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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coe_comap (f : C(α, β)) (U : opens β) : ↑(comap f U) = f ⁻¹' U | rfl | lemma | topological_space.opens.coe_comap | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comp (g : C(β, γ)) (f : C(α, β)) :
comap (g.comp f) = (comap f).comp (comap g) | rfl | lemma | topological_space.opens.comap_comp | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comap (g : C(β, γ)) (f : C(α, β)) (U : opens γ) :
comap f (comap g U) = comap (g.comp f) U | rfl | lemma | topological_space.opens.comap_comap | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_injective [t0_space β] : injective (comap : C(α, β) → frame_hom (opens β) (opens α)) | λ f g h, continuous_map.ext $ λ a, inseparable.eq $ inseparable_iff_forall_open.2 $ λ s hs,
have comap f ⟨s, hs⟩ = comap g ⟨s, hs⟩, from fun_like.congr_fun h ⟨_, hs⟩,
show a ∈ f ⁻¹' s ↔ a ∈ g ⁻¹' s, from set.ext_iff.1 (coe_inj.2 this) a | lemma | topological_space.opens.comap_injective | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"continuous_map.ext",
"frame_hom",
"fun_like.congr_fun",
"inseparable.eq",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.homeomorph.opens_congr (f : α ≃ₜ β) : opens α ≃o opens β | { to_fun := opens.comap f.symm.to_continuous_map,
inv_fun := opens.comap f.to_continuous_map,
left_inv := by { intro U, ext1, exact f.to_equiv.preimage_symm_preimage _ },
right_inv := by { intro U, ext1, exact f.to_equiv.symm_preimage_preimage _ },
map_rel_iff' := λ U V, by simp only [← set_like.coe_subset_coe]... | def | homeomorph.opens_congr | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"inv_fun",
"set_like.coe_subset_coe"
] | A homeomorphism induces an order-preserving equivalence on open sets, by taking comaps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.homeomorph.opens_congr_symm (f : α ≃ₜ β) :
f.opens_congr.symm = f.symm.opens_congr | rfl | lemma | homeomorph.opens_congr_symm | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_nhds_of (x : α) extends opens α | (mem' : x ∈ carrier) | structure | topological_space.open_nhds_of | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | The open neighborhoods of a point. See also `opens` or `nhds`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_opens_injective : injective (to_opens : open_nhds_of x → opens α) | | ⟨_, _⟩ ⟨_, _⟩ rfl := rfl | lemma | topological_space.open_nhds_of.to_opens_injective | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
can_lift_set : can_lift (set α) (open_nhds_of x) coe (λ s, is_open s ∧ x ∈ s) | ⟨λ s hs, ⟨⟨⟨s, hs.1⟩, hs.2⟩, rfl⟩⟩ | instance | topological_space.open_nhds_of.can_lift_set | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"can_lift",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem (U : open_nhds_of x) : x ∈ U | U.mem' | lemma | topological_space.open_nhds_of.mem | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open (U : open_nhds_of x) : is_open (U : set α) | U.is_open' | lemma | topological_space.open_nhds_of.is_open | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_nhds : (𝓝 x).has_basis (λ U : open_nhds_of x, true) coe | (nhds_basis_opens x).to_has_basis (λ U hU, ⟨⟨⟨U, hU.2⟩, hU.1⟩, trivial, subset.rfl⟩)
(λ U _, ⟨U, ⟨⟨U.mem, U.is_open⟩, subset.rfl⟩⟩) | lemma | topological_space.open_nhds_of.basis_nhds | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (f : C(α, β)) (x : α) : lattice_hom (open_nhds_of (f x)) (open_nhds_of x) | { to_fun := λ U, ⟨opens.comap f U.1, U.mem⟩,
map_sup' := λ U V, rfl,
map_inf' := λ U V, rfl } | def | topological_space.open_nhds_of.comap | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"lattice_hom"
] | Preimage of an open neighborhood of `f x` under a continuous map `f` as a `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
opens_find_tac : expr → option auto_cases_tac | | `(topological_space.opens _) := tac_cases
| _ := none | def | tactic.auto_cases.opens_find_tac | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | Find an `auto_cases_tac` which matches `topological_space.opens`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
auto_cases_opens : tactic string | auto_cases tactic.auto_cases.opens_find_tac | def | tactic.auto_cases_opens | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"tactic.auto_cases.opens_find_tac"
] | A version of `tactic.auto_cases` that works for `topological_space.opens`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
clopen_upper_set (α : Type*) [topological_space α] [has_le α] extends clopens α | (upper' : is_upper_set carrier) | structure | clopen_upper_set | topology.sets | src/topology/sets/order.lean | [
"order.upper_lower.basic",
"topology.sets.closeds"
] | [
"is_upper_set",
"topological_space"
] | The type of clopen upper sets of a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper (s : clopen_upper_set α) : is_upper_set (s : set α) | s.upper' | lemma | clopen_upper_set.upper | topology.sets | src/topology/sets/order.lean | [
"order.upper_lower.basic",
"topology.sets.closeds"
] | [
"clopen_upper_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
clopen (s : clopen_upper_set α) : is_clopen (s : set α) | s.clopen' | lemma | clopen_upper_set.clopen | topology.sets | src/topology/sets/order.lean | [
"order.upper_lower.basic",
"topology.sets.closeds"
] | [
"clopen_upper_set",
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_upper_set (s : clopen_upper_set α) : upper_set α | ⟨s, s.upper⟩ | def | clopen_upper_set.to_upper_set | topology.sets | src/topology/sets/order.lean | [
"order.upper_lower.basic",
"topology.sets.closeds"
] | [
"clopen_upper_set",
"upper_set"
] | Reinterpret a upper clopen as an upper set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {s t : clopen_upper_set α} (h : (s : set α) = t) : s = t | set_like.ext' h | lemma | clopen_upper_set.ext | topology.sets | src/topology/sets/order.lean | [
"order.upper_lower.basic",
"topology.sets.closeds"
] | [
"clopen_upper_set",
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (s : clopens α) (h) : (mk s h : set α) = s | rfl | lemma | clopen_upper_set.coe_mk | topology.sets | src/topology/sets/order.lean | [
"order.upper_lower.basic",
"topology.sets.closeds"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (s t : clopen_upper_set α) : (↑(s ⊔ t) : set α) = s ∪ t | rfl | lemma | clopen_upper_set.coe_sup | topology.sets | src/topology/sets/order.lean | [
"order.upper_lower.basic",
"topology.sets.closeds"
] | [
"clopen_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (s t : clopen_upper_set α) : (↑(s ⊓ t) : set α) = s ∩ t | rfl | lemma | clopen_upper_set.coe_inf | topology.sets | src/topology/sets/order.lean | [
"order.upper_lower.basic",
"topology.sets.closeds"
] | [
"clopen_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : (↑(⊤ : clopen_upper_set α) : set α) = univ | rfl | lemma | clopen_upper_set.coe_top | topology.sets | src/topology/sets/order.lean | [
"order.upper_lower.basic",
"topology.sets.closeds"
] | [
"clopen_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : (↑(⊥ : clopen_upper_set α) : set α) = ∅ | rfl | lemma | clopen_upper_set.coe_bot | topology.sets | src/topology/sets/order.lean | [
"order.upper_lower.basic",
"topology.sets.closeds"
] | [
"clopen_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sheaf_is_abelian [has_finite_limits D] : abelian (Sheaf J D) | let adj := sheafification_adjunction J D in abelian_of_adjunction _ _ (as_iso adj.counit) adj | instance | category_theory.Sheaf_is_abelian | topology.sheaves | src/topology/sheaves/abelian.lean | [
"category_theory.abelian.functor_category",
"category_theory.preadditive.additive_functor",
"category_theory.preadditive.functor_category",
"category_theory.abelian.transfer",
"category_theory.sites.left_exact"
] | [
"adj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
presheaf_to_Sheaf_additive : (presheaf_to_Sheaf J D).additive | (presheaf_to_Sheaf J D).additive_of_preserves_binary_biproducts | instance | category_theory.presheaf_to_Sheaf_additive | topology.sheaves | src/topology/sheaves/abelian.lean | [
"category_theory.abelian.functor_category",
"category_theory.preadditive.additive_functor",
"category_theory.preadditive.functor_category",
"category_theory.abelian.transfer",
"category_theory.sites.left_exact"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_comp_preserves_limits :
diagram F U ⋙ G ≅ diagram.{v} (F ⋙ G) U | begin
fapply nat_iso.of_components,
rintro ⟨j⟩,
exact (preserves_product.iso _ _),
exact (preserves_product.iso _ _),
rintros ⟨⟩ ⟨⟩ ⟨⟩,
{ ext, simp, dsimp, simp, }, -- non-terminal `simp`, but `squeeze_simp` fails
{ ext,
simp only [limit.lift_π, functor.comp_map, map_lift_pi_comparison, fan.mk_π_app,
... | def | Top.presheaf.sheaf_condition.diagram_comp_preserves_limits | topology.sheaves | src/topology/sheaves/forget.lean | [
"category_theory.limits.preserves.shapes.products",
"topology.sheaves.sheaf_condition.equalizer_products"
] | [] | When `G` preserves limits, the sheaf condition diagram for `F` composed with `G` is
naturally isomorphic to the sheaf condition diagram for `F ⋙ G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cone_fork : G.map_cone (fork.{v} F U) ≅
(cones.postcompose (diagram_comp_preserves_limits G F U).inv).obj (fork (F ⋙ G) U) | cones.ext (iso.refl _) (λ j,
begin
dsimp, simp [diagram_comp_preserves_limits], cases j; dsimp,
{ rw iso.eq_comp_inv,
ext,
simp, dsimp, simp, },
{ rw iso.eq_comp_inv,
ext,
simp, -- non-terminal `simp`, but `squeeze_simp` fails
dsimp,
simp only [limit.lift_π, fan.mk_π_app, ←G.map_comp, limi... | def | Top.presheaf.sheaf_condition.map_cone_fork | topology.sheaves | src/topology/sheaves/forget.lean | [
"category_theory.limits.preserves.shapes.products",
"topology.sheaves.sheaf_condition.equalizer_products"
] | [] | When `G` preserves limits, the image under `G` of the sheaf condition fork for `F`
is the sheaf condition fork for `F ⋙ G`,
postcomposed with the inverse of the natural isomorphism `diagram_comp_preserves_limits`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_iff_is_sheaf_comp :
presheaf.is_sheaf F ↔ presheaf.is_sheaf (F ⋙ G) | begin
rw [presheaf.is_sheaf_iff_is_sheaf_equalizer_products,
presheaf.is_sheaf_iff_is_sheaf_equalizer_products],
split,
{ intros S ι U,
-- We have that the sheaf condition fork for `F` is a limit fork,
obtain ⟨t₁⟩ := S U,
-- and since `G` preserves limits, the image under `G` of this fork is a lim... | lemma | Top.presheaf.is_sheaf_iff_is_sheaf_comp | topology.sheaves | src/topology/sheaves/forget.lean | [
"category_theory.limits.preserves.shapes.products",
"topology.sheaves.sheaf_condition.equalizer_products"
] | [] | If `G : C ⥤ D` is a functor which reflects isomorphisms and preserves limits
(we assume all limits exist in both `C` and `D`),
then checking the sheaf condition for a presheaf `F : presheaf C X`
is equivalent to checking the sheaf condition for `F ⋙ G`.
The important special case is when
`C` is a concrete category wit... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_diagram :
pairwise.diagram U ⋙ opens.map f = pairwise.diagram ((opens.map f).obj ∘ U) | begin
apply functor.hext,
abstract obj_eq {intro i, cases i; refl},
intros i j g, apply subsingleton.helim,
iterate 2 {rw map_diagram.obj_eq},
end | lemma | Top.presheaf.sheaf_condition_pairwise_intersections.map_diagram | topology.sheaves | src/topology/sheaves/functors.lean | [
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_cocone : (opens.map f).map_cocone (pairwise.cocone U)
== pairwise.cocone ((opens.map f).obj ∘ U) | begin
unfold functor.map_cocone cocones.functoriality, dsimp, congr,
iterate 2 {rw map_diagram, rw opens.map_supr},
apply subsingleton.helim, rw [map_diagram, opens.map_supr],
apply proof_irrel_heq,
end | lemma | Top.presheaf.sheaf_condition_pairwise_intersections.map_cocone | topology.sheaves | src/topology/sheaves/functors.lean | [
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [
"proof_irrel_heq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward_sheaf_of_sheaf {F : presheaf C X}
(h : F.is_sheaf_pairwise_intersections) :
(f _* F).is_sheaf_pairwise_intersections | λ ι U, begin
convert h ((opens.map f).obj ∘ U) using 2,
rw ← map_diagram, refl,
change F.map_cone ((opens.map f).map_cocone _).op == _,
congr, iterate 2 {rw map_diagram}, apply map_cocone,
end | theorem | Top.presheaf.sheaf_condition_pairwise_intersections.pushforward_sheaf_of_sheaf | topology.sheaves | src/topology/sheaves/functors.lean | [
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward_sheaf_of_sheaf
{F : X.presheaf C} (h : F.is_sheaf) : (f _* F).is_sheaf | by rw is_sheaf_iff_is_sheaf_pairwise_intersections at h ⊢;
exact sheaf_condition_pairwise_intersections.pushforward_sheaf_of_sheaf f h | theorem | Top.sheaf.pushforward_sheaf_of_sheaf | topology.sheaves | src/topology/sheaves/functors.lean | [
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | The pushforward of a sheaf (by a continuous map) is a sheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushforward (f : X ⟶ Y) : X.sheaf C ⥤ Y.sheaf C | { obj := λ ℱ, ⟨f _* ℱ.1, pushforward_sheaf_of_sheaf f ℱ.2⟩,
map := λ _ _ g, ⟨pushforward_map f g.1⟩ } | def | Top.sheaf.pushforward | topology.sheaves | src/topology/sheaves/functors.lean | [
"topology.sheaves.sheaf_condition.pairwise_intersections"
] | [] | The pushforward functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_of_is_limit [has_limits C] {X : Top} (F : J ⥤ presheaf.{v} C X)
(H : ∀ j, (F.obj j).is_sheaf) {c : cone F} (hc : is_limit c) : c.X.is_sheaf | begin
let F' : J ⥤ sheaf C X := { obj := λ j, ⟨F.obj j, H j⟩, map := λ X Y f, ⟨F.map f⟩ },
let e : F' ⋙ sheaf.forget C X ≅ F := nat_iso.of_components (λ _, iso.refl _) (by tidy),
exact presheaf.is_sheaf_of_iso ((is_limit_of_preserves (sheaf.forget C X)
(limit.is_limit F')).cone_points_iso_of_nat_iso hc e) (... | lemma | Top.is_sheaf_of_is_limit | topology.sheaves | src/topology/sheaves/limits.lean | [
"topology.sheaves.sheaf",
"category_theory.sites.limits",
"category_theory.limits.functor_category"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_is_sheaf [has_limits C] {X : Top} (F : J ⥤ presheaf.{v} C X)
(H : ∀ j, (F.obj j).is_sheaf) : (limit F).is_sheaf | is_sheaf_of_is_limit F H (limit.is_limit F) | lemma | Top.limit_is_sheaf | topology.sheaves | src/topology/sheaves/limits.lean | [
"topology.sheaves.sheaf",
"category_theory.sites.limits",
"category_theory.limits.functor_category"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_locally_surjective (T : ℱ ⟶ 𝒢) | category_theory.is_locally_surjective (opens.grothendieck_topology X) T | def | Top.presheaf.is_locally_surjective | topology.sheaves | src/topology/sheaves/locally_surjective.lean | [
"topology.sheaves.presheaf",
"topology.sheaves.stalks",
"category_theory.sites.surjective"
] | [
"category_theory.is_locally_surjective",
"opens.grothendieck_topology"
] | A map of presheaves `T : ℱ ⟶ 𝒢` is **locally surjective** if for any open set `U`,
section `t` over `U`, and `x ∈ U`, there exists an open set `x ∈ V ⊆ U` and a section `s` over `V`
such that `$T_*(s_V) = t|_V$`.
See `is_locally_surjective_iff` below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_locally_surjective_iff (T : ℱ ⟶ 𝒢) :
is_locally_surjective T ↔
∀ U t (x ∈ U), ∃ V (ι : V ⟶ U), (∃ s, T.app _ s = t |_ₕ ι) ∧ x ∈ V | iff.rfl | lemma | Top.presheaf.is_locally_surjective_iff | topology.sheaves | src/topology/sheaves/locally_surjective.lean | [
"topology.sheaves.presheaf",
"topology.sheaves.stalks",
"category_theory.sites.surjective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_surjective_iff_surjective_on_stalks (T : ℱ ⟶ 𝒢) :
is_locally_surjective T ↔
∀ (x : X), function.surjective ((stalk_functor C x).map T) | begin
split; intro hT,
{ /- human proof:
Let g ∈ Γₛₜ 𝒢 x be a germ. Represent it on an open set U ⊆ X
as ⟨t, U⟩. By local surjectivity, pass to a smaller open set V
on which there exists s ∈ Γ_ ℱ V mapping to t |_ V.
Then the germ of s maps to g -/
-- Let g ∈ Γₛₜ 𝒢 x be a germ.
intros x g... | lemma | Top.presheaf.locally_surjective_iff_surjective_on_stalks | topology.sheaves | src/topology/sheaves/locally_surjective.lean | [
"topology.sheaves.presheaf",
"topology.sheaves.stalks",
"category_theory.sites.surjective"
] | [] | An equivalent condition for a map of presheaves to be locally surjective
is for all the induced maps on stalks to be surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prelocal_predicate | (pred : Π {U : opens X}, (Π x : U, T x) → Prop)
(res : ∀ {U V : opens X} (i : U ⟶ V) (f : Π x : V, T x) (h : pred f), pred (λ x : U, f (i x))) | structure | Top.prelocal_predicate | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | Given a topological space `X : Top` and a type family `T : X → Type`,
a `P : prelocal_predicate T` consists of:
* a family of predicates `P.pred`, one for each `U : opens X`, of the form `(Π x : U, T x) → Prop`
* a proof that if `f : Π x : V, T x` satisfies the predicate on `V : opens X`, then
the restriction of `f` ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_prelocal (T : Top.{v}) : prelocal_predicate (λ x : X, T) | { pred := λ U f, continuous f,
res := λ U V i f h, continuous.comp h (opens.open_embedding_of_le i.le).continuous, } | def | Top.continuous_prelocal | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [
"continuous",
"continuous.comp"
] | Continuity is a "prelocal" predicate on functions to a fixed topological space `T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_prelocal_predicate (T : Top.{v}) : inhabited (prelocal_predicate (λ x : X, T)) | ⟨continuous_prelocal X T⟩ | instance | Top.inhabited_prelocal_predicate | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | Satisfying the inhabited linter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_predicate extends prelocal_predicate T | (locality : ∀ {U : opens X} (f : Π x : U, T x)
(w : ∀ x : U, ∃ (V : opens X) (m : x.1 ∈ V) (i : V ⟶ U), pred (λ x : V, f (i x : U))), pred f) | structure | Top.local_predicate | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | Given a topological space `X : Top` and a type family `T : X → Type`,
a `P : local_predicate T` consists of:
* a family of predicates `P.pred`, one for each `U : opens X`, of the form `(Π x : U, T x) → Prop`
* a proof that if `f : Π x : V, T x` satisfies the predicate on `V : opens X`, then
the restriction of `f` to ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_local (T : Top.{v}) : local_predicate (λ x : X, T) | { locality := λ U f w,
begin
apply continuous_iff_continuous_at.2,
intro x,
specialize w x,
rcases w with ⟨V, m, i, w⟩,
dsimp at w,
rw continuous_iff_continuous_at at w,
specialize w ⟨x, m⟩,
simpa using (opens.open_embedding_of_le i.le).continuous_at_iff.1 w,
end,
..conti... | def | Top.continuous_local | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [
"continuous_iff_continuous_at"
] | Continuity is a "local" predicate on functions to a fixed topological space `T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_local_predicate (T : Top.{v}) : inhabited (local_predicate _) | ⟨continuous_local X T⟩ | instance | Top.inhabited_local_predicate | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | Satisfying the inhabited linter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prelocal_predicate.sheafify {T : X → Type v} (P : prelocal_predicate T) : local_predicate T | { pred := λ U f, ∀ x : U, ∃ (V : opens X) (m : x.1 ∈ V) (i : V ⟶ U), P.pred (λ x : V, f (i x : U)),
res := λ V U i f w x,
begin
specialize w (i x),
rcases w with ⟨V', m', i', p⟩,
refine ⟨V ⊓ V', ⟨x.2,m'⟩, opens.inf_le_left _ _, _⟩,
convert P.res (opens.inf_le_right V V') _ p,
end,
locality := λ ... | def | Top.prelocal_predicate.sheafify | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | Given a `P : prelocal_predicate`, we can always construct a `local_predicate`
by asking that the condition from `P` holds locally near every point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prelocal_predicate.sheafify_of {T : X → Type v} {P : prelocal_predicate T}
{U : opens X} {f : Π x : U, T x} (h : P.pred f) :
P.sheafify.pred f | λ x, ⟨U, x.2, 𝟙 _, by { convert h, ext ⟨y, w⟩, refl, }⟩ | lemma | Top.prelocal_predicate.sheafify_of | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subpresheaf_to_Types (P : prelocal_predicate T) : presheaf (Type v) X | { obj := λ U, { f : Π x : unop U, T x // P.pred f },
map := λ U V i f, ⟨λ x, f.1 (i.unop x), P.res i.unop f.1 f.2⟩ }. | def | Top.subpresheaf_to_Types | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | The subpresheaf of dependent functions on `X` satisfying the "pre-local" predicate `P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype : subpresheaf_to_Types P ⟶ presheaf_to_Types X T | { app := λ U f, f.1 } | def | Top.subpresheaf_to_Types.subtype | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | The natural transformation including the subpresheaf of functions satisfying a local predicate
into the presheaf of all functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf (P : local_predicate T) :
(subpresheaf_to_Types P.to_prelocal_predicate).is_sheaf | presheaf.is_sheaf_of_is_sheaf_unique_gluing_types _ $ λ ι U sf sf_comp, begin
-- We show the sheaf condition in terms of unique gluing.
-- First we obtain a family of sections for the underlying sheaf of functions,
-- by forgetting that the prediacte holds
let sf' : Π i : ι, (presheaf_to_Types X T).obj (op (U i... | lemma | Top.subpresheaf_to_Types.is_sheaf | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | The functions satisfying a local predicate satisfy the sheaf condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsheaf_to_Types (P : local_predicate T) : sheaf (Type v) X | ⟨subpresheaf_to_Types P.to_prelocal_predicate, subpresheaf_to_Types.is_sheaf P⟩ | def | Top.subsheaf_to_Types | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | The subsheaf of the sheaf of all dependently typed functions satisfying the local predicate `P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_to_fiber (P : local_predicate T) (x : X) :
(subsheaf_to_Types P).presheaf.stalk x ⟶ T x | begin
refine colimit.desc _
{ X := T x, ι := { app := λ U f, _, naturality' := _ } },
{ exact f.1 ⟨x, (unop U).2⟩, },
{ tidy, }
end | def | Top.stalk_to_fiber | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | There is a canonical map from the stalk to the original fiber, given by evaluating sections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_to_fiber_germ (P : local_predicate T) (U : opens X) (x : U) (f) :
stalk_to_fiber P x ((subsheaf_to_Types P).presheaf.germ x f) = f.1 x | begin
dsimp [presheaf.germ, stalk_to_fiber],
cases x,
simp,
refl,
end | lemma | Top.stalk_to_fiber_germ | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_to_fiber_surjective (P : local_predicate T) (x : X)
(w : ∀ (t : T x), ∃ (U : open_nhds x) (f : Π y : U.1, T y) (h : P.pred f), f ⟨x, U.2⟩ = t) :
function.surjective (stalk_to_fiber P x) | λ t,
begin
rcases w t with ⟨U, f, h, rfl⟩,
fsplit,
{ exact (subsheaf_to_Types P).presheaf.germ ⟨x, U.2⟩ ⟨f, h⟩, },
{ exact stalk_to_fiber_germ _ U.1 ⟨x, U.2⟩ ⟨f, h⟩, }
end | lemma | Top.stalk_to_fiber_surjective | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | The `stalk_to_fiber` map is surjective at `x` if
every point in the fiber `T x` has an allowed section passing through it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_to_fiber_injective (P : local_predicate T) (x : X)
(w : ∀ (U V : open_nhds x) (fU : Π y : U.1, T y) (hU : P.pred fU)
(fV : Π y : V.1, T y) (hV : P.pred fV) (e : fU ⟨x, U.2⟩ = fV ⟨x, V.2⟩),
∃ (W : open_nhds x) (iU : W ⟶ U) (iV : W ⟶ V), ∀ (w : W.1), fU (iU w : U.1) = fV (iV w : V.1)) :
function.injecti... | λ tU tV h,
begin
-- We promise to provide all the ingredients of the proof later:
let Q :
∃ (W : (open_nhds x)ᵒᵖ) (s : Π w : (unop W).1, T w) (hW : P.pred s),
tU = (subsheaf_to_Types P).presheaf.germ ⟨x, (unop W).2⟩ ⟨s, hW⟩ ∧
tV = (subsheaf_to_Types P).presheaf.germ ⟨x, (unop W).2⟩ ⟨s, hW⟩ := _,
{... | lemma | Top.stalk_to_fiber_injective | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | The `stalk_to_fiber` map is injective at `x` if any two allowed sections which agree at `x`
agree on some neighborhood of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subpresheaf_continuous_prelocal_iso_presheaf_to_Top (T : Top.{v}) :
subpresheaf_to_Types (continuous_prelocal X T) ≅ presheaf_to_Top X T | nat_iso.of_components
(λ X,
{ hom := by { rintro ⟨f, c⟩, exact ⟨f, c⟩, },
inv := by { rintro ⟨f, c⟩, exact ⟨f, c⟩, },
hom_inv_id' := by { ext ⟨f, p⟩ x, refl, },
inv_hom_id' := by { ext ⟨f, p⟩ x, refl, }, })
(by tidy) | def | Top.subpresheaf_continuous_prelocal_iso_presheaf_to_Top | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | Some repackaging:
the presheaf of functions satisfying `continuous_prelocal` is just the same thing as
the presheaf of continuous functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sheaf_to_Top (T : Top.{v}) : sheaf (Type v) X | ⟨presheaf_to_Top X T,
presheaf.is_sheaf_of_iso (subpresheaf_continuous_prelocal_iso_presheaf_to_Top T)
(subpresheaf_to_Types.is_sheaf (continuous_local X T))⟩ | def | Top.sheaf_to_Top | topology.sheaves | src/topology/sheaves/local_predicate.lean | [
"topology.sheaves.sheaf_of_functions",
"topology.sheaves.stalks",
"topology.local_homeomorph",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | The sheaf of continuous functions on `X` with values in a space `T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submonoid_presheaf [∀ X : C, mul_one_class X]
[∀ (X Y : C), monoid_hom_class (X ⟶ Y) X Y] (F : X.presheaf C) | (obj : ∀ U, submonoid (F.obj U))
(map : ∀ {U V : (opens X)ᵒᵖ} (i : U ⟶ V), (obj U) ≤ (obj V).comap (F.map i)) | structure | Top.presheaf.submonoid_presheaf | topology.sheaves | src/topology/sheaves/operations.lean | [
"algebra.category.Ring.instances",
"algebra.category.Ring.filtered_colimits",
"ring_theory.localization.basic",
"topology.sheaves.stalks"
] | [
"monoid_hom_class",
"mul_one_class",
"submonoid"
] | A subpresheaf with a submonoid structure on each of the components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submonoid_presheaf.localization_presheaf :
X.presheaf CommRing | { obj := λ U, CommRing.of $ localization (G.obj U),
map := λ U V i, CommRing.of_hom $ is_localization.map _ (F.map i) (G.map i),
map_id' := λ U, begin
apply is_localization.ring_hom_ext (G.obj U),
any_goals { dsimp, apply_instance },
refine (is_localization.map_comp _).trans _,
rw F.map_id,
refl... | def | Top.presheaf.submonoid_presheaf.localization_presheaf | topology.sheaves | src/topology/sheaves/operations.lean | [
"algebra.category.Ring.instances",
"algebra.category.Ring.filtered_colimits",
"ring_theory.localization.basic",
"topology.sheaves.stalks"
] | [
"CommRing",
"CommRing.of",
"CommRing.of_hom",
"is_localization.map",
"is_localization.map_comp",
"is_localization.map_comp_map",
"is_localization.ring_hom_ext",
"localization"
] | The localization of a presheaf of `CommRing`s with respect to a `submonoid_presheaf`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submonoid_presheaf.to_localization_presheaf :
F ⟶ G.localization_presheaf | { app := λ U, CommRing.of_hom $ algebra_map (F.obj U) (localization $ G.obj U),
naturality' := λ U V i, (is_localization.map_comp (G.map i)).symm } | def | Top.presheaf.submonoid_presheaf.to_localization_presheaf | topology.sheaves | src/topology/sheaves/operations.lean | [
"algebra.category.Ring.instances",
"algebra.category.Ring.filtered_colimits",
"ring_theory.localization.basic",
"topology.sheaves.stalks"
] | [
"CommRing.of_hom",
"algebra_map",
"is_localization.map_comp",
"localization"
] | The map into the localization presheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submonoid_presheaf_of_stalk (S : ∀ x : X, submonoid (F.stalk x)) :
F.submonoid_presheaf | { obj := λ U, ⨅ x : (unop U), submonoid.comap (F.germ x) (S x),
map := λ U V i, begin
intros s hs,
simp only [submonoid.mem_comap, submonoid.mem_infi] at ⊢ hs,
intro x,
change (F.map i.unop.op ≫ F.germ x) s ∈ _,
rw F.germ_res,
exact hs _,
end } | def | Top.presheaf.submonoid_presheaf_of_stalk | topology.sheaves | src/topology/sheaves/operations.lean | [
"algebra.category.Ring.instances",
"algebra.category.Ring.filtered_colimits",
"ring_theory.localization.basic",
"topology.sheaves.stalks"
] | [
"submonoid",
"submonoid.comap",
"submonoid.mem_comap",
"submonoid.mem_infi"
] | Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
total_quotient_presheaf : X.presheaf CommRing.{w} | (F.submonoid_presheaf_of_stalk (λ x, (F.stalk x)⁰)).localization_presheaf | def | Top.presheaf.total_quotient_presheaf | topology.sheaves | src/topology/sheaves/operations.lean | [
"algebra.category.Ring.instances",
"algebra.category.Ring.filtered_colimits",
"ring_theory.localization.basic",
"topology.sheaves.stalks"
] | [] | The localization of a presheaf of `CommRing`s at locally non-zero-divisor sections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_total_quotient_presheaf : F ⟶ F.total_quotient_presheaf | submonoid_presheaf.to_localization_presheaf _ | def | Top.presheaf.to_total_quotient_presheaf | topology.sheaves | src/topology/sheaves/operations.lean | [
"algebra.category.Ring.instances",
"algebra.category.Ring.filtered_colimits",
"ring_theory.localization.basic",
"topology.sheaves.stalks"
] | [] | The map into the presheaf of total quotient rings | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
presheaf (X : Top.{w}) : Type (max u v w) | (opens X)ᵒᵖ ⥤ C | def | Top.presheaf | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | The category of `C`-valued presheaves on a (bundled) topological space `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_attr : user_attribute (tactic unit → tactic unit) unit | { name := `sheaf_restrict,
descr := "tag lemmas to use in `Top.presheaf.restrict_tac`",
cache_cfg :=
{ mk_cache := λ ns, pure $ λ t, do
{ ctx <- tactic.local_context,
ctx.any_of (tactic.focus1 ∘ (tactic.apply' >=> (λ _, tactic.done)) >=> (λ _, t)) <|>
ns.any_of (tactic.focus1 ∘ (tactic.re... | def | Top.presheaf.restrict_attr | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"tactic.apply'"
] | Tag lemmas to use in `Top.presheaf.restrict_tac`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_tac : Π (n : ℕ), tactic unit | | 0 := tactic.fail "`restrict_tac` failed"
| (n + 1) := monad.join (restrict_attr.get_cache <*> pure tactic.done) <|>
`[apply' le_trans, mjoin (restrict_attr.get_cache <*> pure (restrict_tac n))] | def | Top.presheaf.restrict_tac | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | A tactic to discharge goals of type `U ≤ V` for `Top.presheaf.restrict_open` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_tac' | restrict_tac 3 | def | Top.presheaf.restrict_tac' | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | A tactic to discharge goals of type `U ≤ V` for `Top.presheaf.restrict_open`.
Defaults to three iterations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict {X : Top} {C : Type*} [category C] [concrete_category C]
{F : X.presheaf C} {V : opens X} (x : F.obj (op V)) {U : opens X} (h : U ⟶ V) : F.obj (op U) | F.map h.op x | def | Top.presheaf.restrict | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"Top"
] | The restriction of a section along an inclusion of open sets.
For `x : F.obj (op V)`, we provide the notation `x |_ₕ i` (`h` stands for `hom`) for `i : U ⟶ V`,
and the notation `x |_ₗ U ⟪i⟫` (`l` stands for `le`) for `i : U ≤ V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_open {X : Top} {C : Type*} [category C] [concrete_category C]
{F : X.presheaf C} {V : opens X} (x : F.obj (op V)) (U : opens X)
(e : U ≤ V . Top.presheaf.restrict_tac') : F.obj (op U) | x |_ₗ U ⟪e⟫ | abbreviation | Top.presheaf.restrict_open | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"Top",
"Top.presheaf.restrict_tac'"
] | The restriction of a section along an inclusion of open sets.
For `x : F.obj (op V)`, we provide the notation `x |_ U`, where the proof `U ≤ V` is inferred by
the tactic `Top.presheaf.restrict_tac'` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_restrict {X : Top} {C : Type*} [category C] [concrete_category C]
{F : X.presheaf C} {U V W : opens X} (e₁ : U ≤ V) (e₂ : V ≤ W) (x : F.obj (op W)) :
x |_ V |_ U = x |_ U | by { delta restrict_open restrict, rw [← comp_apply, ← functor.map_comp], refl } | lemma | Top.presheaf.restrict_restrict | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_restrict {X : Top} {C : Type*} [category C] [concrete_category C]
{F G : X.presheaf C} (e : F ⟶ G) {U V : opens X} (h : U ≤ V) (x : F.obj (op V)) :
e.app _ (x |_ U) = (e.app _ x) |_ U | by { delta restrict_open restrict, rw [← comp_apply, nat_trans.naturality, comp_apply] } | lemma | Top.presheaf.map_restrict | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward_obj {X Y : Top.{w}} (f : X ⟶ Y) (ℱ : X.presheaf C) : Y.presheaf C | (opens.map f).op ⋙ ℱ | def | Top.presheaf.pushforward_obj | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | Pushforward a presheaf on `X` along a continuous map `f : X ⟶ Y`, obtaining a presheaf
on `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushforward_obj_obj {X Y : Top.{w}} (f : X ⟶ Y) (ℱ : X.presheaf C) (U : (opens Y)ᵒᵖ) :
(f _* ℱ).obj U = ℱ.obj ((opens.map f).op.obj U) | rfl | lemma | Top.presheaf.pushforward_obj_obj | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward_obj_map {X Y : Top.{w}} (f : X ⟶ Y) (ℱ : X.presheaf C)
{U V : (opens Y)ᵒᵖ} (i : U ⟶ V) :
(f _* ℱ).map i = ℱ.map ((opens.map f).op.map i) | rfl | lemma | Top.presheaf.pushforward_obj_map | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward_eq {X Y : Top.{w}} {f g : X ⟶ Y} (h : f = g) (ℱ : X.presheaf C) :
f _* ℱ ≅ g _* ℱ | iso_whisker_right (nat_iso.op (opens.map_iso f g h).symm) ℱ | def | Top.presheaf.pushforward_eq | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | An equality of continuous maps induces a natural isomorphism between the pushforwards of a presheaf
along those maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushforward_eq' {X Y : Top.{w}} {f g : X ⟶ Y} (h : f = g) (ℱ : X.presheaf C) :
f _* ℱ = g _* ℱ | by rw h | lemma | Top.presheaf.pushforward_eq' | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward_eq_hom_app
{X Y : Top.{w}} {f g : X ⟶ Y} (h : f = g) (ℱ : X.presheaf C) (U) :
(pushforward_eq h ℱ).hom.app U =
ℱ.map (begin dsimp [functor.op], apply quiver.hom.op, apply eq_to_hom, rw h, end) | by simp [pushforward_eq] | lemma | Top.presheaf.pushforward_eq_hom_app | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"quiver.hom.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward_eq'_hom_app
{X Y : Top.{w}} {f g : X ⟶ Y} (h : f = g) (ℱ : X.presheaf C) (U) :
nat_trans.app (eq_to_hom (pushforward_eq' h ℱ)) U = ℱ.map (eq_to_hom (by rw h)) | by simpa [eq_to_hom_map] | lemma | Top.presheaf.pushforward_eq'_hom_app | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward_eq_rfl {X Y : Top.{w}} (f : X ⟶ Y) (ℱ : X.presheaf C) (U) :
(pushforward_eq (rfl : f = f) ℱ).hom.app (op U) = 𝟙 _ | begin
dsimp [pushforward_eq],
simp,
end | lemma | Top.presheaf.pushforward_eq_rfl | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward_eq_eq {X Y : Top.{w}} {f g : X ⟶ Y} (h₁ h₂ : f = g) (ℱ : X.presheaf C) :
ℱ.pushforward_eq h₁ = ℱ.pushforward_eq h₂ | rfl | lemma | Top.presheaf.pushforward_eq_eq | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : (𝟙 X) _* ℱ ≅ ℱ | (iso_whisker_right (nat_iso.op (opens.map_id X).symm) ℱ) ≪≫ functor.left_unitor _ | def | Top.presheaf.pushforward.id | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | The natural isomorphism between the pushforward of a presheaf along the identity continuous map
and the original presheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_eq : (𝟙 X) _* ℱ = ℱ | by { unfold pushforward_obj, rw opens.map_id_eq, erw functor.id_comp } | lemma | Top.presheaf.pushforward.id_eq | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_hom_app' (U) (p) :
(id ℱ).hom.app (op ⟨U, p⟩) = ℱ.map (𝟙 (op ⟨U, p⟩)) | by { dsimp [id], simp, } | lemma | Top.presheaf.pushforward.id_hom_app' | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_hom_app (U) :
(id ℱ).hom.app U = ℱ.map (eq_to_hom (opens.op_map_id_obj U)) | begin
-- was `tidy`
induction U using opposite.rec,
cases U,
rw [id_hom_app'],
congr
end | lemma | Top.presheaf.pushforward.id_hom_app | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_inv_app' (U) (p) : (id ℱ).inv.app (op ⟨U, p⟩) = ℱ.map (𝟙 (op ⟨U, p⟩)) | by { dsimp [id], simp, } | lemma | Top.presheaf.pushforward.id_inv_app' | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {Y Z : Top.{w}} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g) _* ℱ ≅ g _* (f _* ℱ) | iso_whisker_right (nat_iso.op (opens.map_comp f g).symm) ℱ | def | Top.presheaf.pushforward.comp | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | The natural isomorphism between
the pushforward of a presheaf along the composition of two continuous maps and
the corresponding pushforward of a pushforward. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_eq {Y Z : Top.{w}} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g) _* ℱ = g _* (f _* ℱ) | rfl | lemma | Top.presheaf.pushforward.comp_eq | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"comp_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_hom_app {Y Z : Top.{w}} (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
(comp ℱ f g).hom.app U = 𝟙 _ | by { dsimp [comp], tidy, } | lemma | Top.presheaf.pushforward.comp_hom_app | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_inv_app {Y Z : Top.{w}} (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
(comp ℱ f g).inv.app U = 𝟙 _ | by { dsimp [comp], tidy, } | lemma | Top.presheaf.pushforward.comp_inv_app | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward_map {X Y : Top.{w}} (f : X ⟶ Y) {ℱ 𝒢 : X.presheaf C} (α : ℱ ⟶ 𝒢) :
f _* ℱ ⟶ f _* 𝒢 | { app := λ U, α.app _,
naturality' := λ U V i, by { erw α.naturality, refl, } } | def | Top.presheaf.pushforward_map | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | A morphism of presheaves gives rise to a morphisms of the pushforwards of those presheaves. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_obj {X Y : Top.{v}} (f : X ⟶ Y) (ℱ : Y.presheaf C) : X.presheaf C | (Lan (opens.map f).op).obj ℱ | def | Top.presheaf.pullback_obj | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | Pullback a presheaf on `Y` along a continuous map `f : X ⟶ Y`, obtaining a presheaf on `X`.
This is defined in terms of left Kan extensions, which is just a fancy way of saying
"take the colimits over the open sets whose preimage contains U". | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_map {X Y : Top.{v}} (f : X ⟶ Y) {ℱ 𝒢 : Y.presheaf C} (α : ℱ ⟶ 𝒢) :
pullback_obj f ℱ ⟶ pullback_obj f 𝒢 | (Lan (opens.map f).op).map α | def | Top.presheaf.pullback_map | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | Pulling back along continuous maps is functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_obj_obj_of_image_open {X Y : Top.{v}} (f : X ⟶ Y) (ℱ : Y.presheaf C) (U : opens X)
(H : is_open (f '' U)) : (pullback_obj f ℱ).obj (op U) ≅ ℱ.obj (op ⟨_, H⟩) | begin
let x : costructured_arrow (opens.map f).op (op U) := begin
refine @costructured_arrow.mk _ _ _ _ _ (op (opens.mk (f '' U) H)) _ _,
exact ((@hom_of_le _ _ _ ((opens.map f).obj ⟨_, H⟩) (set.image_preimage.le_u_l _)).op),
end,
have hx : is_terminal x :=
{ lift := λ s,
begin
fapply costruct... | def | Top.presheaf.pullback_obj_obj_of_image_open | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"is_open",
"lift",
"set.image_subset"
] | If `f '' U` is open, then `f⁻¹ℱ U ≅ ℱ (f '' U)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : pullback_obj (𝟙 _) ℱ ≅ ℱ | nat_iso.of_components
(λ U, pullback_obj_obj_of_image_open (𝟙 _) ℱ (unop U) (by simpa using U.unop.2) ≪≫
ℱ.map_iso (eq_to_iso (by simp)))
(λ U V i,
begin
ext, simp,
erw colimit.pre_desc_assoc,
erw colimit.ι_desc_assoc,
erw colimit.ι_desc_assoc,
dsimp, simp only [←ℱ.map_comp], co... | def | Top.presheaf.pullback.id | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | The pullback along the identity is isomorphic to the original presheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_inv_app (U : opens Y) :
(id ℱ).inv.app (op U) = colimit.ι (Lan.diagram (opens.map (𝟙 Y)).op ℱ (op U))
(@costructured_arrow.mk _ _ _ _ _ (op U) _ (eq_to_hom (by simp))) | begin
rw [← category.id_comp ((id ℱ).inv.app (op U)), ← nat_iso.app_inv, iso.comp_inv_eq],
dsimp [id],
rw colimit.ι_desc_assoc,
dsimp,
rw [← ℱ.map_comp, ← ℱ.map_id], refl,
end | lemma | Top.presheaf.pullback.id_inv_app | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward {X Y : Top.{w}} (f : X ⟶ Y) : X.presheaf C ⥤ Y.presheaf C | { obj := pushforward_obj f,
map := @pushforward_map _ _ X Y f } | def | Top.presheaf.pushforward | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | The pushforward functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushforward_map_app' {X Y : Top.{w}} (f : X ⟶ Y)
{ℱ 𝒢 : X.presheaf C} (α : ℱ ⟶ 𝒢) {U : (opens Y)ᵒᵖ} :
((pushforward C f).map α).app U = α.app (op $ (opens.map f).obj U.unop) | rfl | lemma | Top.presheaf.pushforward_map_app' | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_pushforward {X : Top.{w}} : pushforward C (𝟙 X) = 𝟭 (X.presheaf C) | begin
apply category_theory.functor.ext,
{ intros,
ext U,
have h := f.congr, erw h (opens.op_map_id_obj U),
simpa [eq_to_hom_map], },
{ intros, apply pushforward.id_eq },
end | lemma | Top.presheaf.id_pushforward | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"category_theory.functor.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
presheaf_equiv_of_iso {X Y : Top} (H : X ≅ Y) :
X.presheaf C ≌ Y.presheaf C | equivalence.congr_left (opens.map_map_iso H).symm.op | def | Top.presheaf.presheaf_equiv_of_iso | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"Top"
] | A homeomorphism of spaces gives an equivalence of categories of presheaves. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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