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)

• toTop : TopCat

  The underlying topological space of an object of CompHaus.

• is_compact : CompactSpace ↑s.toTop

  The underlying topological space is compact.

• is_hausdorff : T2Space ↑s.toTop

  The underlying topological space is T2.

The type of Compact Hausdorff topological spaces.

instance CompHaus.instInhabitedCompHaus : Inhabited CompHaus

instance CompHaus.instCoeSortCompHausType : CoeSort CompHaus (Type u_1)

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

instance CompHaus.instT2SpaceαTopologicalSpaceToTopTopologicalSpace_coe {X : CompHaus} : T2Space ↑X.toTop

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

instance CompHaus.concreteCategory : CategoryTheory.ConcreteCategory CompHaus

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.

@[simp]

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

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

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

instance CompHaus.instT2SpaceObjCompHausToQuiverToCategoryStructCategoryTypeTypesToPrefunctorForgetConcreteCategoryInstTopologicalSpaceObjCompHausToQuiverToCategoryStructCategoryTypeTypesToPrefunctorForgetConcreteCategory (X : CompHaus) : T2Space ((CategoryTheory.forget CompHaus).obj X)

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.

@[simp]

theorem CompHaus.isoOfHomeo_inv {X : CompHaus} {Y : CompHaus} (f : ↑X.toTop ≃ₜ ↑Y.toTop) : (CompHaus.isoOfHomeo f).inv = ContinuousMap.mk ↑(Homeomorph.symm f)

@[simp]

theorem CompHaus.isoOfHomeo_hom {X : CompHaus} {Y : CompHaus} (f : ↑X.toTop ≃ₜ ↑Y.toTop) : (CompHaus.isoOfHomeo f).hom = ContinuousMap.mk ↑f

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

Construct an isomorphism from a homeomorphism.

@[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) : ↑(Homeomorph.symm (CompHaus.homeoOfIso f)) 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.

@[simp]

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

@[simp]

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

@[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.

instance instFullCompHausCategoryTopCatInstTopCatLargeCategoryCompHausToTop : CategoryTheory.Full compHausToTop

instance instFaithfulCompHausCategoryTopCatInstTopCatLargeCategoryCompHausToTop : CategoryTheory.Faithful compHausToTop

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

instance instT2SpaceαTopologicalSpaceObjCompHausToQuiverToCategoryStructCategoryTopCatInstTopCatLargeCategoryToPrefunctorCompHausToTopTopologicalSpace_coe (X : CompHaus) : T2Space ↑(compHausToTop.obj X)

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

@[simp]

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

@[simp]

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

def stoneCechObj (X : TopCat) : CompHaus

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

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.

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.

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.

noncomputable instance compHausToTop.createsLimits : CategoryTheory.CreatesLimits compHausToTop

instance CompHaus.hasLimits : CategoryTheory.Limits.HasLimits CompHaus

instance CompHaus.hasColimits : CategoryTheory.Limits.HasColimits CompHaus

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.

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.

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