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to_pushforward_of_iso {X Y : Top} (H : X ≅ Y) {ℱ : X.presheaf C} {𝒢 : Y.presheaf C}
(α : H.hom _* ℱ ⟶ 𝒢) : ℱ ⟶ H.inv _* 𝒢 | (presheaf_equiv_of_iso _ H).to_adjunction.hom_equiv ℱ 𝒢 α | def | Top.presheaf.to_pushforward_of_iso | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"Top"
] | If `H : X ≅ Y` is a homeomorphism,
then given an `H _* ℱ ⟶ 𝒢`, we may obtain an `ℱ ⟶ H ⁻¹ _* 𝒢`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_pushforward_of_iso_app {X Y : Top} (H₁ : X ≅ Y) {ℱ : X.presheaf C} {𝒢 : Y.presheaf C}
(H₂ : H₁.hom _* ℱ ⟶ 𝒢) (U : (opens X)ᵒᵖ) :
(to_pushforward_of_iso H₁ H₂).app U =
ℱ.map (eq_to_hom (by simp [opens.map, set.preimage_preimage])) ≫
H₂.app (op ((opens.map H₁.inv).obj (unop U))) | begin
delta to_pushforward_of_iso,
simp only [equiv.to_fun_as_coe, nat_trans.comp_app, equivalence.equivalence_mk'_unit,
eq_to_hom_map, eq_to_hom_op, eq_to_hom_trans, presheaf_equiv_of_iso_unit_iso_hom_app_app,
equivalence.to_adjunction, equivalence.equivalence_mk'_counit,
presheaf_equiv_of_iso_inverse_... | lemma | Top.presheaf.to_pushforward_of_iso_app | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"Top",
"equiv.to_fun_as_coe",
"set.preimage_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushforward_to_of_iso {X Y : Top} (H₁ : X ≅ Y) {ℱ : Y.presheaf C} {𝒢 : X.presheaf C}
(H₂ : ℱ ⟶ H₁.hom _* 𝒢) : H₁.inv _* ℱ ⟶ 𝒢 | ((presheaf_equiv_of_iso _ H₁.symm).to_adjunction.hom_equiv ℱ 𝒢).symm H₂ | def | Top.presheaf.pushforward_to_of_iso | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"Top"
] | If `H : X ≅ Y` is a homeomorphism,
then given an `H _* ℱ ⟶ 𝒢`, we may obtain an `ℱ ⟶ H ⁻¹ _* 𝒢`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushforward_to_of_iso_app {X Y : Top} (H₁ : X ≅ Y) {ℱ : Y.presheaf C} {𝒢 : X.presheaf C}
(H₂ : ℱ ⟶ H₁.hom _* 𝒢) (U : (opens X)ᵒᵖ) :
(pushforward_to_of_iso H₁ H₂).app U =
H₂.app (op ((opens.map H₁.inv).obj (unop U))) ≫
𝒢.map (eq_to_hom (by simp [opens.map, set.preimage_preimage])) | by simpa [pushforward_to_of_iso, equivalence.to_adjunction] | lemma | Top.presheaf.pushforward_to_of_iso_app | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [
"Top",
"set.preimage_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback {X Y : Top.{v}} (f : X ⟶ Y) : Y.presheaf C ⥤ X.presheaf C | Lan (opens.map f).op | def | Top.presheaf.pullback | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_obj_eq_pullback_obj {C} [category C] [has_colimits C] {X Y : Top.{w}}
(f : X ⟶ Y) (ℱ : Y.presheaf C) : (pullback C f).obj ℱ = pullback_obj f ℱ | rfl | lemma | Top.presheaf.pullback_obj_eq_pullback_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_pullback_adjunction {X Y : Top.{v}} (f : X ⟶ Y) :
pullback C f ⊣ pushforward C f | Lan.adjunction _ _ | def | Top.presheaf.pushforward_pullback_adjunction | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | The pullback and pushforward along a continuous map are adjoint to each other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_hom_iso_pushforward_inv {X Y : Top.{v}} (H : X ≅ Y) :
pullback C H.hom ≅ pushforward C H.inv | adjunction.left_adjoint_uniq
(pushforward_pullback_adjunction C H.hom)
(presheaf_equiv_of_iso C H.symm).to_adjunction | def | Top.presheaf.pullback_hom_iso_pushforward_inv | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | Pulling back along a homeomorphism is the same as pushing forward along its inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_inv_iso_pushforward_hom {X Y : Top.{v}} (H : X ≅ Y) :
pullback C H.inv ≅ pushforward C H.hom | adjunction.left_adjoint_uniq
(pushforward_pullback_adjunction C H.inv)
(presheaf_equiv_of_iso C H).to_adjunction | def | Top.presheaf.pullback_inv_iso_pushforward_hom | topology.sheaves | src/topology/sheaves/presheaf.lean | [
"category_theory.limits.kan_extension",
"topology.category.Top.opens",
"category_theory.adjunction.opposites"
] | [] | Pulling back along the inverse of a homeomorphism is the same as pushing forward along it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
presheaf_to_Types (T : X → Type v) : X.presheaf (Type v) | { obj := λ U, Π x : (unop U), T x,
map := λ U V i g, λ (x : unop V), g (i.unop x),
map_id' := λ U, by { ext g ⟨x, hx⟩, refl },
map_comp' := λ U V W i j, rfl } | def | Top.presheaf_to_Types | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [] | The presheaf of dependently typed functions on `X`, with fibres given by a type family `T`.
There is no requirement that the functions are continuous, here. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
presheaf_to_Types_obj
{T : X → Type v} {U : (opens X)ᵒᵖ} :
(presheaf_to_Types X T).obj U = Π x : (unop U), T x | rfl | lemma | Top.presheaf_to_Types_obj | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
presheaf_to_Types_map
{T : X → Type v} {U V : (opens X)ᵒᵖ} {i : U ⟶ V} {f} :
(presheaf_to_Types X T).map i f = λ x, f (i.unop x) | rfl | lemma | Top.presheaf_to_Types_map | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
presheaf_to_Type (T : Type v) : X.presheaf (Type v) | { obj := λ U, (unop U) → T,
map := λ U V i g, g ∘ i.unop,
map_id' := λ U, by { ext g ⟨x, hx⟩, refl },
map_comp' := λ U V W i j, rfl } | def | Top.presheaf_to_Type | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
presheaf_to_Type_obj
{T : Type v} {U : (opens X)ᵒᵖ} :
(presheaf_to_Type X T).obj U = ((unop U) → T) | rfl | lemma | Top.presheaf_to_Type_obj | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
presheaf_to_Type_map
{T : Type v} {U V : (opens X)ᵒᵖ} {i : U ⟶ V} {f} :
(presheaf_to_Type X T).map i f = f ∘ i.unop | rfl | lemma | Top.presheaf_to_Type_map | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
presheaf_to_Top (T : Top.{v}) : X.presheaf (Type v) | (opens.to_Top X).op ⋙ (yoneda.obj T) | def | Top.presheaf_to_Top | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [] | The presheaf of continuous functions on `X` with values in fixed target topological space
`T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
presheaf_to_Top_obj (T : Top.{v}) (U : (opens X)ᵒᵖ) :
(presheaf_to_Top X T).obj U = ((opens.to_Top X).obj (unop U) ⟶ T) | rfl | lemma | Top.presheaf_to_Top_obj | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_functions (X : Top.{v}ᵒᵖ) (R : TopCommRing.{v}) : CommRing.{v} | CommRing.of (unop X ⟶ (forget₂ TopCommRing Top).obj R) | def | Top.continuous_functions | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [
"CommRing.of",
"Top",
"TopCommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback {X Y : Topᵒᵖ} (f : X ⟶ Y) (R : TopCommRing) :
continuous_functions X R ⟶ continuous_functions Y R | { to_fun := λ g, f.unop ≫ g,
map_one' := rfl,
map_zero' := rfl,
map_add' := by tidy,
map_mul' := by tidy } | def | Top.continuous_functions.pullback | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [
"TopCommRing"
] | Pulling back functions into a topological ring along a continuous map is a ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (X : Top.{u}ᵒᵖ) {R S : TopCommRing.{u}} (φ : R ⟶ S) :
continuous_functions X R ⟶ continuous_functions X S | { to_fun := λ g, g ≫ ((forget₂ TopCommRing Top).map φ),
map_one' := by ext; exact φ.1.map_one,
map_zero' := by ext; exact φ.1.map_zero,
map_add' := by intros; ext; apply φ.1.map_add,
map_mul' := by intros; ext; apply φ.1.map_mul } | def | Top.continuous_functions.map | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [
"Top",
"TopCommRing"
] | A homomorphism of topological rings can be postcomposed with functions from a source space `X`;
this is a ring homomorphism (with respect to the pointwise ring operations on functions). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CommRing_yoneda : TopCommRing.{u} ⥤ (Top.{u}ᵒᵖ ⥤ CommRing.{u}) | { obj := λ R,
{ obj := λ X, continuous_functions X R,
map := λ X Y f, continuous_functions.pullback f R,
map_id' := λ X, by { ext, refl },
map_comp' := λ X Y Z f g, rfl },
map := λ R S φ,
{ app := λ X, continuous_functions.map X φ,
naturality' := λ X Y f, rfl },
map_id' := λ X, by { ext, refl },... | def | Top.CommRing_yoneda | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [] | An upgraded version of the Yoneda embedding, observing that the continuous maps
from `X : Top` to `R : TopCommRing` form a commutative ring, functorial in both `X` and `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
presheaf_to_TopCommRing (T : TopCommRing.{v}) :
X.presheaf CommRing.{v} | (opens.to_Top X).op ⋙ (CommRing_yoneda.obj T) | def | Top.presheaf_to_TopCommRing | topology.sheaves | src/topology/sheaves/presheaf_of_functions.lean | [
"category_theory.yoneda",
"topology.sheaves.presheaf",
"topology.category.TopCommRing",
"topology.continuous_function.algebra"
] | [] | The presheaf (of commutative rings), consisting of functions on an open set `U ⊆ X` with
values in some topological commutative ring `T`.
For example, we could construct the presheaf of continuous complex valued functions of `X` as
```
presheaf_to_TopCommRing X (TopCommRing.of ℂ)
```
(this requires `import topology.in... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_of_is_terminal_of_indiscrete {X : Top.{w}} (hind : X.str = ⊤) (F : presheaf C X)
(it : is_terminal $ F.obj $ op ⊥) : F.is_sheaf | λ c U s hs, begin
obtain rfl | hne := eq_or_ne U ⊥,
{ intros _ _, rw @exists_unique_iff_exists _ ⟨λ _ _, _⟩,
{ refine ⟨it.from _, λ U hU hs, is_terminal.hom_ext _ _ _⟩, rwa le_bot_iff.1 hU.le },
{ apply it.hom_ext } },
{ convert presieve.is_sheaf_for_top_sieve _, rw ←sieve.id_mem_iff_eq_top,
have := (... | lemma | Top.presheaf.is_sheaf_of_is_terminal_of_indiscrete | topology.sheaves | src/topology/sheaves/punit.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [
"eq_or_ne",
"exists_unique_iff_exists",
"is_empty_or_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_sheaf_iff_is_terminal_of_indiscrete {X : Top.{w}} (hind : X.str = ⊤)
(F : presheaf C X) : F.is_sheaf ↔ nonempty (is_terminal $ F.obj $ op ⊥) | ⟨λ h, ⟨sheaf.is_terminal_of_empty ⟨F, h⟩⟩, λ ⟨it⟩, is_sheaf_of_is_terminal_of_indiscrete hind F it⟩ | lemma | Top.presheaf.is_sheaf_iff_is_terminal_of_indiscrete | topology.sheaves | src/topology/sheaves/punit.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_sheaf_on_punit_of_is_terminal (F : presheaf C (Top.of punit))
(it : is_terminal $ F.obj $ op ⊥) : F.is_sheaf | is_sheaf_of_is_terminal_of_indiscrete (@subsingleton.elim (topological_space punit) _ _ _) F it | lemma | Top.presheaf.is_sheaf_on_punit_of_is_terminal | topology.sheaves | src/topology/sheaves/punit.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [
"Top.of",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_sheaf_on_punit_iff_is_terminal (F : presheaf C (Top.of punit)) :
F.is_sheaf ↔ nonempty (is_terminal $ F.obj $ op ⊥) | ⟨λ h, ⟨sheaf.is_terminal_of_empty ⟨F, h⟩⟩, λ ⟨it⟩, is_sheaf_on_punit_of_is_terminal F it⟩ | lemma | Top.presheaf.is_sheaf_on_punit_iff_is_terminal | topology.sheaves | src/topology/sheaves/punit.lean | [
"topology.sheaves.sheaf_condition.sites"
] | [
"Top.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_sheaf (F : presheaf.{w v u} C X) : Prop | presheaf.is_sheaf (opens.grothendieck_topology X) F | def | Top.presheaf.is_sheaf | topology.sheaves | src/topology/sheaves/sheaf.lean | [
"topology.sheaves.presheaf",
"category_theory.sites.sheaf",
"category_theory.sites.spaces"
] | [
"opens.grothendieck_topology"
] | The sheaf condition has several different equivalent formulations.
The official definition chosen here is in terms of grothendieck topologies so that the results on
sites could be applied here easily, and this condition does not require additional constraints on
the value category.
The equivalent formulations of the sh... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_unit (F : presheaf (category_theory.discrete unit) X) : F.is_sheaf | λ x U S hS x hx, ⟨eq_to_hom (subsingleton.elim _ _), by tidy, by tidy⟩ | lemma | Top.presheaf.is_sheaf_unit | topology.sheaves | src/topology/sheaves/sheaf.lean | [
"topology.sheaves.presheaf",
"category_theory.sites.sheaf",
"category_theory.sites.spaces"
] | [
"category_theory.discrete"
] | The presheaf valued in `unit` over any topological space is a sheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_iso_iff {F G : presheaf C X} (α : F ≅ G) : F.is_sheaf ↔ G.is_sheaf | presheaf.is_sheaf_of_iso_iff α | lemma | Top.presheaf.is_sheaf_iso_iff | topology.sheaves | src/topology/sheaves/sheaf.lean | [
"topology.sheaves.presheaf",
"category_theory.sites.sheaf",
"category_theory.sites.spaces"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_sheaf_of_iso {F G : presheaf C X} (α : F ≅ G) (h : F.is_sheaf) : G.is_sheaf | (is_sheaf_iso_iff α).1 h | lemma | Top.presheaf.is_sheaf_of_iso | topology.sheaves | src/topology/sheaves/sheaf.lean | [
"topology.sheaves.presheaf",
"category_theory.sites.sheaf",
"category_theory.sites.spaces"
] | [] | Transfer the sheaf condition across an isomorphism of presheaves. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sheaf : Type (max u v w) | Sheaf (opens.grothendieck_topology X) C | def | Top.sheaf | topology.sheaves | src/topology/sheaves/sheaf.lean | [
"topology.sheaves.presheaf",
"category_theory.sites.sheaf",
"category_theory.sites.spaces"
] | [
"opens.grothendieck_topology"
] | A `sheaf C X` is a presheaf of objects from `C` over a (bundled) topological space `X`,
satisfying the sheaf condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sheaf.presheaf (F : X.sheaf C) : Top.presheaf C X | F.1 | abbreviation | Top.sheaf.presheaf | topology.sheaves | src/topology/sheaves/sheaf.lean | [
"topology.sheaves.presheaf",
"category_theory.sites.sheaf",
"category_theory.sites.spaces"
] | [
"Top.presheaf"
] | The underlying presheaf of a sheaf | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sheaf_inhabited : inhabited (sheaf (category_theory.discrete punit) X) | ⟨⟨functor.star _, presheaf.is_sheaf_unit _⟩⟩ | instance | Top.sheaf_inhabited | topology.sheaves | src/topology/sheaves/sheaf.lean | [
"topology.sheaves.presheaf",
"category_theory.sites.sheaf",
"category_theory.sites.spaces"
] | [
"category_theory.discrete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget : Top.sheaf C X ⥤ Top.presheaf C X | Sheaf_to_presheaf _ _ | def | Top.sheaf.forget | topology.sheaves | src/topology/sheaves/sheaf.lean | [
"topology.sheaves.presheaf",
"category_theory.sites.sheaf",
"category_theory.sites.spaces"
] | [
"Top.presheaf",
"Top.sheaf"
] | The forgetful functor from sheaves to presheaves. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_app (F : sheaf C X) (t) : (𝟙 F : F ⟶ F).1.app t = 𝟙 _ | rfl | lemma | Top.sheaf.id_app | topology.sheaves | src/topology/sheaves/sheaf.lean | [
"topology.sheaves.presheaf",
"category_theory.sites.sheaf",
"category_theory.sites.spaces"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_app {F G H : sheaf C X} (f : F ⟶ G) (g : G ⟶ H) (t) :
(f ≫ g).1.app t = f.1.app t ≫ g.1.app t | rfl | lemma | Top.sheaf.comp_app | topology.sheaves | src/topology/sheaves/sheaf.lean | [
"topology.sheaves.presheaf",
"category_theory.sites.sheaf",
"category_theory.sites.spaces"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_germ : prelocal_predicate (λ x, F.stalk x) | { pred := λ U f, ∃ (g : F.obj (op U)), ∀ x : U, f x = F.germ x g,
res := λ V U i f ⟨g, p⟩, ⟨F.map i.op g, λ x, (p (i x)).trans (F.germ_res_apply _ _ _).symm⟩, } | def | Top.presheaf.sheafify.is_germ | topology.sheaves | src/topology/sheaves/sheafify.lean | [
"topology.sheaves.local_predicate",
"topology.sheaves.stalks"
] | [] | The prelocal predicate on functions into the stalks, asserting that the function is equal to a germ. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_locally_germ : local_predicate (λ x, F.stalk x) | (is_germ F).sheafify | def | Top.presheaf.sheafify.is_locally_germ | topology.sheaves | src/topology/sheaves/sheafify.lean | [
"topology.sheaves.local_predicate",
"topology.sheaves.stalks"
] | [] | The local predicate on functions into the stalks,
asserting that the function is locally equal to a germ. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sheafify : sheaf (Type v) X | subsheaf_to_Types (sheafify.is_locally_germ F) | def | Top.presheaf.sheafify | topology.sheaves | src/topology/sheaves/sheafify.lean | [
"topology.sheaves.local_predicate",
"topology.sheaves.stalks"
] | [] | The sheafification of a `Type` valued presheaf, defined as the functions into the stalks which
are locally equal to germs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_sheafify : F ⟶ F.sheafify.1 | { app := λ U f, ⟨λ x, F.germ x f, prelocal_predicate.sheafify_of ⟨f, λ x, rfl⟩⟩,
naturality' := λ U U' f, by { ext x ⟨u, m⟩, exact germ_res_apply F f.unop ⟨u, m⟩ x } } | def | Top.presheaf.to_sheafify | topology.sheaves | src/topology/sheaves/sheafify.lean | [
"topology.sheaves.local_predicate",
"topology.sheaves.stalks"
] | [] | The morphism from a presheaf to its sheafification,
sending each section to its germs.
(This forms the unit of the adjunction.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_to_fiber (x : X) : F.sheafify.presheaf.stalk x ⟶ F.stalk x | stalk_to_fiber (sheafify.is_locally_germ F) x | def | Top.presheaf.stalk_to_fiber | topology.sheaves | src/topology/sheaves/sheafify.lean | [
"topology.sheaves.local_predicate",
"topology.sheaves.stalks"
] | [] | The natural morphism from the stalk of the sheafification to the original stalk.
In `sheafify_stalk_iso` we show this is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_to_fiber_surjective (x : X) : function.surjective (F.stalk_to_fiber x) | begin
apply stalk_to_fiber_surjective,
intro t,
obtain ⟨U, m, s, rfl⟩ := F.germ_exist _ t,
{ use ⟨U, m⟩,
fsplit,
{ exact λ y, F.germ y s, },
{ exact ⟨prelocal_predicate.sheafify_of ⟨s, (λ _, rfl)⟩, rfl⟩, }, },
end | lemma | Top.presheaf.stalk_to_fiber_surjective | topology.sheaves | src/topology/sheaves/sheafify.lean | [
"topology.sheaves.local_predicate",
"topology.sheaves.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_to_fiber_injective (x : X) : function.injective (F.stalk_to_fiber x) | begin
apply stalk_to_fiber_injective,
intros,
rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩,
rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩,
have wUx := wU ⟨x, mU⟩,
dsimp at wUx, erw wUx at e, clear wUx,
have wVx := wV ⟨x, mV⟩,
dsimp at wVx, erw wVx at e, clear wVx,
rcases F.germ_eq x mU mV gU gV e with ... | lemma | Top.presheaf.stalk_to_fiber_injective | topology.sheaves | src/topology/sheaves/sheafify.lean | [
"topology.sheaves.local_predicate",
"topology.sheaves.stalks"
] | [
"category_theory.types_comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sheafify_stalk_iso (x : X) : F.sheafify.presheaf.stalk x ≅ F.stalk x | (equiv.of_bijective _ ⟨stalk_to_fiber_injective _ _, stalk_to_fiber_surjective _ _⟩).to_iso | def | Top.presheaf.sheafify_stalk_iso | topology.sheaves | src/topology/sheaves/sheafify.lean | [
"topology.sheaves.local_predicate",
"topology.sheaves.stalks"
] | [
"equiv.of_bijective"
] | The isomorphism betweeen a stalk of the sheafification and the original stalk. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Types_is_sheaf (T : X → Type u) : (presheaf_to_Types X T).is_sheaf | is_sheaf_of_is_sheaf_unique_gluing_types _ $ λ ι U sf hsf,
-- We use the sheaf condition in terms of unique gluing
-- U is a family of open sets, indexed by `ι` and `sf` is a compatible family of sections.
-- In the informal comments below, I'll just write `U` to represent the union.
begin
-- Our first goal is to def... | lemma | Top.presheaf.to_Types_is_sheaf | topology.sheaves | src/topology/sheaves/sheaf_of_functions.lean | [
"topology.sheaves.presheaf_of_functions",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [
"supr"
] | We show that the presheaf of functions to a type `T`
(no continuity assumptions, just plain functions)
form a sheaf.
In fact, the proof is identical when we do this for dependent functions to a type family `T`,
so we do the more general case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Type_is_sheaf (T : Type u) : (presheaf_to_Type X T).is_sheaf | to_Types_is_sheaf X (λ _, T) | lemma | Top.presheaf.to_Type_is_sheaf | topology.sheaves | src/topology/sheaves/sheaf_of_functions.lean | [
"topology.sheaves.presheaf_of_functions",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | The presheaf of not-necessarily-continuous functions to
a target type `T` satsifies the sheaf condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sheaf_to_Types (T : X → Type u) : sheaf (Type u) X | ⟨presheaf_to_Types X T, presheaf.to_Types_is_sheaf _ _⟩ | def | Top.sheaf_to_Types | topology.sheaves | src/topology/sheaves/sheaf_of_functions.lean | [
"topology.sheaves.presheaf_of_functions",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | The sheaf of not-necessarily-continuous functions on `X` with values in type family
`T : X → Type u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sheaf_to_Type (T : Type u) : sheaf (Type u) X | ⟨presheaf_to_Type X T, presheaf.to_Type_is_sheaf _ _⟩ | def | Top.sheaf_to_Type | topology.sheaves | src/topology/sheaves/sheaf_of_functions.lean | [
"topology.sheaves.presheaf_of_functions",
"topology.sheaves.sheaf_condition.unique_gluing"
] | [] | The sheaf of not-necessarily-continuous functions on `X` with values in a type `T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf : presheaf C X | { obj := λ U, if p₀ ∈ unop U then A else terminal C,
map := λ U V i, if h : p₀ ∈ unop V
then eq_to_hom $ by erw [if_pos h, if_pos (le_of_hom i.unop h)]
else ((if_neg h).symm.rec terminal_is_terminal).from _,
map_id' := λ U, (em (p₀ ∈ U.unop)).elim (λ h, dif_pos h)
(λ h, ((if_neg h).symm.rec terminal_is_... | def | skyscraper_presheaf | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"em",
"hom_ext"
] | A skyscraper presheaf is a presheaf supported at a single point: if `p₀ ∈ X` is a specified
point, then the skyscraper presheaf `𝓕` with value `A` is defined by `U ↦ A` if `p₀ ∈ U` and
`U ↦ *` if `p₀ ∉ A` where `*` is some terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf_eq_pushforward
[hd : Π (U : opens (Top.of punit.{u+1})), decidable (punit.star ∈ U)] :
skyscraper_presheaf p₀ A =
continuous_map.const (Top.of punit) p₀ _* skyscraper_presheaf punit.star A | by convert_to @skyscraper_presheaf X p₀
(λ U, hd $ (opens.map $ continuous_map.const _ p₀).obj U) C _ _ A = _; congr <|> refl | lemma | skyscraper_presheaf_eq_pushforward | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"Top.of",
"continuous_map.const",
"skyscraper_presheaf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skyscraper_presheaf_functor.map' {a b : C} (f : a ⟶ b) :
skyscraper_presheaf p₀ a ⟶ skyscraper_presheaf p₀ b | { app := λ U, if h : p₀ ∈ U.unop
then eq_to_hom (if_pos h) ≫ f ≫ eq_to_hom (if_pos h).symm
else ((if_neg h).symm.rec terminal_is_terminal).from _,
naturality' := λ U V i,
begin
simp only [skyscraper_presheaf_map], by_cases hV : p₀ ∈ V.unop,
{ have hU : p₀ ∈ U.unop := le_of_hom i.unop hV, split_ifs,
... | def | skyscraper_presheaf_functor.map' | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"hom_ext",
"skyscraper_presheaf"
] | Taking skyscraper presheaf at a point is functorial: `c ↦ skyscraper p₀ c` defines a functor by
sending every `f : a ⟶ b` to the natural transformation `α` defined as: `α(U) = f : a ⟶ b` if
`p₀ ∈ U` and the unique morphism to a terminal object in `C` if `p₀ ∉ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf_functor.map'_id {a : C} :
skyscraper_presheaf_functor.map' p₀ (𝟙 a) = 𝟙 _ | begin
ext1, ext1, simp only [skyscraper_presheaf_functor.map'_app, nat_trans.id_app], split_ifs,
{ simp only [category.id_comp, category.comp_id, eq_to_hom_trans, eq_to_hom_refl], },
{ apply ((if_neg h).symm.rec terminal_is_terminal).hom_ext, },
end | lemma | skyscraper_presheaf_functor.map'_id | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"hom_ext",
"skyscraper_presheaf_functor.map'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skyscraper_presheaf_functor.map'_comp {a b c : C} (f : a ⟶ b) (g : b ⟶ c) :
skyscraper_presheaf_functor.map' p₀ (f ≫ g) =
skyscraper_presheaf_functor.map' p₀ f ≫ skyscraper_presheaf_functor.map' p₀ g | begin
ext1, ext1, simp only [skyscraper_presheaf_functor.map'_app, nat_trans.comp_app], split_ifs,
{ simp only [category.assoc, eq_to_hom_trans_assoc, eq_to_hom_refl, category.id_comp], },
{ apply ((if_neg h).symm.rec terminal_is_terminal).hom_ext, },
end | lemma | skyscraper_presheaf_functor.map'_comp | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"hom_ext",
"skyscraper_presheaf_functor.map'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skyscraper_presheaf_functor : C ⥤ presheaf C X | { obj := skyscraper_presheaf p₀,
map := λ _ _, skyscraper_presheaf_functor.map' p₀,
map_id' := λ _, skyscraper_presheaf_functor.map'_id p₀,
map_comp' := λ _ _ _, skyscraper_presheaf_functor.map'_comp p₀ } | def | skyscraper_presheaf_functor | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"skyscraper_presheaf",
"skyscraper_presheaf_functor.map'",
"skyscraper_presheaf_functor.map'_comp",
"skyscraper_presheaf_functor.map'_id"
] | Taking skyscraper presheaf at a point is functorial: `c ↦ skyscraper p₀ c` defines a functor by
sending every `f : a ⟶ b` to the natural transformation `α` defined as: `α(U) = f : a ⟶ b` if
`p₀ ∈ U` and the unique morphism to a terminal object in `C` if `p₀ ∉ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf_cocone_of_specializes {y : X} (h : p₀ ⤳ y) :
cocone ((open_nhds.inclusion y).op ⋙ skyscraper_presheaf p₀ A) | { X := A,
ι := { app := λ U, eq_to_hom $ if_pos $ h.mem_open U.unop.1.2 U.unop.2,
naturality' := λ U V inc, begin
change dite _ _ _ ≫ _ = _, rw dif_pos,
{ erw [category.comp_id, eq_to_hom_trans], refl },
{ exact h.mem_open V.unop.1.2 V.unop.2 },
end } } | def | skyscraper_presheaf_cocone_of_specializes | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"skyscraper_presheaf"
] | The cocone at `A` for the stalk functor of `skyscraper_presheaf p₀ A` when `y ∈ closure {p₀}` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf_cocone_is_colimit_of_specializes
{y : X} (h : p₀ ⤳ y) : is_colimit (skyscraper_presheaf_cocone_of_specializes p₀ A h) | { desc := λ c, eq_to_hom (if_pos trivial).symm ≫ c.ι.app (op ⊤),
fac' := λ c U, begin
rw ← c.w (hom_of_le $ (le_top : unop U ≤ _)).op,
change _ ≫ _ ≫ dite _ _ _ ≫ _ = _,
rw dif_pos,
{ simpa only [skyscraper_presheaf_cocone_of_specializes_ι_app,
eq_to_hom_trans_assoc, eq_to_hom_refl, category.i... | def | skyscraper_presheaf_cocone_is_colimit_of_specializes | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"le_top",
"skyscraper_presheaf_cocone_of_specializes"
] | The cocone at `A` for the stalk functor of `skyscraper_presheaf p₀ A` when `y ∈ closure {p₀}` is a
colimit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf_stalk_of_specializes [has_colimits C]
{y : X} (h : p₀ ⤳ y) : (skyscraper_presheaf p₀ A).stalk y ≅ A | colimit.iso_colimit_cocone ⟨_, skyscraper_presheaf_cocone_is_colimit_of_specializes p₀ A h⟩ | def | skyscraper_presheaf_stalk_of_specializes | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"skyscraper_presheaf",
"skyscraper_presheaf_cocone_is_colimit_of_specializes"
] | If `y ∈ closure {p₀}`, then the stalk of `skyscraper_presheaf p₀ A` at `y` is `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf_cocone (y : X) :
cocone ((open_nhds.inclusion y).op ⋙ skyscraper_presheaf p₀ A) | { X := terminal C,
ι :=
{ app := λ U, terminal.from _,
naturality' := λ U V inc, terminal_is_terminal.hom_ext _ _ } } | def | skyscraper_presheaf_cocone | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"skyscraper_presheaf"
] | The cocone at `*` for the stalk functor of `skyscraper_presheaf p₀ A` when `y ∉ closure {p₀}` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf_cocone_is_colimit_of_not_specializes
{y : X} (h : ¬p₀ ⤳ y) : is_colimit (skyscraper_presheaf_cocone p₀ A y) | let h1 : ∃ (U : open_nhds y), p₀ ∉ U.1 :=
let ⟨U, ho, h₀, hy⟩ := not_specializes_iff_exists_open.mp h in ⟨⟨⟨U, ho⟩, h₀⟩, hy⟩ in
{ desc := λ c, eq_to_hom (if_neg h1.some_spec).symm ≫ c.ι.app (op h1.some),
fac' := λ c U, begin
change _ = c.ι.app (op U.unop),
simp only [← c.w (hom_of_le $ @inf_le_left _ _ h1.s... | def | skyscraper_presheaf_cocone_is_colimit_of_not_specializes | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"hom_ext",
"inf_le_left",
"inf_le_right",
"skyscraper_presheaf_cocone"
] | The cocone at `*` for the stalk functor of `skyscraper_presheaf p₀ A` when `y ∉ closure {p₀}` is a
colimit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf_stalk_of_not_specializes [has_colimits C]
{y : X} (h : ¬p₀ ⤳ y) : (skyscraper_presheaf p₀ A).stalk y ≅ terminal C | colimit.iso_colimit_cocone ⟨_, skyscraper_presheaf_cocone_is_colimit_of_not_specializes _ A h⟩ | def | skyscraper_presheaf_stalk_of_not_specializes | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"skyscraper_presheaf",
"skyscraper_presheaf_cocone_is_colimit_of_not_specializes"
] | If `y ∉ closure {p₀}`, then the stalk of `skyscraper_presheaf p₀ A` at `y` is isomorphic to a
terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf_stalk_of_not_specializes_is_terminal
[has_colimits C] {y : X} (h : ¬p₀ ⤳ y) : is_terminal ((skyscraper_presheaf p₀ A).stalk y) | is_terminal.of_iso terminal_is_terminal $ (skyscraper_presheaf_stalk_of_not_specializes _ _ h).symm | def | skyscraper_presheaf_stalk_of_not_specializes_is_terminal | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"skyscraper_presheaf",
"skyscraper_presheaf_stalk_of_not_specializes"
] | If `y ∉ closure {p₀}`, then the stalk of `skyscraper_presheaf p₀ A` at `y` is a terminal object | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf_is_sheaf : (skyscraper_presheaf p₀ A).is_sheaf | by classical; exact (presheaf.is_sheaf_iso_iff
(eq_to_iso $ skyscraper_presheaf_eq_pushforward p₀ A)).mpr
(sheaf.pushforward_sheaf_of_sheaf _ (presheaf.is_sheaf_on_punit_of_is_terminal _
(by { dsimp, rw if_neg, exact terminal_is_terminal, exact set.not_mem_empty punit.star }))) | lemma | skyscraper_presheaf_is_sheaf | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"set.not_mem_empty",
"skyscraper_presheaf",
"skyscraper_presheaf_eq_pushforward"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skyscraper_sheaf : sheaf C X | ⟨skyscraper_presheaf p₀ A, skyscraper_presheaf_is_sheaf _ _⟩ | def | skyscraper_sheaf | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"skyscraper_presheaf_is_sheaf"
] | The skyscraper presheaf supported at `p₀` with value `A` is the sheaf that assigns `A` to all opens
`U` that contain `p₀` and assigns `*` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_sheaf_functor : C ⥤ sheaf C X | { obj := λ c, skyscraper_sheaf p₀ c,
map := λ a b f, Sheaf.hom.mk $ (skyscraper_presheaf_functor p₀).map f,
map_id' := λ c, Sheaf.hom.ext _ _ $ (skyscraper_presheaf_functor p₀).map_id _,
map_comp' := λ _ _ _ f g, Sheaf.hom.ext _ _ $ (skyscraper_presheaf_functor p₀).map_comp _ _ } | def | skyscraper_sheaf_functor | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"map_comp",
"map_id",
"skyscraper_presheaf_functor",
"skyscraper_sheaf"
] | Taking skyscraper sheaf at a point is functorial: `c ↦ skyscraper p₀ c` defines a functor by
sending every `f : a ⟶ b` to the natural transformation `α` defined as: `α(U) = f : a ⟶ b` if
`p₀ ∈ U` and the unique morphism to a terminal object in `C` if `p₀ ∉ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_skyscraper_presheaf {𝓕 : presheaf C X} {c : C} (f : 𝓕.stalk p₀ ⟶ c) :
𝓕 ⟶ skyscraper_presheaf p₀ c | { app := λ U, if h : p₀ ∈ U.unop
then 𝓕.germ ⟨p₀, h⟩ ≫ f ≫ eq_to_hom (if_pos h).symm
else ((if_neg h).symm.rec terminal_is_terminal).from _,
naturality' := λ U V inc,
begin
dsimp, by_cases hV : p₀ ∈ V.unop,
{ have hU : p₀ ∈ U.unop := le_of_hom inc.unop hV, split_ifs,
erw [←category.assoc, 𝓕.... | def | stalk_skyscraper_presheaf_adjunction_auxs.to_skyscraper_presheaf | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"hom_ext",
"skyscraper_presheaf"
] | If `f : 𝓕.stalk p₀ ⟶ c`, then a natural transformation `𝓕 ⟶ skyscraper_presheaf p₀ c` can be
defined by: `𝓕.germ p₀ ≫ f : 𝓕(U) ⟶ c` if `p₀ ∈ U` and the unique morphism to a terminal object
if `p₀ ∉ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_stalk {𝓕 : presheaf C X} {c : C} (f : 𝓕 ⟶ skyscraper_presheaf p₀ c) :
𝓕.stalk p₀ ⟶ c | let χ : cocone ((open_nhds.inclusion p₀).op ⋙ 𝓕) := cocone.mk c $
{ app := λ U, f.app (op U.unop.1) ≫ eq_to_hom (if_pos U.unop.2),
naturality' := λ U V inc,
begin
dsimp, erw [category.comp_id, ←category.assoc, comp_eq_to_hom_iff, category.assoc,
eq_to_hom_trans, f.naturality, skyscraper_presheaf_map],
... | def | stalk_skyscraper_presheaf_adjunction_auxs.from_stalk | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"skyscraper_presheaf"
] | If `f : 𝓕 ⟶ skyscraper_presheaf p₀ c` is a natural transformation, then there is a morphism
`𝓕.stalk p₀ ⟶ c` defined as the morphism from colimit to cocone at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_skyscraper_from_stalk {𝓕 : presheaf C X} {c : C} (f : 𝓕 ⟶ skyscraper_presheaf p₀ c) :
to_skyscraper_presheaf p₀ (from_stalk _ f) = f | nat_trans.ext _ _ $ funext $ λ U, (em (p₀ ∈ U.unop)).elim
(λ h, by { dsimp, split_ifs, erw [←category.assoc, colimit.ι_desc, category.assoc,
eq_to_hom_trans, eq_to_hom_refl, category.comp_id], refl }) $
λ h, by { dsimp, split_ifs, apply ((if_neg h).symm.rec terminal_is_terminal).hom_ext } | lemma | stalk_skyscraper_presheaf_adjunction_auxs.to_skyscraper_from_stalk | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"em",
"hom_ext",
"skyscraper_presheaf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_stalk_to_skyscraper {𝓕 : presheaf C X} {c : C} (f : 𝓕.stalk p₀ ⟶ c) :
from_stalk p₀ (to_skyscraper_presheaf _ f) = f | colimit.hom_ext $ λ U, by { erw [colimit.ι_desc], dsimp, rw dif_pos U.unop.2, rw [category.assoc,
category.assoc, eq_to_hom_trans, eq_to_hom_refl, category.comp_id, presheaf.germ], congr' 3,
apply_fun opposite.unop using unop_injective, rw [unop_op], ext, refl } | lemma | stalk_skyscraper_presheaf_adjunction_auxs.from_stalk_to_skyscraper | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"opposite.unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit :
𝟭 (presheaf C X) ⟶ presheaf.stalk_functor C p₀ ⋙ skyscraper_presheaf_functor p₀ | { app := λ 𝓕, to_skyscraper_presheaf _ $ 𝟙 _,
naturality' := λ 𝓕 𝓖 f,
begin
ext U, dsimp, split_ifs,
{ simp only [category.id_comp, ←category.assoc], rw [comp_eq_to_hom_iff],
simp only [category.assoc, eq_to_hom_trans, eq_to_hom_refl, category.comp_id],
erw [colimit.ι_map], refl, },
{ ap... | def | stalk_skyscraper_presheaf_adjunction_auxs.unit | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"hom_ext",
"skyscraper_presheaf_functor"
] | The unit in `presheaf.stalk ⊣ skyscraper_presheaf_functor` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit :
(skyscraper_presheaf_functor p₀ ⋙ (presheaf.stalk_functor C p₀ : presheaf C X ⥤ C)) ⟶ 𝟭 C | { app := λ c, (skyscraper_presheaf_stalk_of_specializes p₀ c specializes_rfl).hom,
naturality' := λ x y f, colimit.hom_ext $ λ U,
begin
erw [←category.assoc, colimit.ι_map, colimit.iso_colimit_cocone_ι_hom_assoc,
skyscraper_presheaf_cocone_of_specializes_ι_app, category.assoc, colimit.ι_desc,
whiske... | def | stalk_skyscraper_presheaf_adjunction_auxs.counit | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"category_theory.functor.id_map",
"skyscraper_presheaf_functor",
"skyscraper_presheaf_stalk_of_specializes",
"specializes_rfl"
] | The counit in `presheaf.stalk ⊣ skyscraper_presheaf_functor` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skyscraper_presheaf_stalk_adjunction [has_colimits C] :
(presheaf.stalk_functor C p₀ : presheaf C X ⥤ C) ⊣ skyscraper_presheaf_functor p₀ | { hom_equiv := λ c 𝓕,
{ to_fun := to_skyscraper_presheaf _,
inv_fun := from_stalk _,
left_inv := from_stalk_to_skyscraper _,
right_inv := to_skyscraper_from_stalk _ },
unit := stalk_skyscraper_presheaf_adjunction_auxs.unit _,
counit := stalk_skyscraper_presheaf_adjunction_auxs.counit _,
hom_equiv_u... | def | skyscraper_presheaf_stalk_adjunction | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"equiv.coe_fn_mk",
"equiv.coe_fn_symm_mk",
"hom_ext",
"inv_fun",
"skyscraper_presheaf_functor",
"stalk_skyscraper_presheaf_adjunction_auxs.counit",
"stalk_skyscraper_presheaf_adjunction_auxs.unit"
] | `skyscraper_presheaf_functor` is the right adjoint of `presheaf.stalk_functor` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_skyscraper_sheaf_adjunction [has_colimits C] :
sheaf.forget C X ⋙ presheaf.stalk_functor _ p₀ ⊣ skyscraper_sheaf_functor p₀ | { hom_equiv := λ 𝓕 c,
⟨λ f, ⟨to_skyscraper_presheaf p₀ f⟩, λ g, from_stalk p₀ g.1, from_stalk_to_skyscraper p₀,
λ g, by { ext1, apply to_skyscraper_from_stalk }⟩,
unit :=
{ app := λ 𝓕, ⟨(stalk_skyscraper_presheaf_adjunction_auxs.unit p₀).app 𝓕.1⟩,
naturality' := λ 𝓐 𝓑 ⟨f⟩,
by { ext1, apply (stal... | def | stalk_skyscraper_sheaf_adjunction | topology.sheaves | src/topology/sheaves/skyscraper.lean | [
"topology.sheaves.punit",
"topology.sheaves.stalks",
"topology.sheaves.functors"
] | [
"skyscraper_presheaf_stalk_adjunction",
"skyscraper_sheaf_functor",
"stalk_skyscraper_presheaf_adjunction_auxs.counit",
"stalk_skyscraper_presheaf_adjunction_auxs.unit"
] | Taking stalks of a sheaf is the left adjoint functor to `skyscraper_sheaf_functor` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_functor (x : X) : X.presheaf C ⥤ C | ((whiskering_left _ _ C).obj (open_nhds.inclusion x).op) ⋙ colim | def | Top.presheaf.stalk_functor | 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... | [] | Stalks are functorial with respect to morphisms of presheaves over a fixed `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk (ℱ : X.presheaf C) (x : X) : C | (stalk_functor C x).obj ℱ | def | Top.presheaf.stalk | 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... | [] | The stalk of a presheaf `F` at a point `x` is calculated as the colimit of the functor
nbhds x ⥤ opens F.X ⥤ C | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_functor_obj (ℱ : X.presheaf C) (x : X) :
(stalk_functor C x).obj ℱ = ℱ.stalk x | rfl | lemma | Top.presheaf.stalk_functor_obj | 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 | |
germ (F : X.presheaf C) {U : opens X} (x : U) : F.obj (op U) ⟶ stalk F x | colimit.ι ((open_nhds.inclusion x.1).op ⋙ F) (op ⟨U, x.2⟩) | def | Top.presheaf.germ | 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... | [] | The germ of a section of a presheaf over an open at a point of that open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
germ_res (F : X.presheaf C) {U V : opens X} (i : U ⟶ V) (x : U) :
F.map i.op ≫ germ F x = germ F (i x : V) | let i' : (⟨U, x.2⟩ : open_nhds x.1) ⟶ ⟨V, (i x : V).2⟩ := i in
colimit.w ((open_nhds.inclusion x.1).op ⋙ F) i'.op | lemma | Top.presheaf.germ_res | 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_hom_ext (F : X.presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}
(ih : ∀ (U : opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ | colimit.hom_ext $ λ U, by { induction U using opposite.rec, cases U with U hxU, exact ih U hxU } | lemma | Top.presheaf.stalk_hom_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... | [
"ih",
"opposite.rec"
] | A morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its
composition with the `germ` morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_functor_map_germ {F G : X.presheaf C} (U : opens X) (x : U)
(f : F ⟶ G) : germ F x ≫ (stalk_functor C x.1).map f = f.app (op U) ≫ germ G x | colimit.ι_map (whisker_left ((open_nhds.inclusion x.1).op) f) (op ⟨U, x.2⟩) | lemma | Top.presheaf.stalk_functor_map_germ | 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_pushforward (f : X ⟶ Y) (F : X.presheaf C) (x : X) : (f _* F).stalk (f x) ⟶ F.stalk x | begin
-- This is a hack; Lean doesn't like to elaborate the term written directly.
transitivity,
swap,
exact colimit.pre _ (open_nhds.map f x).op,
exact colim.map (whisker_right (nat_trans.op (open_nhds.inclusion_map_iso f x).inv) F),
end | def | Top.presheaf.stalk_pushforward | 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... | [] | For a presheaf `F` on a space `X`, a continuous map `f : X ⟶ Y` induces a morphisms between the
stalk of `f _ * F` at `f x` and the stalk of `F` at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_pushforward_germ (f : X ⟶ Y) (F : X.presheaf C) (U : opens Y)
(x : (opens.map f).obj U) :
(f _* F).germ ⟨f x, x.2⟩ ≫ F.stalk_pushforward C f x = F.germ x | begin
rw [stalk_pushforward, germ, colimit.ι_map_assoc, colimit.ι_pre, whisker_right_app],
erw [category_theory.functor.map_id, category.id_comp],
refl,
end | lemma | Top.presheaf.stalk_pushforward_germ | 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 | |
id (ℱ : X.presheaf C) (x : X) :
ℱ.stalk_pushforward C (𝟙 X) x = (stalk_functor C x).map ((pushforward.id ℱ).hom) | begin
dsimp [stalk_pushforward, stalk_functor],
ext1,
tactic.op_induction',
rcases j with ⟨⟨_, _⟩, _⟩,
rw [colimit.ι_map_assoc, colimit.ι_map, colimit.ι_pre, whisker_left_app, whisker_right_app,
pushforward.id_hom_app, eq_to_hom_map, eq_to_hom_refl],
dsimp,
-- FIXME A simp lemma which unfortunately... | lemma | Top.presheaf.stalk_pushforward.id | 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... | [
"tactic.op_induction'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (ℱ : X.presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
ℱ.stalk_pushforward C (f ≫ g) x =
((f _* ℱ).stalk_pushforward C g (f x)) ≫ (ℱ.stalk_pushforward C f x) | begin
dsimp [stalk_pushforward, stalk_functor],
ext U,
induction U using opposite.rec,
rcases U with ⟨⟨_, _⟩, _⟩,
simp only [colimit.ι_map_assoc, colimit.ι_pre_assoc,
whisker_right_app, category.assoc],
dsimp,
-- FIXME: Some of these are simp lemmas, but don't fire successfully:
erw [catego... | lemma | Top.presheaf.stalk_pushforward.comp | 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 | |
stalk_pushforward_iso_of_open_embedding {f : X ⟶ Y} (hf : open_embedding f)
(F : X.presheaf C) (x : X) : is_iso (F.stalk_pushforward _ f x) | begin
haveI := functor.initial_of_adjunction (hf.is_open_map.adjunction_nhds x),
convert is_iso.of_iso ((functor.final.colimit_iso (hf.is_open_map.functor_nhds x).op
((open_nhds.inclusion (f x)).op ⋙ f _* F) : _).symm ≪≫ colim.map_iso _),
swap,
{ fapply nat_iso.of_components,
{ intro U,
ref... | lemma | Top.presheaf.stalk_pushforward.stalk_pushforward_iso_of_open_embedding | 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... | [
"open_embedding",
"set.image_preimage_subset",
"set.preimage_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_pullback_hom (f : X ⟶ Y) (F : Y.presheaf C) (x : X) :
F.stalk (f x) ⟶ (pullback_obj f F).stalk x | (stalk_functor _ (f x)).map ((pushforward_pullback_adjunction C f).unit.app F) ≫
stalk_pushforward _ _ _ x | def | Top.presheaf.stalk_pullback_hom | 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... | [] | The morphism `ℱ_{f x} ⟶ (f⁻¹ℱ)ₓ` that factors through `(f_*f⁻¹ℱ)_{f x}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
germ_to_pullback_stalk (f : X ⟶ Y) (F : Y.presheaf C) (U : opens X) (x : U) :
(pullback_obj f F).obj (op U) ⟶ F.stalk (f x) | colimit.desc (Lan.diagram (opens.map f).op F (op U))
{ X := F.stalk (f x),
ι := { app := λ V, F.germ ⟨f x, V.hom.unop.le x.2⟩,
naturality' := λ _ _ i, by { erw category.comp_id, exact F.germ_res i.left.unop _ } } } | def | Top.presheaf.germ_to_pullback_stalk | 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... | [] | The morphism `(f⁻¹ℱ)(U) ⟶ ℱ_{f(x)}` for some `U ∋ x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_pullback_inv (f : X ⟶ Y) (F : Y.presheaf C) (x : X) :
(pullback_obj f F).stalk x ⟶ F.stalk (f x) | colimit.desc ((open_nhds.inclusion x).op ⋙ presheaf.pullback_obj f F)
{ X := F.stalk (f x),
ι := { app := λ U, F.germ_to_pullback_stalk _ f (unop U).1 ⟨x, (unop U).2⟩,
naturality' := λ _ _ _, by { erw [colimit.pre_desc, category.comp_id], congr } } } | def | Top.presheaf.stalk_pullback_inv | 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... | [] | The morphism `(f⁻¹ℱ)ₓ ⟶ ℱ_{f(x)}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_pullback_iso (f : X ⟶ Y) (F : Y.presheaf C) (x : X) :
F.stalk (f x) ≅ (pullback_obj f F).stalk x | { hom := stalk_pullback_hom _ _ _ _,
inv := stalk_pullback_inv _ _ _ _,
hom_inv_id' :=
begin
delta stalk_pullback_hom stalk_pullback_inv stalk_functor presheaf.pullback stalk_pushforward
germ_to_pullback_stalk germ,
ext j,
induction j using opposite.rec,
cases j,
simp only [topological_s... | def | Top.presheaf.stalk_pullback_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",
"topological_space.open_nhds.inclusion_map_iso_inv"
] | The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_specializes (F : X.presheaf C) {x y : X} (h : x ⤳ y) : F.stalk y ⟶ F.stalk x | begin
refine colimit.desc _ ⟨_,λ U, _,_⟩,
{ exact colimit.ι ((open_nhds.inclusion x).op ⋙ F)
(op ⟨(unop U).1, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩) },
{ intros U V i,
dsimp,
rw category.comp_id,
let U' : open_nhds x := ⟨_, (specializes_iff_forall_open.mp h _ (unop U)... | def | Top.presheaf.stalk_specializes | 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... | [] | If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
germ_stalk_specializes (F : X.presheaf C) {U : opens X} {y : U} {x : X} (h : x ⤳ y) :
F.germ y ≫ F.stalk_specializes h =
F.germ (⟨x, h.mem_open U.is_open y.prop⟩ : U) | colimit.ι_desc _ _ | lemma | Top.presheaf.germ_stalk_specializes | 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 | |
germ_stalk_specializes' (F : X.presheaf C) {U : opens X} {x y : X} (h : x ⤳ y) (hy : y ∈ U) :
F.germ ⟨y, hy⟩ ≫ F.stalk_specializes h =
F.germ ⟨x, h.mem_open U.is_open hy⟩ | colimit.ι_desc _ _ | lemma | Top.presheaf.germ_stalk_specializes' | 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_specializes_refl {C : Type*} [category C] [limits.has_colimits C]
{X : Top} (F : X.presheaf C) (x : X) :
F.stalk_specializes (specializes_refl x) = 𝟙 _ | F.stalk_hom_ext $ λ _ _, by { dsimp, simpa } | lemma | Top.presheaf.stalk_specializes_refl | 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... | [
"Top",
"specializes_refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_specializes_comp {C : Type*} [category C] [limits.has_colimits C]
{X : Top} (F : X.presheaf C)
{x y z : X} (h : x ⤳ y) (h' : y ⤳ z) :
F.stalk_specializes h' ≫ F.stalk_specializes h = F.stalk_specializes (h.trans h') | F.stalk_hom_ext $ λ _ _, by simp | lemma | Top.presheaf.stalk_specializes_comp | 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... | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_specializes_stalk_functor_map {F G : X.presheaf C} (f : F ⟶ G) {x y : X} (h : x ⤳ y) :
F.stalk_specializes h ≫ (stalk_functor C x).map f =
(stalk_functor C y).map f ≫ G.stalk_specializes h | by { ext, delta stalk_functor, simpa [stalk_specializes] } | lemma | Top.presheaf.stalk_specializes_stalk_functor_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... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_specializes_stalk_pushforward (f : X ⟶ Y) (F : X.presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalk_specializes (f.map_specializes h) ≫ F.stalk_pushforward _ f x =
F.stalk_pushforward _ f y ≫ F.stalk_specializes h | by { ext, delta stalk_pushforward, simpa [stalk_specializes] } | lemma | Top.presheaf.stalk_specializes_stalk_pushforward | 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_congr {X : Top} {C : Type*} [category C] [has_colimits C]
(F : X.presheaf C) {x y : X}
(e : inseparable x y) : F.stalk x ≅ F.stalk y | ⟨F.stalk_specializes e.ge, F.stalk_specializes e.le, by simp, by simp⟩ | def | Top.presheaf.stalk_congr | 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... | [
"Top",
"inseparable"
] | The stalks are isomorphic on inseparable points | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
germ_ext (F : X.presheaf C) {U V : opens X} {x : X} {hxU : x ∈ U} {hxV : x ∈ V}
(W : opens X) (hxW : x ∈ W) (iWU : W ⟶ U) (iWV : W ⟶ V) {sU : F.obj (op U)} {sV : F.obj (op V)}
(ih : F.map iWU.op sU = F.map iWV.op sV) :
F.germ ⟨x, hxU⟩ sU = F.germ ⟨x, hxV⟩ sV | by erw [← F.germ_res iWU ⟨x, hxW⟩,
← F.germ_res iWV ⟨x, hxW⟩, comp_apply, comp_apply, ih] | lemma | Top.presheaf.germ_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... | [
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
germ_exist (F : X.presheaf C) (x : X) (t : stalk F x) :
∃ (U : opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t | begin
obtain ⟨U, s, e⟩ := types.jointly_surjective.{v v} _
(is_colimit_of_preserves (forget C) (colimit.is_colimit _)) t,
revert s e,
rw [(show U = op (unop U), from rfl)],
generalize : unop U = V, clear U,
cases V with V m,
intros s e,
exact ⟨V, m, s, e⟩,
end | lemma | Top.presheaf.germ_exist | 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... | [] | For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
germ_eq (F : X.presheaf C) {U V : opens X} (x : X) (mU : x ∈ U) (mV : x ∈ V)
(s : F.obj (op U)) (t : F.obj (op V))
(h : germ F ⟨x, mU⟩ s = germ F ⟨x, mV⟩ t) :
∃ (W : opens X) (m : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V), F.map iU.op s = F.map iV.op t | begin
obtain ⟨W, iU, iV, e⟩ := (types.filtered_colimit.is_colimit_eq_iff.{v v} _
(is_colimit_of_preserves _ (colimit.is_colimit ((open_nhds.inclusion x).op ⋙ F)))).mp h,
exact ⟨(unop W).1, (unop W).2, iU.unop, iV.unop, e⟩,
end | lemma | Top.presheaf.germ_eq | 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_map_injective_of_app_injective {F G : presheaf C X} (f : F ⟶ G)
(h : ∀ U : opens X, function.injective (f.app (op U))) (x : X) :
function.injective ((stalk_functor C x).map f) | λ s t hst,
begin
rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩,
rcases germ_exist F x t with ⟨U₂, hxU₂, t, rfl⟩,
simp only [stalk_functor_map_germ_apply _ ⟨x,_⟩] at hst,
obtain ⟨W, hxW, iWU₁, iWU₂, heq⟩ := G.germ_eq x hxU₁ hxU₂ _ _ hst,
rw [← comp_apply, ← comp_apply, ← f.naturality, ← f.naturality, comp_ap... | lemma | Top.presheaf.stalk_functor_map_injective_of_app_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 |
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