fact
stringlengths 6
3.84k
| type
stringclasses 11
values | library
stringclasses 32
values | imports
listlengths 1
14
| filename
stringlengths 20
95
| symbolic_name
stringlengths 1
90
| docstring
stringlengths 7
20k
⌀ |
|---|---|---|---|---|---|---|
@[simps]
functorNhds (h : IsOpenMap f) (x : X) : OpenNhds x ⥤ OpenNhds (f x) where
obj U := ⟨h.functor.obj U.1, ⟨x, U.2, rfl⟩⟩
map i := h.functor.map i
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
functorNhds
|
An open map `f : X ⟶ Y` induces a functor `OpenNhds x ⥤ OpenNhds (f x)`.
|
adjunctionNhds (h : IsOpenMap f) (x : X) : IsOpenMap.functorNhds h x ⊣ OpenNhds.map f x where
unit := { app := fun _ => homOfLE fun x hxU => ⟨x, hxU, rfl⟩ }
counit := { app := fun _ => homOfLE fun _ ⟨_, hfxV, hxy⟩ => hxy ▸ hfxV }
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
adjunctionNhds
|
An open map `f : X ⟶ Y` induces an adjunction between `OpenNhds x` and `OpenNhds (f x)`.
|
@[simps]
functorNhds (h : IsInducing f) (x : X) :
OpenNhds x ⥤ OpenNhds (f x) where
obj U := ⟨h.functor.obj U.1, (h.mem_functorObj_iff U.1).mpr U.2⟩
map := h.functor.map
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
functorNhds
|
An inducing map `f : X ⟶ Y` induces a functor `open_nhds x ⥤ open_nhds (f x)`.
|
adjunctionNhds (h : IsInducing f) (x : X) :
OpenNhds.map f x ⊣ h.functorNhds x where
unit := { app := fun U => homOfLE (h.adjunction.unit.app U.1).le }
counit := { app := fun U => homOfLE (h.adjunction.counit.app U.1).le }
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
adjunctionNhds
|
An inducing map `f : X ⟶ Y` induces an adjunction between `open_nhds x` and `open_nhds (f x)`.
|
opensHom.instFunLike : FunLike (U ⟶ V) U V where
coe f := Set.inclusion f.le
coe_injective' := by rintro ⟨⟨_⟩⟩ _ _; congr!
|
instance
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
opensHom.instFunLike
| null |
apply_def (f : U ⟶ V) (x : U) : f x = ⟨x, f.le x.2⟩ := rfl
@[simp] lemma apply_mk (f : U ⟶ V) (x : X) (hx) : f ⟨x, hx⟩ = ⟨x, f.le hx⟩ := rfl
@[simp] lemma val_apply (f : U ⟶ V) (x : U) : (f x : X) = x := rfl
@[simp, norm_cast] lemma coe_id (f : U ⟶ U) : ⇑f = id := rfl
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
apply_def
| null |
id_apply (f : U ⟶ U) (x : U) : f x = x := rfl
@[simp] lemma comp_apply (f : U ⟶ V) (g : V ⟶ W) (x : U) : (f ≫ g) x = g (f x) := rfl
/-!
We now construct as morphisms various inclusions of open sets.
-/
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
id_apply
| null |
noncomputable infLELeft (U V : Opens X) : U ⊓ V ⟶ U :=
inf_le_left.hom
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
infLELeft
|
The inclusion `U ⊓ V ⟶ U` as a morphism in the category of open sets.
|
noncomputable infLERight (U V : Opens X) : U ⊓ V ⟶ V :=
inf_le_right.hom
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
infLERight
|
The inclusion `U ⊓ V ⟶ V` as a morphism in the category of open sets.
|
noncomputable leSupr {ι : Type*} (U : ι → Opens X) (i : ι) : U i ⟶ iSup U :=
(le_iSup U i).hom
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
leSupr
|
The inclusion `U i ⟶ iSup U` as a morphism in the category of open sets.
|
noncomputable botLE (U : Opens X) : ⊥ ⟶ U :=
bot_le.hom
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
botLE
|
The inclusion `⊥ ⟶ U` as a morphism in the category of open sets.
|
noncomputable leTop (U : Opens X) : U ⟶ ⊤ :=
le_top.hom
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
leTop
|
The inclusion `U ⟶ ⊤` as a morphism in the category of open sets.
|
infLELeft_apply (U V : Opens X) (x) :
(infLELeft U V) x = ⟨x.1, (@inf_le_left _ _ U V : _ ≤ _) x.2⟩ :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
infLELeft_apply
| null |
infLELeft_apply_mk (U V : Opens X) (x) (m) :
(infLELeft U V) ⟨x, m⟩ = ⟨x, (@inf_le_left _ _ U V : _ ≤ _) m⟩ :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
infLELeft_apply_mk
| null |
leSupr_apply_mk {ι : Type*} (U : ι → Opens X) (i : ι) (x) (m) :
(leSupr U i) ⟨x, m⟩ = ⟨x, (le_iSup U i :) m⟩ :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
leSupr_apply_mk
| null |
toTopCat (X : TopCat.{u}) : Opens X ⥤ TopCat where
obj U := TopCat.of U
map i := TopCat.ofHom ⟨fun x ↦ ⟨x.1, i.le x.2⟩,
IsEmbedding.subtypeVal.continuous_iff.2 continuous_induced_dom⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
toTopCat
|
The functor from open sets in `X` to `TopCat`,
realising each open set as a topological space itself.
|
toTopCat_map (X : TopCat.{u}) {U V : Opens X} {f : U ⟶ V} {x} {h} :
((toTopCat X).map f) ⟨x, h⟩ = ⟨x, f.le h⟩ :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
toTopCat_map
| null |
@[simps! -fullyApplied]
inclusion' {X : TopCat.{u}} (U : Opens X) : (toTopCat X).obj U ⟶ X :=
TopCat.ofHom
{ toFun := _
continuous_toFun := continuous_subtype_val }
@[simp]
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
inclusion'
|
The inclusion map from an open subset to the whole space, as a morphism in `TopCat`.
|
coe_inclusion' {X : TopCat} {U : Opens X} :
(inclusion' U : U → X) = Subtype.val := rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
coe_inclusion'
| null |
isOpenEmbedding {X : TopCat.{u}} (U : Opens X) : IsOpenEmbedding (inclusion' U) :=
U.2.isOpenEmbedding_subtypeVal
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
isOpenEmbedding
| null |
inclusionTopIso (X : TopCat.{u}) : (toTopCat X).obj ⊤ ≅ X where
hom := inclusion' ⊤
inv := TopCat.ofHom ⟨fun x => ⟨x, trivial⟩, continuous_def.2 fun _ ⟨_, hS, hSU⟩ => hSU ▸ hS⟩
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
inclusionTopIso
|
The inclusion of the top open subset (i.e. the whole space) is an isomorphism.
|
map (f : X ⟶ Y) : Opens Y ⥤ Opens X where
obj U := ⟨f ⁻¹' (U : Set Y), U.isOpen.preimage f.hom.continuous⟩
map i := ⟨⟨fun _ h => i.le h⟩⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map
|
`Opens.map f` gives the functor from open sets in Y to open set in X,
given by taking preimages under f.
|
map_coe (f : X ⟶ Y) (U : Opens Y) : ((map f).obj U : Set X) = f ⁻¹' (U : Set Y) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_coe
| null |
map_obj (f : X ⟶ Y) (U) (p) : (map f).obj ⟨U, p⟩ = ⟨f ⁻¹' U, p.preimage f.hom.continuous⟩ :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_obj
| null |
map_homOfLE (f : X ⟶ Y) {U V : Opens Y} (e : U ≤ V) :
(TopologicalSpace.Opens.map f).map (homOfLE e) =
homOfLE (show (Opens.map f).obj U ≤ (Opens.map f).obj V from fun _ hx ↦ e hx) :=
rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_homOfLE
| null |
map_id_obj (U : Opens X) : (map (𝟙 X)).obj U = U :=
let ⟨_, _⟩ := U
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_id_obj
| null |
map_id_obj' (U) (p) : (map (𝟙 X)).obj ⟨U, p⟩ = ⟨U, p⟩ :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_id_obj'
| null |
map_id_obj_unop (U : (Opens X)ᵒᵖ) : (map (𝟙 X)).obj (unop U) = unop U := by
simp
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_id_obj_unop
| null |
op_map_id_obj (U : (Opens X)ᵒᵖ) : (map (𝟙 X)).op.obj U = U := by simp
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
op_map_id_obj
| null |
map_top (f : X ⟶ Y) : (Opens.map f).obj ⊤ = ⊤ := rfl
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_top
| null |
noncomputable leMapTop (f : X ⟶ Y) (U : Opens X) : U ⟶ (map f).obj ⊤ :=
leTop U
@[simp]
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
leMapTop
|
The inclusion `U ⟶ (map f).obj ⊤` as a morphism in the category of open sets.
|
map_comp_obj (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
(map (f ≫ g)).obj U = (map f).obj ((map g).obj U) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_comp_obj
| null |
map_comp_obj' (f : X ⟶ Y) (g : Y ⟶ Z) (U) (p) :
(map (f ≫ g)).obj ⟨U, p⟩ = (map f).obj ((map g).obj ⟨U, p⟩) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_comp_obj'
| null |
map_comp_map (f : X ⟶ Y) (g : Y ⟶ Z) {U V} (i : U ⟶ V) :
(map (f ≫ g)).map i = (map f).map ((map g).map i) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_comp_map
| null |
map_comp_obj_unop (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
(map (f ≫ g)).obj (unop U) = (map f).obj ((map g).obj (unop U)) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_comp_obj_unop
| null |
op_map_comp_obj (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
(map (f ≫ g)).op.obj U = (map f).op.obj ((map g).op.obj U) :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
op_map_comp_obj
| null |
map_iSup (f : X ⟶ Y) {ι : Type*} (U : ι → Opens Y) :
(map f).obj (iSup U) = iSup ((map f).obj ∘ U) := by
ext1; rw [iSup_def, iSup_def, map_obj]
dsimp; rw [Set.preimage_iUnion]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_iSup
| null |
@[simps]
mapId : map (𝟙 X) ≅ 𝟭 (Opens X) where
hom := { app := fun U => eqToHom (map_id_obj U) }
inv := { app := fun U => eqToHom (map_id_obj U).symm }
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
mapId
|
The functor `Opens X ⥤ Opens X` given by taking preimages under the identity function
is naturally isomorphic to the identity functor.
|
map_id_eq : map (𝟙 X) = 𝟭 (Opens X) := by
rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_id_eq
| null |
@[simps]
mapComp (f : X ⟶ Y) (g : Y ⟶ Z) : map (f ≫ g) ≅ map g ⋙ map f where
hom := { app := fun U => eqToHom (map_comp_obj f g U) }
inv := { app := fun U => eqToHom (map_comp_obj f g U).symm }
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
mapComp
|
The natural isomorphism between taking preimages under `f ≫ g`, and the composite
of taking preimages under `g`, then preimages under `f`.
|
map_comp_eq (f : X ⟶ Y) (g : Y ⟶ Z) : map (f ≫ g) = map g ⋙ map f :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_comp_eq
| null |
mapIso (f g : X ⟶ Y) (h : f = g) : map f ≅ map g :=
NatIso.ofComponents fun U => eqToIso (by rw [congr_arg map h])
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
mapIso
|
If two continuous maps `f g : X ⟶ Y` are equal,
then the functors `Opens Y ⥤ Opens X` they induce are isomorphic.
|
map_eq (f g : X ⟶ Y) (h : f = g) : map f = map g := by
subst h
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_eq
| null |
mapIso_refl (f : X ⟶ Y) (h) : mapIso f f h = Iso.refl (map _) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
mapIso_refl
| null |
mapIso_hom_app (f g : X ⟶ Y) (h : f = g) (U : Opens Y) :
(mapIso f g h).hom.app U = eqToHom (by rw [h]) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
mapIso_hom_app
| null |
mapIso_inv_app (f g : X ⟶ Y) (h : f = g) (U : Opens Y) :
(mapIso f g h).inv.app U = eqToHom (by rw [h]) :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
mapIso_inv_app
| null |
@[simps]
mapMapIso {X Y : TopCat.{u}} (H : X ≅ Y) : Opens Y ≌ Opens X where
functor := map H.hom
inverse := map H.inv
unitIso := NatIso.ofComponents fun U => eqToIso (by simp [map, Set.preimage_preimage])
counitIso := NatIso.ofComponents fun U => eqToIso (by simp [map, Set.preimage_preimage])
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
mapMapIso
|
A homeomorphism of spaces gives an equivalence of categories of open sets.
TODO: define `OrderIso.equivalence`, use it.
|
@[simps obj_coe]
IsOpenMap.functor {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) : Opens X ⥤ Opens Y where
obj U := ⟨f '' (U : Set X), hf (U : Set X) U.2⟩
map h := ⟨⟨Set.image_mono h.down.down⟩⟩
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
IsOpenMap.functor
|
An open map `f : X ⟶ Y` induces a functor `Opens X ⥤ Opens Y`.
|
IsOpenMap.adjunction {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) :
hf.functor ⊣ Opens.map f where
unit := { app := fun _ => homOfLE fun x hxU => ⟨x, hxU, rfl⟩ }
counit := { app := fun _ => homOfLE fun _ ⟨_, hfxV, hxy⟩ => hxy ▸ hfxV }
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
IsOpenMap.adjunction
|
An open map `f : X ⟶ Y` induces an adjunction between `Opens X` and `Opens Y`.
|
IsOpenMap.functorFullOfMono {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) [H : Mono f] :
hf.functor.Full where
map_surjective i :=
⟨homOfLE fun x hx => by
obtain ⟨y, hy, eq⟩ := i.le ⟨x, hx, rfl⟩
exact (TopCat.mono_iff_injective f).mp H eq ▸ hy, rfl⟩
|
instance
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
IsOpenMap.functorFullOfMono
| null |
IsOpenMap.functor_faithful {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) :
hf.functor.Faithful where
|
instance
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
IsOpenMap.functor_faithful
| null |
Topology.IsOpenEmbedding.functor_obj_injective {X Y : TopCat} {f : X ⟶ Y}
(hf : IsOpenEmbedding f) : Function.Injective hf.isOpenMap.functor.obj :=
fun _ _ e ↦ Opens.ext (Set.image_injective.mpr hf.injective (congr_arg (↑· : Opens Y → Set Y) e))
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
Topology.IsOpenEmbedding.functor_obj_injective
| null |
@[nolint unusedArguments]
functorObj {X Y : TopCat} {f : X ⟶ Y} (_ : IsInducing f) (U : Opens X) : Opens Y :=
sSup { s : Opens Y | (Opens.map f).obj s = U }
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
functorObj
|
Given an inducing map `X ⟶ Y` and some `U : Opens X`, this is the union of all open sets
whose preimage is `U`. This is right adjoint to `Opens.map`.
|
map_functorObj {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f)
(U : Opens X) :
(Opens.map f).obj (hf.functorObj U) = U := by
apply le_antisymm
· rintro x ⟨_, ⟨s, rfl⟩, _, ⟨rfl : _ = U, rfl⟩, hx : f x ∈ s⟩; exact hx
· intro x hx
obtain ⟨U, hU⟩ := U
obtain ⟨t, ht, rfl⟩ := hf.isOpen_iff.mp hU
exact Opens.mem_sSup.mpr ⟨⟨_, ht⟩, rfl, hx⟩
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_functorObj
| null |
mem_functorObj_iff {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) (U : Opens X)
{x : X} : f x ∈ hf.functorObj U ↔ x ∈ U := by
conv_rhs => rw [← hf.map_functorObj U]
rfl
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
mem_functorObj_iff
| null |
le_functorObj_iff {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) {U : Opens X}
{V : Opens Y} : V ≤ hf.functorObj U ↔ (Opens.map f).obj V ≤ U := by
obtain ⟨U, hU⟩ := U
obtain ⟨t, ht, rfl⟩ := hf.isOpen_iff.mp hU
constructor
· exact fun i x hx ↦ (hf.mem_functorObj_iff ((Opens.map f).obj ⟨t, ht⟩)).mp (i hx)
· intro h x hx
refine Opens.mem_sSup.mpr ⟨⟨_, V.2.union ht⟩, Opens.ext ?_, Set.mem_union_left t hx⟩
dsimp
rwa [Set.union_eq_right]
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
le_functorObj_iff
| null |
opensGI {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) :
GaloisInsertion (Opens.map f).obj hf.functorObj :=
⟨_, fun _ _ ↦ hf.le_functorObj_iff.symm, fun U ↦ (hf.map_functorObj U).ge, fun _ _ ↦ rfl⟩
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
opensGI
|
An inducing map `f : X ⟶ Y` induces a Galois insertion between `Opens Y` and `Opens X`.
|
@[simps]
functor {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) :
Opens X ⥤ Opens Y where
obj := hf.functorObj
map {U V} h := homOfLE (hf.le_functorObj_iff.mpr ((hf.map_functorObj U).trans_le h.le))
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
functor
|
An inducing map `f : X ⟶ Y` induces a functor `Opens X ⥤ Opens Y`.
|
adjunction {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) :
Opens.map f ⊣ hf.functor :=
hf.opensGI.gc.adjunction
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
adjunction
|
An inducing map `f : X ⟶ Y` induces an adjunction between `Opens Y` and `Opens X`.
|
@[simp]
isOpenEmbedding_obj_top {X : TopCat} (U : Opens X) :
U.isOpenEmbedding.isOpenMap.functor.obj ⊤ = U := by
ext1
exact Set.image_univ.trans Subtype.range_coe
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
isOpenEmbedding_obj_top
| null |
inclusion'_map_eq_top {X : TopCat} (U : Opens X) : (Opens.map U.inclusion').obj U = ⊤ := by
ext1
exact Subtype.coe_preimage_self _
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
inclusion'_map_eq_top
| null |
adjunction_counit_app_self {X : TopCat} (U : Opens X) :
U.isOpenEmbedding.isOpenMap.adjunction.counit.app U = eqToHom (by simp) := Subsingleton.elim _ _
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
adjunction_counit_app_self
| null |
inclusion'_top_functor (X : TopCat) :
(@Opens.isOpenEmbedding X ⊤).isOpenMap.functor = map (inclusionTopIso X).inv := by
refine CategoryTheory.Functor.ext ?_ ?_
· intro U
ext x
exact ⟨fun ⟨⟨_, _⟩, h, rfl⟩ => h, fun h => ⟨⟨x, trivial⟩, h, rfl⟩⟩
· subsingleton
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
inclusion'_top_functor
| null |
functor_obj_map_obj {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) (U : Opens Y) :
hf.functor.obj ((Opens.map f).obj U) = hf.functor.obj ⊤ ⊓ U := by
ext
constructor
· rintro ⟨x, hx, rfl⟩
exact ⟨⟨x, trivial, rfl⟩, hx⟩
· rintro ⟨⟨x, -, rfl⟩, hx⟩
exact ⟨x, hx, rfl⟩
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
functor_obj_map_obj
| null |
set_range_inclusion' {X : TopCat} (U : Opens X) :
Set.range (inclusion' U) = (U : Set X) := by
ext x
constructor
· rintro ⟨x, rfl⟩
exact x.2
· intro h
exact ⟨⟨x, h⟩, rfl⟩
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
set_range_inclusion'
| null |
functor_map_eq_inf {X : TopCat} (U V : Opens X) :
U.isOpenEmbedding.isOpenMap.functor.obj ((Opens.map U.inclusion').obj V) = V ⊓ U := by
ext1
simp only [IsOpenMap.coe_functor_obj, map_coe, coe_inf,
Set.image_preimage_eq_inter_range, set_range_inclusion' U]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
functor_map_eq_inf
| null |
map_functor_eq' {X U : TopCat} (f : U ⟶ X) (hf : IsOpenEmbedding f) (V) :
((Opens.map f).obj <| hf.isOpenMap.functor.obj V) = V :=
Opens.ext <| Set.preimage_image_eq _ hf.injective
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_functor_eq'
| null |
map_functor_eq {X : TopCat} {U : Opens X} (V : Opens U) :
((Opens.map U.inclusion').obj <| U.isOpenEmbedding.isOpenMap.functor.obj V) = V :=
TopologicalSpace.Opens.map_functor_eq' _ U.isOpenEmbedding V
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
map_functor_eq
| null |
adjunction_counit_map_functor {X : TopCat} {U : Opens X} (V : Opens U) :
U.isOpenEmbedding.isOpenMap.adjunction.counit.app (U.isOpenEmbedding.isOpenMap.functor.obj V) =
eqToHom (by dsimp; rw [map_functor_eq V]) := by
subsingleton
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Category.GaloisConnection",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
adjunction_counit_map_functor
| null |
noncomputable disk (n : ℕ) : TopCat.{u} :=
TopCat.of <| ULift <| Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) 1
|
def
|
Topology
|
[
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Category.TopCat.Basic"
] |
Mathlib/Topology/Category/TopCat/Sphere.lean
|
disk
|
The `n`-disk is the set of points in ℝⁿ whose norm is at most `1`,
endowed with the subspace topology.
|
noncomputable diskBoundary (n : ℕ) : TopCat.{u} :=
TopCat.of <| ULift <| Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1
|
def
|
Topology
|
[
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Category.TopCat.Basic"
] |
Mathlib/Topology/Category/TopCat/Sphere.lean
|
diskBoundary
|
The boundary of the `n`-disk.
|
noncomputable sphere (n : ℕ) : TopCat.{u} :=
diskBoundary (n + 1)
|
def
|
Topology
|
[
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Category.TopCat.Basic"
] |
Mathlib/Topology/Category/TopCat/Sphere.lean
|
sphere
|
The `n`-sphere is the set of points in ℝⁿ⁺¹ whose norm equals `1`,
endowed with the subspace topology.
|
diskBoundaryInclusion (n : ℕ) : ∂𝔻 n ⟶ 𝔻 n :=
ofHom
{ toFun := fun ⟨p, hp⟩ ↦ ⟨p, le_of_eq hp⟩
continuous_toFun := ⟨fun t ⟨s, ⟨r, hro, hrs⟩, hst⟩ ↦ by
rw [isOpen_induced_iff, ← hst, ← hrs]
tauto⟩ }
|
def
|
Topology
|
[
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Category.TopCat.Basic"
] |
Mathlib/Topology/Category/TopCat/Sphere.lean
|
diskBoundaryInclusion
|
`𝔻 n` denotes the `n`-disk. -/
scoped prefix:arg "𝔻 " => disk
/-- `∂𝔻 n` denotes the boundary of the `n`-disk. -/
scoped prefix:arg "∂𝔻 " => diskBoundary
/-- `𝕊 n` denotes the `n`-sphere. -/
scoped prefix:arg "𝕊 " => sphere
/-- The inclusion `∂𝔻 n ⟶ 𝔻 n` of the boundary of the `n`-disk.
|
uliftFunctor : TopCat.{u} ⥤ TopCat.{max u v} where
obj X := TopCat.of (ULift.{v} X)
map {X Y} f := ofHom ⟨ULift.map f, by continuity⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Homeomorph.Lemmas"
] |
Mathlib/Topology/Category/TopCat/ULift.lean
|
uliftFunctor
|
The functor which sends a topological space in `Type u` to a homeomorphic
space in `Type (max u v)`.
|
uliftFunctorObjHomeo (X : TopCat.{u}) : X ≃ₜ uliftFunctor.{v}.obj X :=
Homeomorph.ulift.symm
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Homeomorph.Lemmas"
] |
Mathlib/Topology/Category/TopCat/ULift.lean
|
uliftFunctorObjHomeo
|
Given `X : TopCat.{u}`, this is the homeomorphism `X ≃ₜ uliftFunctor.{v}.obj X`.
|
uliftFunctorObjHomeo_naturality_apply {X Y : TopCat.{u}} (f : X ⟶ Y) (x : X) :
uliftFunctor.{v}.map f (X.uliftFunctorObjHomeo x) =
Y.uliftFunctorObjHomeo (f x) := rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Homeomorph.Lemmas"
] |
Mathlib/Topology/Category/TopCat/ULift.lean
|
uliftFunctorObjHomeo_naturality_apply
| null |
uliftFunctorObjHomeo_symm_naturality_apply {X Y : TopCat.{u}} (f : X ⟶ Y)
(x : uliftFunctor.{v}.obj X) :
Y.uliftFunctorObjHomeo.symm (uliftFunctor.{v}.map f x) =
f (X.uliftFunctorObjHomeo.symm x) :=
rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Homeomorph.Lemmas"
] |
Mathlib/Topology/Category/TopCat/ULift.lean
|
uliftFunctorObjHomeo_symm_naturality_apply
| null |
@[simps!]
uliftFunctorCompForgetIso : uliftFunctor.{v, u} ⋙ forget TopCat.{max u v} ≅
forget TopCat.{u} ⋙ CategoryTheory.uliftFunctor.{v, u} := Iso.refl _
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Homeomorph.Lemmas"
] |
Mathlib/Topology/Category/TopCat/ULift.lean
|
uliftFunctorCompForgetIso
|
The `ULift` functor on categories of topological spaces is compatible
with the one defined on categories of types.
|
uliftFunctorFullyFaithful : uliftFunctor.{v, u}.FullyFaithful where
preimage f := ofHom ⟨ULift.down ∘ f ∘ ULift.up, by continuity⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.Topology.Homeomorph.Lemmas"
] |
Mathlib/Topology/Category/TopCat/ULift.lean
|
uliftFunctorFullyFaithful
|
The `ULift` functor on categories of topological spaces is fully faithful.
|
@[simps]
yonedaPresheaf : Cᵒᵖ ⥤ Type (max w w') where
obj X := C(F.obj (unop X), Y)
map f g := ContinuousMap.comp g (F.map f.unop).hom
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Limits.Preserves.Finite",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Products"
] |
Mathlib/Topology/Category/TopCat/Yoneda.lean
|
yonedaPresheaf
|
A universe polymorphic "Yoneda presheaf" on `C` given by continuous maps into a topological space
`Y`.
|
@[simps]
yonedaPresheaf' : TopCat.{w}ᵒᵖ ⥤ Type (max w w') where
obj X := C((unop X).1, Y)
map f g := ContinuousMap.comp g f.unop.hom
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Limits.Preserves.Finite",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Products"
] |
Mathlib/Topology/Category/TopCat/Yoneda.lean
|
yonedaPresheaf'
|
A universe polymorphic Yoneda presheaf on `TopCat` given by continuous maps into a topological
space `Y`.
|
comp_yonedaPresheaf' : yonedaPresheaf F Y = F.op ⋙ yonedaPresheaf' Y := rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Limits.Preserves.Finite",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Products"
] |
Mathlib/Topology/Category/TopCat/Yoneda.lean
|
comp_yonedaPresheaf'
| null |
piComparison_fac {α : Type} (X : α → TopCat) :
piComparison (yonedaPresheaf'.{w, w'} Y) (fun x ↦ op (X x)) =
(yonedaPresheaf' Y).map ((opCoproductIsoProduct X).inv ≫ (TopCat.sigmaIsoSigma X).inv.op) ≫
(equivEquivIso (sigmaEquiv Y (fun x ↦ (X x).1))).inv ≫ (Types.productIso _).inv := by
rw [← Category.assoc, Iso.eq_comp_inv]
ext
simp only [yonedaPresheaf', unop_op, piComparison, types_comp_apply,
Types.productIso_hom_comp_eval_apply, Types.pi_lift_π_apply, comp_apply, TopCat.coe_of,
unop_comp, Quiver.Hom.unop_op, sigmaEquiv, equivEquivIso_hom, Equiv.toIso_inv,
Equiv.coe_fn_symm_mk, comp_assoc, sigmaMk_apply, ← opCoproductIsoProduct_inv_comp_ι]
rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Limits.Preserves.Finite",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Products"
] |
Mathlib/Topology/Category/TopCat/Yoneda.lean
|
piComparison_fac
| null |
hypothesis.
In this section we define the relevant projection maps and prove some compatibility results.
|
inductive
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
hypothesis.
| null |
Proj : (I → Bool) → (I → Bool) :=
fun c i ↦ if J i then c i else false
@[simp]
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
Proj
|
The projection mapping everything that satisfies `J i` to itself, and everything else to `false`
|
continuous_proj :
Continuous (Proj J : (I → Bool) → (I → Bool)) := by
dsimp +unfoldPartialApp [Proj]
apply continuous_pi
intro i
split
· apply continuous_apply
· apply continuous_const
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
continuous_proj
| null |
π : Set (I → Bool) := (Proj J) '' C
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
π
|
The image of `Proj π J`
|
@[simps!]
ProjRestrict : C → π C J :=
Set.MapsTo.restrict (Proj J) _ _ (Set.mapsTo_image _ _)
@[simp]
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
ProjRestrict
|
The restriction of `Proj π J` to a subset, mapping to its image.
|
continuous_projRestrict : Continuous (ProjRestrict C J) :=
Continuous.restrict _ (continuous_proj _)
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
continuous_projRestrict
| null |
proj_eq_self {x : I → Bool} (h : ∀ i, x i ≠ false → J i) : Proj J x = x := by
ext i
simp only [Proj, ite_eq_left_iff]
contrapose!
simpa only [ne_comm] using h i
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
proj_eq_self
| null |
proj_prop_eq_self (hh : ∀ i x, x ∈ C → x i ≠ false → J i) : π C J = C := by
ext x
refine ⟨fun ⟨y, hy, h⟩ ↦ ?_, fun h ↦ ⟨x, h, ?_⟩⟩
· rwa [← h, proj_eq_self]; exact (hh · y hy)
· rw [proj_eq_self]; exact (hh · x h)
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
proj_prop_eq_self
| null |
proj_comp_of_subset (h : ∀ i, J i → K i) : (Proj J ∘ Proj K) =
(Proj J : (I → Bool) → (I → Bool)) := by
ext x i; dsimp [Proj]; simp_all
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
proj_comp_of_subset
| null |
proj_eq_of_subset (h : ∀ i, J i → K i) : π (π C K) J = π C J := by
ext x
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· obtain ⟨y, ⟨z, hz, rfl⟩, rfl⟩ := h
refine ⟨z, hz, (?_ : _ = (Proj J ∘ Proj K) z)⟩
rw [proj_comp_of_subset J K h]
· obtain ⟨y, hy, rfl⟩ := h
dsimp [π]
rw [← Set.image_comp]
refine ⟨y, hy, ?_⟩
rw [proj_comp_of_subset J K h]
variable {J K L}
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
proj_eq_of_subset
| null |
@[simps!]
ProjRestricts (h : ∀ i, J i → K i) : π C K → π C J :=
Homeomorph.setCongr (proj_eq_of_subset C J K h) ∘ ProjRestrict (π C K) J
@[simp]
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
ProjRestricts
|
A variant of `ProjRestrict` with domain of the form `π C K`
|
continuous_projRestricts (h : ∀ i, J i → K i) : Continuous (ProjRestricts C h) :=
Continuous.comp (Homeomorph.continuous _) (continuous_projRestrict _ _)
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
continuous_projRestricts
| null |
surjective_projRestricts (h : ∀ i, J i → K i) : Function.Surjective (ProjRestricts C h) :=
(Homeomorph.surjective _).comp (Set.surjective_mapsTo_image_restrict _ _)
variable (J) in
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
surjective_projRestricts
| null |
projRestricts_eq_id : ProjRestricts C (fun i (h : J i) ↦ h) = id := by
ext ⟨x, y, hy, rfl⟩ i
simp +contextual only [π, Proj, ProjRestricts_coe, id_eq, if_true]
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
projRestricts_eq_id
| null |
projRestricts_eq_comp (hJK : ∀ i, J i → K i) (hKL : ∀ i, K i → L i) :
ProjRestricts C hJK ∘ ProjRestricts C hKL = ProjRestricts C (fun i ↦ hKL i ∘ hJK i) := by
ext x i
simp only [π, Proj, Function.comp_apply, ProjRestricts_coe]
simp_all
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
projRestricts_eq_comp
| null |
projRestricts_comp_projRestrict (h : ∀ i, J i → K i) :
ProjRestricts C h ∘ ProjRestrict C K = ProjRestrict C J := by
ext x i
simp only [π, Proj, Function.comp_apply, ProjRestricts_coe, ProjRestrict_coe]
simp_all
variable (J)
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
projRestricts_comp_projRestrict
| null |
iso_map : C(π C J, (IndexFunctor.obj C J)) :=
⟨fun x ↦ ⟨fun i ↦ x.val i.val, by
rcases x with ⟨x, y, hy, rfl⟩
refine ⟨y, hy, ?_⟩
ext ⟨i, hi⟩
simp [precomp, Proj, hi]⟩, by
refine Continuous.subtype_mk (continuous_pi fun i ↦ ?_) _
exact (continuous_apply i.val).comp continuous_subtype_val⟩
|
def
|
Topology
|
[
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.Topology.Category.Profinite.Product",
"Mathlib.Topology.LocallyConstant.Algebra"
] |
Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean
|
iso_map
|
The objectwise map in the isomorphism `spanFunctor ≅ Profinite.indexFunctor`.
|
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