The category of Compact Hausdorff Spaces

We construct the category of compact Hausdorff spaces. The type of compact Hausdorff spaces is denoted CompHaus, and it is endowed with a category instance making it a full subcategory of TopCat. The fully faithful functor CompHaus ⥤ TopCat is denoted compHausToTop.

Note: The file Topology/Category/Compactum.lean provides the equivalence between Compactum, which is defined as the category of algebras for the ultrafilter monad, and CompHaus. CompactumToCompHaus is the functor from Compactum to CompHaus which is proven to be an equivalence of categories in CompactumToCompHaus.isEquivalence. See topology/category/Compactum.lean for a more detailed discussion where these definitions are introduced.

structure CompHaus : Type (u_1 + 1)

The type of Compact Hausdorff topological spaces.

• toTop : TopCat

  The underlying topological space of an object of CompHaus.

• is_compact : CompactSpace ↑self.toTop

  The underlying topological space is compact.

• is_hausdorff : T2Space ↑self.toTop

  The underlying topological space is T2.

theorem CompHaus.is_compact (self : CompHaus) : CompactSpace ↑self.toTop

The underlying topological space is compact.

theorem CompHaus.is_hausdorff (self : CompHaus) : T2Space ↑self.toTop

The underlying topological space is T2.

instance CompHaus.instInhabited : Inhabited CompHaus

Equations

• CompHaus.instInhabited = { default := CompHaus.mk { α := PEmpty.{u_1 + 1} , str := inferInstance } }

instance CompHaus.instCoeSortType : CoeSort CompHaus (Type u_1)

Equations

• CompHaus.instCoeSortType = { coe := fun (X : CompHaus) => ↑X.toTop }

instance CompHaus.instCompactSpaceαTopologicalSpaceToTop {X : CompHaus} : CompactSpace ↑X.toTop

Equations

• ⋯ = ⋯

instance CompHaus.instT2SpaceαTopologicalSpaceToTop {X : CompHaus} : T2Space ↑X.toTop

Equations

• ⋯ = ⋯

instance CompHaus.category : CategoryTheory.Category.{u_1, u_1 + 1} CompHaus

Equations

• CompHaus.category = CategoryTheory.InducedCategory.category CompHaus.toTop

instance CompHaus.concreteCategory : CategoryTheory.ConcreteCategory CompHaus

Equations

• CompHaus.concreteCategory = CategoryTheory.InducedCategory.concreteCategory CompHaus.toTop

def CompHaus.of (X : Type u_1) [TopologicalSpace X] [CompactSpace X] [T2Space X] : CompHaus

A constructor for objects of the category CompHaus, taking a type, and bundling the compact Hausdorff topology found by typeclass inference.

Equations

• CompHaus.of X = CompHaus.mk (TopCat.of X)

@[simp]

theorem CompHaus.coe_of (X : Type u_1) [TopologicalSpace X] [CompactSpace X] [T2Space X] : ↑(CompHaus.of X).toTop = X

instance CompHaus.instTopologicalSpaceObjForget (X : CompHaus) : TopologicalSpace ((CategoryTheory.forget CompHaus).obj X)

Equations

• X.instTopologicalSpaceObjForget = let_fun this := inferInstance; this

instance CompHaus.instCompactSpaceObjForget (X : CompHaus) : CompactSpace ((CategoryTheory.forget CompHaus).obj X)

Equations

• ⋯ = ⋯

instance CompHaus.instT2SpaceObjForget (X : CompHaus) : T2Space ((CategoryTheory.forget CompHaus).obj X)

Equations

• ⋯ = ⋯

theorem CompHaus.isClosedMap {X : CompHaus} {Y : CompHaus} (f : X ⟶ Y) : IsClosedMap ⇑f

Any continuous function on compact Hausdorff spaces is a closed map.

theorem CompHaus.isIso_of_bijective {X : CompHaus} {Y : CompHaus} (f : X ⟶ Y) (bij : Function.Bijective ⇑f) : CategoryTheory.IsIso f

Any continuous bijection of compact Hausdorff spaces is an isomorphism.

noncomputable def CompHaus.isoOfBijective {X : CompHaus} {Y : CompHaus} (f : X ⟶ Y) (bij : Function.Bijective ⇑f) : X ≅ Y

Any continuous bijection of compact Hausdorff spaces induces an isomorphism.

Equations

• CompHaus.isoOfBijective f bij = CategoryTheory.asIso f

@[simp]

theorem CompHaus.isoOfHomeo_inv {X : CompHaus} {Y : CompHaus} (f : ↑X.toTop ≃ₜ ↑Y.toTop) : (CompHaus.isoOfHomeo f).inv = { toFun := ⇑f.symm, continuous_toFun := ⋯ }

@[simp]

theorem CompHaus.isoOfHomeo_hom {X : CompHaus} {Y : CompHaus} (f : ↑X.toTop ≃ₜ ↑Y.toTop) : (CompHaus.isoOfHomeo f).hom = { toFun := ⇑f, continuous_toFun := ⋯ }

def CompHaus.isoOfHomeo {X : CompHaus} {Y : CompHaus} (f : ↑X.toTop ≃ₜ ↑Y.toTop) : X ≅ Y

Construct an isomorphism from a homeomorphism.

Equations

• CompHaus.isoOfHomeo f = { hom := { toFun := ⇑f, continuous_toFun := ⋯ }, inv := { toFun := ⇑f.symm, continuous_toFun := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ }

@[simp]

theorem CompHaus.homeoOfIso_apply {X : CompHaus} {Y : CompHaus} (f : X ≅ Y) (a : (CategoryTheory.forget CompHaus).obj X) : (CompHaus.homeoOfIso f) a = f.hom a

@[simp]

theorem CompHaus.homeoOfIso_symm_apply {X : CompHaus} {Y : CompHaus} (f : X ≅ Y) (a : (CategoryTheory.forget CompHaus).obj Y) : (CompHaus.homeoOfIso f).symm a = f.inv a

def CompHaus.homeoOfIso {X : CompHaus} {Y : CompHaus} (f : X ≅ Y) : ↑X.toTop ≃ₜ ↑Y.toTop

Construct a homeomorphism from an isomorphism.

Equations

• CompHaus.homeoOfIso f = { toFun := ⇑f.hom, invFun := ⇑f.inv, left_inv := ⋯, right_inv := ⋯, continuous_toFun := ⋯, continuous_invFun := ⋯ }

@[simp]

theorem CompHaus.isoEquivHomeo_apply {X : CompHaus} {Y : CompHaus} (f : X ≅ Y) : CompHaus.isoEquivHomeo f = CompHaus.homeoOfIso f

@[simp]

theorem CompHaus.isoEquivHomeo_symm_apply {X : CompHaus} {Y : CompHaus} (f : ↑X.toTop ≃ₜ ↑Y.toTop) : CompHaus.isoEquivHomeo.symm f = CompHaus.isoOfHomeo f

def CompHaus.isoEquivHomeo {X : CompHaus} {Y : CompHaus} : (X ≅ Y) ≃ (↑X.toTop ≃ₜ ↑Y.toTop)

The equivalence between isomorphisms in CompHaus and homeomorphisms of topological spaces.

Equations

• CompHaus.isoEquivHomeo = { toFun := CompHaus.homeoOfIso, invFun := CompHaus.isoOfHomeo, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem compHausToTop_map : ∀ {X Y : CategoryTheory.InducedCategory TopCat CompHaus.toTop} (f : X ⟶ Y), compHausToTop.map f = f

@[simp]

theorem compHausToTop_obj (self : CompHaus) : compHausToTop.obj self = self.toTop

def compHausToTop : CategoryTheory.Functor CompHaus TopCat

The fully faithful embedding of CompHaus in TopCat.

Equations

• compHausToTop = CategoryTheory.inducedFunctor CompHaus.toTop

instance instFullCompHausTopCatCompHausToTop : compHausToTop.Full

Equations

• instFullCompHausTopCatCompHausToTop = ⋯

instance instFaithfulCompHausTopCatCompHausToTop : compHausToTop.Faithful

Equations

• instFaithfulCompHausTopCatCompHausToTop = ⋯

instance instCompactSpaceαTopologicalSpaceObjCompHausTopCatCompHausToTop (X : CompHaus) : CompactSpace ↑(compHausToTop.obj X)

Equations

• ⋯ = ⋯

instance instT2SpaceαTopologicalSpaceObjCompHausTopCatCompHausToTop (X : CompHaus) : T2Space ↑(compHausToTop.obj X)

Equations

• ⋯ = ⋯

instance CompHaus.forget_reflectsIsomorphisms : (CategoryTheory.forget CompHaus).ReflectsIsomorphisms

Equations

• CompHaus.forget_reflectsIsomorphisms = ⋯

@[simp]

theorem stoneCechObj_toTop_str_IsOpen (X : TopCat) (s : Set (Quotient (t2Setoid (PreStoneCech ↑X)))) : TopologicalSpace.IsOpen s = IsOpen (Quotient.mk' ⁻¹' s)

@[simp]

theorem stoneCechObj_toTop_α (X : TopCat) : ↑(stoneCechObj X).toTop = StoneCech ↑X

def stoneCechObj (X : TopCat) : CompHaus

(Implementation) The object part of the compactification functor from topological spaces to compact Hausdorff spaces.

Equations

• stoneCechObj X = CompHaus.of (StoneCech ↑X)

noncomputable def stoneCechEquivalence (X : TopCat) (Y : CompHaus) : (stoneCechObj X ⟶ Y) ≃ (X ⟶ compHausToTop.obj Y)

(Implementation) The bijection of homsets to establish the reflective adjunction of compact Hausdorff spaces in topological spaces.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def topToCompHaus : CategoryTheory.Functor TopCat CompHaus

The Stone-Cech compactification functor from topological spaces to compact Hausdorff spaces, left adjoint to the inclusion functor.

Equations

• topToCompHaus = CategoryTheory.Adjunction.leftAdjointOfEquiv stoneCechEquivalence topToCompHaus.proof_1

theorem topToCompHaus_obj (X : TopCat) : ↑(topToCompHaus.obj X).toTop = StoneCech ↑X

noncomputable instance compHausToTop.reflective : CategoryTheory.Reflective compHausToTop

The category of compact Hausdorff spaces is reflective in the category of topological spaces.

Equations

• compHausToTop.reflective = CategoryTheory.Reflective.mk topToCompHaus (CategoryTheory.Adjunction.adjunctionOfEquivLeft stoneCechEquivalence topToCompHaus.proof_1)

noncomputable instance compHausToTop.createsLimits : CategoryTheory.CreatesLimits compHausToTop

Equations

• compHausToTop.createsLimits = CategoryTheory.monadicCreatesLimits compHausToTop

instance CompHaus.hasLimits : CategoryTheory.Limits.HasLimits CompHaus

Equations

• CompHaus.hasLimits = ⋯

instance CompHaus.hasColimits : CategoryTheory.Limits.HasColimits CompHaus

Equations

• CompHaus.hasColimits = ⋯

def CompHaus.limitCone {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CompHaus) : CategoryTheory.Limits.Cone F

An explicit limit cone for a functor F : J ⥤ CompHaus, defined in terms of TopCat.limitCone.

Equations

• CompHaus.limitCone F = { pt := CompHaus.mk (TopCat.limitCone (F.comp compHausToTop)).pt, π := { app := fun (j : J) => (TopCat.limitCone (F.comp compHausToTop)).π.app j, naturality := ⋯ } }

def CompHaus.limitConeIsLimit {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CompHaus) : CategoryTheory.Limits.IsLimit (CompHaus.limitCone F)

The limit cone CompHaus.limitCone F is indeed a limit cone.

Equations

• One or more equations did not get rendered due to their size.

theorem CompHaus.epi_iff_surjective {X : CompHaus} {Y : CompHaus} (f : X ⟶ Y) : CategoryTheory.Epi f ↔ Function.Surjective ⇑f

theorem CompHaus.mono_iff_injective {X : CompHaus} {Y : CompHaus} (f : X ⟶ Y) : CategoryTheory.Mono f ↔ Function.Injective ⇑f