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noncomputable
isoindexConeLift :
@Profinite.of C _ (by rwa [← isCompact_iff_compactSpace]) _ _ ≅
(Profinite.limitCone.{u, u} (indexFunctor hC)).pt :=
asIso <| (Profinite.limitConeIsLimit.{u, u} _).lift (indexCone hC)
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Profinite/Product.lean
|
isoindexConeLift
|
The canonical map from `C` to the explicit limit as an isomorphism.
|
noncomputable
asLimitindexConeIso : indexCone hC ≅ Profinite.limitCone.{u, u} _ :=
Limits.Cones.ext (isoindexConeLift hC) fun _ => rfl
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Profinite/Product.lean
|
asLimitindexConeIso
|
The isomorphism of cones induced by `isoindexConeLift`.
|
noncomputable
indexCone_isLimit : CategoryTheory.Limits.IsLimit (indexCone hC) :=
Limits.IsLimit.ofIsoLimit (Profinite.limitConeIsLimit _) (asLimitindexConeIso hC).symm
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Profinite/Product.lean
|
indexCone_isLimit
|
`indexCone` is a limit cone.
|
projective_ultrafilter (X : Type u) : Projective (of <| Ultrafilter X) where
factors {Y Z} f g hg := by
rw [epi_iff_surjective] at hg
obtain ⟨g', hg'⟩ := hg.hasRightInverse
let t : X → Y := g' ∘ f ∘ (pure : X → Ultrafilter X)
let h : Ultrafilter X → Y := Ultrafilter.extend t
have hh : Continuous h := continuous_ultrafilter_extend _
use CompHausLike.ofHom _ ⟨h, hh⟩
apply ConcreteCategory.coe_ext
simp only [h]
convert denseRange_pure.equalizer (g.hom.continuous.comp hh) f.hom.continuous _
have : g.hom ∘ g' = id := hg'.comp_eq_id
rw [comp_assoc, ultrafilter_extend_extends, ← comp_assoc, this, id_comp]
rfl
|
instance
|
Topology
|
[
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.CategoryTheory.Preadditive.Projective.Basic",
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono"
] |
Mathlib/Topology/Category/Profinite/Projective.lean
|
projective_ultrafilter
| null |
projectivePresentation (X : Profinite.{u}) : ProjectivePresentation X where
p := of <| Ultrafilter X
f := CompHausLike.ofHom _ ⟨_, continuous_ultrafilter_extend id⟩
projective := Profinite.projective_ultrafilter X
epi := ConcreteCategory.epi_of_surjective _ fun x =>
⟨(pure x : Ultrafilter X), congr_fun (ultrafilter_extend_extends (𝟙 X)) x⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.CategoryTheory.Preadditive.Projective.Basic",
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono"
] |
Mathlib/Topology/Category/Profinite/Projective.lean
|
projectivePresentation
|
For any profinite `X`, the natural map `Ultrafilter X → X` is a projective presentation.
|
stoneCechObj (X : Type u) : Stonean :=
letI : TopologicalSpace X := ⊥
haveI : DiscreteTopology X := ⟨rfl⟩
haveI : ExtremallyDisconnected (StoneCech X) :=
CompactT2.Projective.extremallyDisconnected StoneCech.projective
of (StoneCech X)
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Stonean.Basic",
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.Topology.Compactification.StoneCech"
] |
Mathlib/Topology/Category/Stonean/Adjunctions.lean
|
stoneCechObj
|
The object part of the compactification functor from types to Stonean spaces.
|
noncomputable stoneCechEquivalence (X : Type u) (Y : Stonean.{u}) :
(stoneCechObj X ⟶ Y) ≃ (X ⟶ ToType Y) := by
letI : TopologicalSpace X := ⊥
haveI : DiscreteTopology X := ⟨rfl⟩
refine fullyFaithfulToCompHaus.homEquiv.trans ?_
exact (_root_.stoneCechEquivalence (TopCat.of X) (toCompHaus.obj Y)).trans
(TopCat.adj₁.homEquiv _ _)
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Stonean.Basic",
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.Topology.Compactification.StoneCech"
] |
Mathlib/Topology/Category/Stonean/Adjunctions.lean
|
stoneCechEquivalence
|
The equivalence of homsets to establish the adjunction between the Stone-Cech compactification
functor and the forgetful functor.
|
noncomputable typeToStonean : Type u ⥤ Stonean.{u} :=
leftAdjointOfEquiv (G := forget _) Stonean.stoneCechEquivalence fun _ _ _ _ _ => rfl
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Stonean.Basic",
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.Topology.Compactification.StoneCech"
] |
Mathlib/Topology/Category/Stonean/Adjunctions.lean
|
typeToStonean
|
The Stone-Cech compactification functor from types to Stonean spaces.
|
noncomputable stoneCechAdjunction : typeToStonean ⊣ (forget Stonean) :=
adjunctionOfEquivLeft (G := forget _) stoneCechEquivalence fun _ _ _ _ _ => rfl
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Stonean.Basic",
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.Topology.Compactification.StoneCech"
] |
Mathlib/Topology/Category/Stonean/Adjunctions.lean
|
stoneCechAdjunction
|
The Stone-Cech compactification functor is left adjoint to the forgetful functor.
|
noncomputable forget.preservesLimits : Limits.PreservesLimits (forget Stonean) :=
rightAdjoint_preservesLimits stoneCechAdjunction
|
instance
|
Topology
|
[
"Mathlib.Topology.Category.Stonean.Basic",
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.Topology.Compactification.StoneCech"
] |
Mathlib/Topology/Category/Stonean/Adjunctions.lean
|
forget.preservesLimits
|
The forgetful functor from Stonean spaces, being a right adjoint, preserves limits.
|
Stonean := CompHausLike (fun X ↦ ExtremallyDisconnected X)
|
abbrev
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
Stonean
|
`Stonean` is the category of extremally disconnected compact Hausdorff spaces.
|
@[simps!]
toStonean (X : CompHaus.{u}) [Projective X] :
Stonean where
toTop := X.toTop
prop := inferInstance
|
def
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
toStonean
|
`Projective` implies `ExtremallyDisconnected`. -/
instance (X : CompHaus.{u}) [Projective X] : ExtremallyDisconnected X := by
apply CompactT2.Projective.extremallyDisconnected
intro A B _ _ _ _ _ _ f g hf hg hsurj
let A' : CompHaus := CompHaus.of A
let B' : CompHaus := CompHaus.of B
let f' : X ⟶ B' := CompHausLike.ofHom _ ⟨f, hf⟩
let g' : A' ⟶ B' := CompHausLike.ofHom _ ⟨g,hg⟩
have : Epi g' := by
rw [CompHaus.epi_iff_surjective]
assumption
obtain ⟨h, hh⟩ := Projective.factors f' g'
refine ⟨h, h.hom.2, ?_⟩
ext t
apply_fun (fun e => e t) at hh
exact hh
/-- `Projective` implies `Stonean`.
|
toCompHaus : Stonean.{u} ⥤ CompHaus.{u} :=
compHausLikeToCompHaus _
|
abbrev
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
toCompHaus
|
The (forgetful) functor from Stonean spaces to compact Hausdorff spaces.
|
fullyFaithfulToCompHaus : toCompHaus.FullyFaithful :=
CompHausLike.fullyFaithfulToCompHausLike _
open CompHausLike
|
abbrev
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
fullyFaithfulToCompHaus
|
The forgetful functor `Stonean ⥤ CompHaus` is fully faithful.
|
of (X : Type*) [TopologicalSpace X] [CompactSpace X] [T2Space X]
[ExtremallyDisconnected X] : Stonean := CompHausLike.of _ X
|
abbrev
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
of
|
Construct a term of `Stonean` from a type endowed with the structure of a
compact, Hausdorff and extremally disconnected topological space.
|
toProfinite : Stonean.{u} ⥤ Profinite.{u} :=
CompHausLike.toCompHausLike (fun _ ↦ inferInstance)
|
abbrev
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
toProfinite
|
The functor from Stonean spaces to profinite spaces.
|
mkFinite (X : Type*) [Finite X] [TopologicalSpace X] [DiscreteTopology X] : Stonean where
toTop := (CompHaus.of X).toTop
prop := by
dsimp
constructor
intro U _
apply isOpen_discrete (closure U)
|
def
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
mkFinite
|
A finite discrete space as a Stonean space.
|
epi_iff_surjective {X Y : Stonean} (f : X ⟶ Y) :
Epi f ↔ Function.Surjective f := by
refine ⟨?_, fun h => ConcreteCategory.epi_of_surjective f h⟩
dsimp [Function.Surjective]
intro h y
by_contra! hy
let C := Set.range f
have hC : IsClosed C := (isCompact_range f.hom.continuous).isClosed
let U := Cᶜ
have hUy : U ∈ 𝓝 y := by
simp only [U, C, Set.mem_range, hy, exists_false, not_false_eq_true, hC.compl_mem_nhds]
obtain ⟨V, hV, hyV, hVU⟩ := isTopologicalBasis_isClopen.mem_nhds_iff.mp hUy
classical
let g : Y ⟶ mkFinite (ULift (Fin 2)) := TopCat.ofHom
⟨(LocallyConstant.ofIsClopen hV).map ULift.up, LocallyConstant.continuous _⟩
let h : Y ⟶ mkFinite (ULift (Fin 2)) := TopCat.ofHom ⟨fun _ => ⟨1⟩, continuous_const⟩
have H : h = g := by
rw [← cancel_epi f]
ext x
apply ULift.ext -- why is `ext` not doing this automatically?
change 1 = ite _ _ _ -- why is `dsimp` not getting me here?
rw [if_neg]
refine mt (hVU ·) ?_ -- what would be an idiomatic tactic for this step?
simpa only [U, Set.mem_compl_iff, Set.mem_range, not_exists, not_forall, not_not]
using exists_apply_eq_apply f x
apply_fun fun e => (e y).down at H
change 1 = ite _ _ _ at H -- why is `dsimp at H` not getting me here?
rw [if_pos hyV] at H
exact one_ne_zero H
|
lemma
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
epi_iff_surjective
|
A morphism in `Stonean` is an epi iff it is surjective.
|
instProjectiveCompHausCompHaus (X : Stonean) : Projective (toCompHaus.obj X) where
factors := by
intro B C φ f _
haveI : ExtremallyDisconnected (toCompHaus.obj X).toTop := X.prop
have hf : Function.Surjective f := by rwa [← CompHaus.epi_iff_surjective]
obtain ⟨f', h⟩ := CompactT2.ExtremallyDisconnected.projective φ.hom.continuous f.hom.continuous
hf
use ofHom _ ⟨f', h.left⟩
ext
exact congr_fun h.right _
|
instance
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
instProjectiveCompHausCompHaus
|
Every Stonean space is projective in `CompHaus`
|
noncomputable
presentation (X : CompHaus) : Stonean where
toTop := (projectivePresentation X).p.1
prop := instExtremallyDisconnectedCarrierToTopTrueOfProjective X.projectivePresentation.p
|
def
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
presentation
|
Every Stonean space is projective in `Profinite` -/
instance (X : Stonean) : Projective (toProfinite.obj X) where
factors := by
intro B C φ f _
haveI : ExtremallyDisconnected (toProfinite.obj X) := X.prop
have hf : Function.Surjective f := by rwa [← Profinite.epi_iff_surjective]
obtain ⟨f', h⟩ := CompactT2.ExtremallyDisconnected.projective φ.hom.continuous f.hom.continuous
hf
use ofHom _ ⟨f', h.left⟩
ext
exact congr_fun h.right _
/-- Every Stonean space is projective in `Stonean`. -/
instance (X : Stonean) : Projective X where
factors := by
intro B C φ f _
haveI : ExtremallyDisconnected X.toTop := X.prop
have hf : Function.Surjective f := by rwa [← Stonean.epi_iff_surjective]
obtain ⟨f', h⟩ := CompactT2.ExtremallyDisconnected.projective φ.hom.continuous f.hom.continuous
hf
use ofHom _ ⟨f', h.left⟩
ext
exact congr_fun h.right _
end Stonean
namespace CompHaus
/-- If `X` is compact Hausdorff, `presentation X` is a Stonean space equipped with an epimorphism
down to `X` (see `CompHaus.presentation.π` and `CompHaus.presentation.epi_π`). It is a
"constructive" witness to the fact that `CompHaus` has enough projectives.
|
noncomputable
presentation.π (X : CompHaus) : Stonean.toCompHaus.obj X.presentation ⟶ X :=
(projectivePresentation X).f
|
def
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
presentation.π
|
The morphism from `presentation X` to `X`.
|
noncomputable
presentation.epi_π (X : CompHaus) : Epi (π X) :=
(projectivePresentation X).epi
|
instance
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
presentation.epi_π
|
The morphism from `presentation X` to `X` is an epimorphism.
|
_root_.Stonean.compHaus (X : Stonean) := Stonean.toCompHaus.obj X
|
abbrev
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
_root_.Stonean.compHaus
|
The underlying `CompHaus` of a `Stonean`.
|
noncomputable
lift {X Y : CompHaus} {Z : Stonean} (e : Z.compHaus ⟶ Y) (f : X ⟶ Y) [Epi f] :
Z.compHaus ⟶ X :=
Projective.factorThru e f
@[simp, reassoc]
|
def
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
lift
|
```
X
|
(f)
|
\/
Z ---(e)---> Y
```
If `Z` is a Stonean space, `f : X ⟶ Y` an epi in `CompHaus` and `e : Z ⟶ Y` is arbitrary, then
`lift e f` is a fixed (but arbitrary) lift of `e` to a morphism `Z ⟶ X`. It exists because
`Z` is a projective object in `CompHaus`.
|
lift_lifts {X Y : CompHaus} {Z : Stonean} (e : Z.compHaus ⟶ Y) (f : X ⟶ Y) [Epi f] :
lift e f ≫ f = e := by simp [lift]
|
lemma
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
lift_lifts
| null |
Gleason (X : CompHaus.{u}) :
Projective X ↔ ExtremallyDisconnected X := by
constructor
· intro h
change ExtremallyDisconnected X.toStonean
infer_instance
· intro h
let X' : Stonean := ⟨X.toTop, inferInstance⟩
change Projective X'.compHaus
apply Stonean.instProjectiveCompHausCompHaus
|
lemma
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
Gleason
| null |
noncomputable
presentation (X : Profinite) : Stonean where
toTop := (profiniteToCompHaus.obj X).projectivePresentation.p.toTop
prop := (profiniteToCompHaus.obj X).presentation.prop
|
def
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
presentation
|
If `X` is profinite, `presentation X` is a Stonean space equipped with an epimorphism down to
`X` (see `Profinite.presentation.π` and `Profinite.presentation.epi_π`).
|
noncomputable
presentation.π (X : Profinite) : Stonean.toProfinite.obj X.presentation ⟶ X :=
(profiniteToCompHaus.obj X).projectivePresentation.f
|
def
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
presentation.π
|
The morphism from `presentation X` to `X`.
|
noncomputable
presentation.epi_π (X : Profinite) : Epi (π X) := by
have := (profiniteToCompHaus.obj X).projectivePresentation.epi
rw [CompHaus.epi_iff_surjective] at this
rw [epi_iff_surjective]
exact this
|
instance
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
presentation.epi_π
|
The morphism from `presentation X` to `X` is an epimorphism.
|
noncomputable
lift {X Y : Profinite} {Z : Stonean} (e : Stonean.toProfinite.obj Z ⟶ Y) (f : X ⟶ Y) [Epi f] :
Stonean.toProfinite.obj Z ⟶ X := Projective.factorThru e f
@[simp, reassoc]
|
def
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
lift
|
```
X
|
(f)
|
\/
Z ---(e)---> Y
```
If `Z` is a Stonean space, `f : X ⟶ Y` an epi in `Profinite` and `e : Z ⟶ Y` is arbitrary,
then `lift e f` is a fixed (but arbitrary) lift of `e` to a morphism `Z ⟶ X`. It is
`CompHaus.lift e f` as a morphism in `Profinite`.
|
lift_lifts {X Y : Profinite} {Z : Stonean} (e : Stonean.toProfinite.obj Z ⟶ Y) (f : X ⟶ Y)
[Epi f] : lift e f ≫ f = e := by simp [lift]
|
lemma
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
lift_lifts
| null |
projective_of_extrDisc {X : Profinite.{u}} (hX : ExtremallyDisconnected X) :
Projective X := by
change Projective (Stonean.toProfinite.obj ⟨X.toTop, inferInstance⟩)
exact inferInstance
|
lemma
|
Topology
|
[
"Mathlib.Topology.ExtremallyDisconnected",
"Mathlib.Topology.Category.CompHaus.Projective",
"Mathlib.Topology.Category.Profinite.Basic"
] |
Mathlib/Topology/Category/Stonean/Basic.lean
|
projective_of_extrDisc
| null |
effectiveEpi_tfae
{B X : Stonean.{u}} (π : X ⟶ B) :
TFAE
[ EffectiveEpi π
, Epi π
, Function.Surjective π
] := by
tfae_have 1 → 2 := fun _ ↦ inferInstance
tfae_have 2 ↔ 3 := epi_iff_surjective π
tfae_have 3 → 1 := fun hπ ↦ ⟨⟨effectiveEpiStruct π hπ⟩⟩
tfae_finish
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular",
"Mathlib.Topology.Category.CompHaus.EffectiveEpi",
"Mathlib.Topology.Category.Stonean.Limits"
] |
Mathlib/Topology/Category/Stonean/EffectiveEpi.lean
|
effectiveEpi_tfae
| null |
noncomputable stoneanToCompHausEffectivePresentation (X : CompHaus) :
Stonean.toCompHaus.EffectivePresentation X where
p := X.presentation
f := CompHaus.presentation.π X
effectiveEpi := ((CompHaus.effectiveEpi_tfae _).out 0 1).mpr (inferInstance : Epi _)
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular",
"Mathlib.Topology.Category.CompHaus.EffectiveEpi",
"Mathlib.Topology.Category.Stonean.Limits"
] |
Mathlib/Topology/Category/Stonean/EffectiveEpi.lean
|
stoneanToCompHausEffectivePresentation
|
An effective presentation of an `X : CompHaus` with respect to the inclusion functor from `Stonean`
|
effectiveEpiFamily_tfae
{α : Type} [Finite α] {B : Stonean.{u}}
(X : α → Stonean.{u}) (π : (a : α) → (X a ⟶ B)) :
TFAE
[ EffectiveEpiFamily X π
, Epi (Sigma.desc π)
, ∀ b : B, ∃ (a : α) (x : X a), π a x = b
] := by
tfae_have 2 → 1
| _ => by
simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
tfae_have 1 → 2 := fun _ ↦ inferInstance
tfae_have 3 ↔ 1 := by
erw [((CompHaus.effectiveEpiFamily_tfae
(fun a ↦ Stonean.toCompHaus.obj (X a)) (fun a ↦ Stonean.toCompHaus.map (π a))).out 2 0 : )]
exact ⟨fun h ↦ Stonean.toCompHaus.finite_effectiveEpiFamily_of_map _ _ h,
fun _ ↦ inferInstance⟩
tfae_finish
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular",
"Mathlib.Topology.Category.CompHaus.EffectiveEpi",
"Mathlib.Topology.Category.Stonean.Limits"
] |
Mathlib/Topology/Category/Stonean/EffectiveEpi.lean
|
effectiveEpiFamily_tfae
| null |
effectiveEpiFamily_of_jointly_surjective
{α : Type} [Finite α] {B : Stonean.{u}}
(X : α → Stonean.{u}) (π : (a : α) → (X a ⟶ B))
(surj : ∀ b : B, ∃ (a : α) (x : X a), π a x = b) :
EffectiveEpiFamily X π :=
((effectiveEpiFamily_tfae X π).out 2 0).mp surj
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular",
"Mathlib.Topology.Category.CompHaus.EffectiveEpi",
"Mathlib.Topology.Category.Stonean.Limits"
] |
Mathlib/Topology/Category/Stonean/EffectiveEpi.lean
|
effectiveEpiFamily_of_jointly_surjective
| null |
extremallyDisconnected_preimage : ExtremallyDisconnected (i ⁻¹' (Set.range f)) where
open_closure U hU := by
have h : IsClopen (i ⁻¹' (Set.range f)) :=
⟨IsClosed.preimage i.hom.continuous (isCompact_range f.hom.continuous).isClosed,
IsOpen.preimage i.hom.continuous hi.isOpen_range⟩
rw [← (closure U).preimage_image_eq Subtype.coe_injective,
← h.1.isClosedEmbedding_subtypeVal.closure_image_eq U]
exact isOpen_induced (ExtremallyDisconnected.open_closure _
(h.2.isOpenEmbedding_subtypeVal.isOpenMap U hU))
|
lemma
|
Topology
|
[
"Mathlib.Topology.Category.CompHausLike.Limits",
"Mathlib.Topology.Category.Stonean.Basic"
] |
Mathlib/Topology/Category/Stonean/Limits.lean
|
extremallyDisconnected_preimage
| null |
extremallyDisconnected_pullback : ExtremallyDisconnected {xy : X × Y | f xy.1 = i xy.2} :=
have := extremallyDisconnected_preimage i hi
let e := (TopCat.pullbackHomeoPreimage i i.hom.2 f hi.isEmbedding).symm
let e' : {xy : X × Y | f xy.1 = i xy.2} ≃ₜ {xy : Y × X | i xy.1 = f xy.2} := by
exact TopCat.homeoOfIso
((TopCat.pullbackIsoProdSubtype f i).symm ≪≫ pullbackSymmetry _ _ ≪≫
(TopCat.pullbackIsoProdSubtype i f))
extremallyDisconnected_of_homeo (e.trans e'.symm)
|
lemma
|
Topology
|
[
"Mathlib.Topology.Category.CompHausLike.Limits",
"Mathlib.Topology.Category.Stonean.Basic"
] |
Mathlib/Topology/Category/Stonean/Limits.lean
|
extremallyDisconnected_pullback
| null |
@[simps! unit counit]
adj₁ : discrete ⊣ forget TopCat.{u} where
unit := { app := fun _ => id }
counit := { app := fun X => TopCat.ofHom (X := discrete.obj X) ⟨id, continuous_bot⟩ }
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.CategoryTheory.Adjunction.Basic"
] |
Mathlib/Topology/Category/TopCat/Adjunctions.lean
|
adj₁
|
Equipping a type with the discrete topology is left adjoint to the forgetful functor
`Top ⥤ Type`.
|
@[simps! unit counit]
adj₂ : forget TopCat.{u} ⊣ trivial where
unit := { app := fun X => TopCat.ofHom (Y := trivial.obj X) ⟨id, continuous_top⟩ }
counit := { app := fun _ => id }
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Basic",
"Mathlib.CategoryTheory.Adjunction.Basic"
] |
Mathlib/Topology/Category/TopCat/Adjunctions.lean
|
adj₂
|
Equipping a type with the trivial topology is right adjoint to the forgetful functor
`Top ⥤ Type`.
|
TopCat where
private mk ::
/-- The underlying type. -/
carrier : Type u
[str : TopologicalSpace carrier]
attribute [instance] TopCat.str
initialize_simps_projections TopCat (-str)
|
structure
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
TopCat
|
The category of topological spaces.
|
of (X : Type u) [TopologicalSpace X] : TopCat :=
⟨X⟩
|
abbrev
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
of
|
The object in `TopCat` associated to a type equipped with the appropriate
typeclasses. This is the preferred way to construct a term of `TopCat`.
|
coe_of (X : Type u) [TopologicalSpace X] : (of X : Type u) = X :=
rfl
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
coe_of
| null |
of_carrier (X : TopCat.{u}) : of X = X := rfl
variable {X} in
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
of_carrier
| null |
@[ext]
Hom (X Y : TopCat.{u}) where
private mk ::
/-- The underlying `ContinuousMap`. -/
hom' : C(X, Y)
|
structure
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
Hom
|
The type of morphisms in `TopCat`.
|
Hom.hom {X Y : TopCat.{u}} (f : Hom X Y) :=
ConcreteCategory.hom (C := TopCat) f
|
abbrev
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
Hom.hom
|
Turn a morphism in `TopCat` back into a `ContinuousMap`.
|
ofHom {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y)) : of X ⟶ of Y :=
ConcreteCategory.ofHom (C := TopCat) f
|
abbrev
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
ofHom
|
Typecheck a `ContinuousMap` as a morphism in `TopCat`.
|
Hom.Simps.hom (X Y : TopCat) (f : Hom X Y) :=
f.hom
initialize_simps_projections Hom (hom' → hom)
/-!
The results below duplicate the `ConcreteCategory` simp lemmas, but we can keep them for `dsimp`.
-/
@[simp]
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
Hom.Simps.hom
|
Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas.
|
hom_id {X : TopCat.{u}} : (𝟙 X : X ⟶ X).hom = ContinuousMap.id X := rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
hom_id
| null |
id_app (X : TopCat.{u}) (x : ↑X) : (𝟙 X : X ⟶ X) x = x := rfl
@[simp] theorem coe_id (X : TopCat.{u}) : (𝟙 X : X → X) = id := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
id_app
| null |
hom_comp {X Y Z : TopCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).hom = g.hom.comp f.hom := rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
hom_comp
| null |
comp_app {X Y Z : TopCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g : X → Z) x = g (f x) := rfl
@[simp] theorem coe_comp {X Y Z : TopCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g : X → Z) = g ∘ f := rfl
@[ext]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
comp_app
| null |
hom_ext {X Y : TopCat} {f g : X ⟶ Y} (hf : f.hom = g.hom) : f = g :=
Hom.ext hf
@[ext]
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
hom_ext
| null |
ext {X Y : TopCat} {f g : X ⟶ Y} (w : ∀ x : X, f x = g x) : f = g :=
ConcreteCategory.hom_ext _ _ w
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
ext
| null |
hom_ofHom {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y)) :
(ofHom f).hom = f := rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
hom_ofHom
| null |
ofHom_hom {X Y : TopCat} (f : X ⟶ Y) :
ofHom (Hom.hom f) = f := rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
ofHom_hom
| null |
ofHom_id {X : Type u} [TopologicalSpace X] : ofHom (ContinuousMap.id X) = 𝟙 (of X) := rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
ofHom_id
| null |
ofHom_comp {X Y Z : Type u} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
(f : C(X, Y)) (g : C(Y, Z)) :
ofHom (g.comp f) = ofHom f ≫ ofHom g :=
rfl
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
ofHom_comp
| null |
ofHom_apply {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y)) (x : X) :
(ofHom f) x = f x := rfl
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
ofHom_apply
| null |
hom_inv_id_apply {X Y : TopCat} (f : X ≅ Y) (x : X) : f.inv (f.hom x) = x := by
simp
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
hom_inv_id_apply
| null |
inv_hom_id_apply {X Y : TopCat} (f : X ≅ Y) (y : Y) : f.hom (f.inv y) = y := by
simp
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
inv_hom_id_apply
| null |
@[simp] coe_of_of {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y]
{f : C(X, Y)} {x} :
@DFunLike.coe (TopCat.of X ⟶ TopCat.of Y) ((CategoryTheory.forget TopCat).obj (TopCat.of X))
(fun _ ↦ (CategoryTheory.forget TopCat).obj (TopCat.of Y)) HasForget.instFunLike
(ofHom f) x =
@DFunLike.coe C(X, Y) X
(fun _ ↦ Y) _
f x :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
coe_of_of
|
Replace a function coercion for a morphism `TopCat.of X ⟶ TopCat.of Y` with the definitionally
equal function coercion for a continuous map `C(X, Y)`.
|
inhabited : Inhabited TopCat :=
⟨TopCat.of Empty⟩
@[deprecated
"Simply remove this from the `simp`/`rw` set: the LHS and RHS are now identical."
(since := "2025-01-30")]
|
instance
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
inhabited
| null |
hom_apply {X Y : TopCat} (f : X ⟶ Y) (x : X) : f x = ContinuousMap.toFun f.hom x := rfl
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
hom_apply
| null |
discrete : Type u ⥤ TopCat.{u} where
obj X := @of X ⊥
map f := @ofHom _ _ ⊥ ⊥ <| @ContinuousMap.mk _ _ ⊥ ⊥ f continuous_bot
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
discrete
|
The discrete topology on any type.
|
trivial : Type u ⥤ TopCat.{u} where
obj X := @of X ⊤
map f := @ofHom _ _ ⊤ ⊤ <| @ContinuousMap.mk _ _ ⊤ ⊤ f continuous_top
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
trivial
|
The trivial topology on any type.
|
@[simps]
isoOfHomeo {X Y : TopCat.{u}} (f : X ≃ₜ Y) : X ≅ Y where
hom := ofHom f
inv := ofHom f.symm
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
isoOfHomeo
|
Any homeomorphisms induces an isomorphism in `Top`.
|
@[simps]
homeoOfIso {X Y : TopCat.{u}} (f : X ≅ Y) : X ≃ₜ Y where
toFun := f.hom
invFun := f.inv
left_inv x := by simp
right_inv x := by simp
continuous_toFun := f.hom.hom.continuous
continuous_invFun := f.inv.hom.continuous
@[simp]
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
homeoOfIso
|
Any isomorphism in `Top` induces a homeomorphism.
|
of_isoOfHomeo {X Y : TopCat.{u}} (f : X ≃ₜ Y) : homeoOfIso (isoOfHomeo f) = f := by
ext
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
of_isoOfHomeo
| null |
of_homeoOfIso {X Y : TopCat.{u}} (f : X ≅ Y) : isoOfHomeo (homeoOfIso f) = f := by
ext
rfl
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
of_homeoOfIso
| null |
isIso_of_bijective_of_isOpenMap {X Y : TopCat.{u}} (f : X ⟶ Y)
(hfbij : Function.Bijective f) (hfcl : IsOpenMap f) : IsIso f :=
let e : X ≃ₜ Y :=
(Equiv.ofBijective f hfbij).toHomeomorphOfContinuousOpen f.hom.continuous hfcl
inferInstanceAs <| IsIso (TopCat.isoOfHomeo e).hom
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
isIso_of_bijective_of_isOpenMap
| null |
isIso_of_bijective_of_isClosedMap {X Y : TopCat.{u}} (f : X ⟶ Y)
(hfbij : Function.Bijective f) (hfcl : IsClosedMap f) : IsIso f :=
let e : X ≃ₜ Y :=
(Equiv.ofBijective f hfbij).toHomeomorphOfContinuousClosed f.hom.continuous hfcl
inferInstanceAs <| IsIso (TopCat.isoOfHomeo e).hom
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
isIso_of_bijective_of_isClosedMap
| null |
isIso_iff_isHomeomorph {X Y : TopCat.{u}} (f : X ⟶ Y) :
IsIso f ↔ IsHomeomorph f :=
⟨fun _ ↦ (homeoOfIso (asIso f)).isHomeomorph,
fun H ↦ isIso_of_bijective_of_isOpenMap _ H.bijective H.isOpenMap⟩
|
lemma
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
isIso_iff_isHomeomorph
| null |
isOpenEmbedding_iff_comp_isIso {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso g] :
IsOpenEmbedding (f ≫ g) ↔ IsOpenEmbedding f :=
(TopCat.homeoOfIso (asIso g)).isOpenEmbedding.of_comp_iff f
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
isOpenEmbedding_iff_comp_isIso
| null |
isOpenEmbedding_iff_comp_isIso' {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso g] :
IsOpenEmbedding (g ∘ f) ↔ IsOpenEmbedding f := by
simp only
exact isOpenEmbedding_iff_comp_isIso f g
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
isOpenEmbedding_iff_comp_isIso'
| null |
isOpenEmbedding_iff_isIso_comp {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] :
IsOpenEmbedding (f ≫ g) ↔ IsOpenEmbedding g := by
constructor
· intro h
convert h.comp (TopCat.homeoOfIso (asIso f).symm).isOpenEmbedding
exact congr_arg (DFunLike.coe ∘ ConcreteCategory.hom) (IsIso.inv_hom_id_assoc f g).symm
· exact fun h => h.comp (TopCat.homeoOfIso (asIso f)).isOpenEmbedding
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
isOpenEmbedding_iff_isIso_comp
| null |
isOpenEmbedding_iff_isIso_comp' {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] :
IsOpenEmbedding (g ∘ f) ↔ IsOpenEmbedding g := by
simp only
exact isOpenEmbedding_iff_isIso_comp f g
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.Elementwise",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Category/TopCat/Basic.lean
|
isOpenEmbedding_iff_isIso_comp'
| null |
noncomputable
effectiveEpiStructOfQuotientMap {B X : TopCat.{u}} (π : X ⟶ B) (hπ : IsQuotientMap π) :
EffectiveEpiStruct π where
/- `IsQuotientMap.lift` gives the required morphism -/
desc e h := ofHom <| hπ.lift e.hom fun a b hab ↦
CategoryTheory.congr_fun (h
(ofHom ⟨fun _ ↦ a, continuous_const⟩)
(ofHom ⟨fun _ ↦ b, continuous_const⟩)
(by ext; exact hab)) a
/- `IsQuotientMap.lift_comp` gives the factorisation -/
fac e h := hom_ext (hπ.lift_comp e.hom
fun a b hab ↦ CategoryTheory.congr_fun (h
(ofHom ⟨fun _ ↦ a, continuous_const⟩)
(ofHom ⟨fun _ ↦ b, continuous_const⟩)
(by ext; exact hab)) a)
/- Uniqueness follows from the fact that `IsQuotientMap.lift` is an equivalence (given by
`IsQuotientMap.liftEquiv`). -/
uniq e h g hm := by
suffices g = ofHom (hπ.liftEquiv ⟨e.hom,
fun a b hab ↦ CategoryTheory.congr_fun (h
(ofHom ⟨fun _ ↦ a, continuous_const⟩)
(ofHom ⟨fun _ ↦ b, continuous_const⟩)
(by ext; exact hab))
a⟩) by assumption
apply hom_ext
rw [hom_ofHom, ← Equiv.symm_apply_eq hπ.liftEquiv]
ext
simp only [IsQuotientMap.liftEquiv_symm_apply_coe, ContinuousMap.comp_apply, ← hm]
rfl
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.EffectiveEpi.RegularEpi",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks"
] |
Mathlib/Topology/Category/TopCat/EffectiveEpi.lean
|
effectiveEpiStructOfQuotientMap
|
Implementation: If `π` is a morphism in `TopCat` which is a quotient map, then it is an effective
epimorphism. The theorem `TopCat.effectiveEpi_iff_isQuotientMap` should be used instead of
this definition.
|
effectiveEpi_iff_isQuotientMap {B X : TopCat.{u}} (π : X ⟶ B) :
EffectiveEpi π ↔ IsQuotientMap π := by
/- The backward direction is given by `effectiveEpiStructOfQuotientMap` above. -/
refine ⟨fun _ ↦ ?_, fun hπ ↦ ⟨⟨effectiveEpiStructOfQuotientMap π hπ⟩⟩⟩
/- Since `TopCat` has pullbacks, `π` is in fact a `RegularEpi`. This means that it exhibits `B` as
a coequalizer of two maps into `X`. It suffices to prove that `π` followed by the isomorphism to
an arbitrary coequalizer is a quotient map. -/
have hπ : RegularEpi π := inferInstance
exact isQuotientMap_of_isColimit_cofork _ hπ.isColimit
|
theorem
|
Topology
|
[
"Mathlib.CategoryTheory.EffectiveEpi.RegularEpi",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks"
] |
Mathlib/Topology/Category/TopCat/EffectiveEpi.lean
|
effectiveEpi_iff_isQuotientMap
|
The effective epimorphisms in `TopCat` are precisely the quotient maps.
|
epi_iff_surjective {X Y : TopCat.{u}} (f : X ⟶ Y) : Epi f ↔ Function.Surjective f := by
suffices Epi f ↔ Epi ((forget TopCat).map f) by
rw [this, CategoryTheory.epi_iff_surjective]
rfl
constructor
· intro
infer_instance
· apply Functor.epi_of_epi_map
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Functor.EpiMono"
] |
Mathlib/Topology/Category/TopCat/EpiMono.lean
|
epi_iff_surjective
| null |
mono_iff_injective {X Y : TopCat.{u}} (f : X ⟶ Y) : Mono f ↔ Function.Injective f := by
suffices Mono f ↔ Mono ((forget TopCat).map f) by
rw [this, CategoryTheory.mono_iff_injective]
rfl
constructor
· intro
infer_instance
· apply Functor.mono_of_mono_map
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Functor.EpiMono"
] |
Mathlib/Topology/Category/TopCat/EpiMono.lean
|
mono_iff_injective
| null |
OpenNhds (x : X) :=
ObjectProperty.FullSubcategory fun U : Opens X => x ∈ U
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
OpenNhds
|
The type of open neighbourhoods of a point `x` in a (bundled) topological space.
|
partialOrder (x : X) : PartialOrder (OpenNhds x) where
le U V := U.1 ≤ V.1
le_refl _ := le_rfl
le_trans _ _ _ := le_trans
le_antisymm _ _ i j := ObjectProperty.FullSubcategory.ext <| le_antisymm i j
|
instance
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
partialOrder
| null |
openNhdsCategory (x : X) : Category.{u} (OpenNhds x) := inferInstance
|
instance
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
openNhdsCategory
| null |
opensNhds.instFunLike : FunLike (U ⟶ V) U.1 V.1 where
coe f := Set.inclusion f.le
coe_injective' := by rintro ⟨⟨_⟩⟩ _ _; congr!
@[simp] lemma apply_mk (f : U ⟶ V) (y : X) (hy) : f ⟨y, hy⟩ = ⟨y, f.le hy⟩ := rfl
@[simp] lemma val_apply (f : U ⟶ V) (y : U.1) : (f y : X) = y := rfl
@[simp, norm_cast] lemma coe_id (f : U ⟶ U) : ⇑f = id := rfl
|
instance
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
opensNhds.instFunLike
| null |
id_apply (f : U ⟶ U) (y : U.1) : f y = y := rfl
@[simp] lemma comp_apply (f : U ⟶ V) (g : V ⟶ W) (x : U.1) : (f ≫ g) x = g (f x) := rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
id_apply
| null |
infLELeft {x : X} (U V : OpenNhds x) : U ⊓ V ⟶ U :=
homOfLE inf_le_left
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
infLELeft
|
The inclusion `U ⊓ V ⟶ U` as a morphism in the category of open sets.
|
infLERight {x : X} (U V : OpenNhds x) : U ⊓ V ⟶ V :=
homOfLE inf_le_right
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
infLERight
|
The inclusion `U ⊓ V ⟶ V` as a morphism in the category of open sets.
|
inclusion (x : X) : OpenNhds x ⥤ Opens X :=
ObjectProperty.ι _
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
inclusion
|
The inclusion functor from open neighbourhoods of `x`
to open sets in the ambient topological space.
|
inclusion_obj (x : X) (U) (p) : (inclusion x).obj ⟨U, p⟩ = U :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
inclusion_obj
| null |
isOpenEmbedding {x : X} (U : OpenNhds x) : IsOpenEmbedding U.1.inclusion' :=
U.1.isOpenEmbedding
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
isOpenEmbedding
| null |
map (x : X) : OpenNhds (f x) ⥤ OpenNhds x where
obj U := ⟨(Opens.map f).obj U.1, U.2⟩
map i := (Opens.map f).map i
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
map
|
The preimage functor from neighborhoods of `f x` to neighborhoods of `x`.
|
map_obj (x : X) (U) (q) : (map f x).obj ⟨U, q⟩ = ⟨(Opens.map f).obj U, q⟩ :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
map_obj
| null |
map_id_obj (x : X) (U) : (map (𝟙 X) x).obj U = U := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
map_id_obj
| null |
map_id_obj' (x : X) (U) (p) (q) : (map (𝟙 X) x).obj ⟨⟨U, p⟩, q⟩ = ⟨⟨U, p⟩, q⟩ :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
map_id_obj'
| null |
map_id_obj_unop (x : X) (U : (OpenNhds x)ᵒᵖ) : (map (𝟙 X) x).obj (unop U) = unop U := by
simp
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
map_id_obj_unop
| null |
op_map_id_obj (x : X) (U : (OpenNhds x)ᵒᵖ) : (map (𝟙 X) x).op.obj U = U := by simp
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
op_map_id_obj
| null |
@[simps! hom_app inv_app]
inclusionMapIso (x : X) : inclusion (f x) ⋙ Opens.map f ≅ map f x ⋙ inclusion x :=
NatIso.ofComponents fun U => { hom := 𝟙 _, inv := 𝟙 _ }
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
inclusionMapIso
|
`Opens.map f` and `OpenNhds.map f` form a commuting square (up to natural isomorphism)
with the inclusion functors into `Opens X`.
|
inclusionMapIso_hom (x : X) : (inclusionMapIso f x).hom = 𝟙 _ :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
inclusionMapIso_hom
| null |
inclusionMapIso_inv (x : X) : (inclusionMapIso f x).inv = 𝟙 _ :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Category.TopCat.Opens",
"Mathlib.Data.Set.Subsingleton"
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
inclusionMapIso_inv
| null |
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