (opens.grothendieck_topology Y) hf.functor | begin
constructor,
rintros 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⟩
end | lemma | is_open_map.cover_preserving | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"is_open_map",
"opens.grothendieck_topology",
"set.mem_image_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Top.presheaf.is_sheaf_of_open_embedding (h : open_embedding f)
(hF : F.is_sheaf) : is_sheaf (h.is_open_map.functor.op ⋙ F) | pullback_is_sheaf_of_cover_preserving h.compatible_preserving h.is_open_map.cover_preserving ⟨_, hF⟩ | lemma | Top.presheaf.is_sheaf_of_open_embedding | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_terminal_of_empty (F : sheaf C X) : limits.is_terminal (F.val.obj (op ⊥)) | F.is_terminal_of_bot_cover ⊥ (by tidy) | def | Top.sheaf.is_terminal_of_empty | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [] | The empty component of a sheaf is terminal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal_of_eq_empty (F : X.sheaf C) {U : opens X} (h : U = ⊥) :
limits.is_terminal (F.val.obj (op U)) | by convert F.is_terminal_of_empty | def | Top.sheaf.is_terminal_of_eq_empty | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [] | A variant of `is_terminal_of_empty` that is easier to `apply`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_hom_equiv_hom :
((induced_functor B).op ⋙ F ⟶ (induced_functor B).op ⋙ F'.1) ≃ (F ⟶ F'.1) | @cover_dense.restrict_hom_equiv_hom _ _ _ _ _ _ _ _ (opens.cover_dense_induced_functor h)
_ F F' | def | Top.sheaf.restrict_hom_equiv_hom | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_hom_app (α : ((induced_functor B).op ⋙ F ⟶ (induced_functor B).op ⋙ F'.1))
(i : ι) : (restrict_hom_equiv_hom F F' h α).app (op (B i)) = α.app (op i) | by { nth_rewrite 1 ← (restrict_hom_equiv_hom F F' h).left_inv α, refl } | lemma | Top.sheaf.extend_hom_app | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext {α β : F ⟶ F'.1} (he : ∀ i, α.app (op (B i)) = β.app (op (B i))) : α = β | by { apply (restrict_hom_equiv_hom F F' h).symm.injective, ext i, exact he i.unop } | lemma | Top.sheaf.hom_ext | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/sites.lean | [
"category_theory.sites.spaces",
"topology.sheaves.sheaf",
"category_theory.sites.dense_subsite"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compatible (sf : Π i : ι, F.obj (op (U i))) : Prop | ∀ i j : ι, F.map (inf_le_left (U i) (U j)).op (sf i) = F.map (inf_le_right (U i) (U j)).op (sf j) | def | Top.presheaf.is_compatible | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [
"inf_le_left",
"inf_le_right"
] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_gluing (sf : Π i : ι, F.obj (op (U i))) (s : F.obj (op (supr U))) : Prop | ∀ i : ι, F.map (opens.le_supr U i).op s = sf i | def | Top.presheaf.is_gluing | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [
"supr"
] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_unique_gluing : Prop | ∀ ⦃ι : Type v⦄ (U : ι → opens X) (sf : Π i : ι, F.obj (op (U i))),
is_compatible F U sf → ∃! s : F.obj (op (supr U)), is_gluing F U sf s | def | Top.presheaf.is_sheaf_unique_gluing | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [
"supr"
] | 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 (supr U))`.
We prove this to be equivalent to the usual one below in
`is_she... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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