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 Mathlib/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 Mathlib/Topology/Category/Compactum.lean for a more detailed discussion where these definitions are introduced.

@[reducible, inline]

abbrev CompHaus : Type (u_1 + 1)

The category of compact Hausdorff spaces.

Equations

• CompHaus = CompHausLike fun (x : TopCat) => True

instance CompHaus.instInhabited : Inhabited CompHaus

Equations

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

instance CompHaus.instCoeSortType : CoeSort CompHaus (Type u_1)

Equations

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

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

Equations

• ⋯ = ⋯

instance CompHaus.instT2SpaceαTopologicalSpaceToTopTrue {X : CompHaus} : T2Space ↑X.toTop

Equations

• ⋯ = ⋯

instance CompHaus.instHasPropTrue (X : Type u_1) [TopologicalSpace X] : CompHausLike.HasProp (fun (x : TopCat) => True) X

Equations

• ⋯ = ⋯

@[reducible, inline]

abbrev 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 = CompHausLike.of (fun (x : TopCat) => True) X

@[reducible, inline]

abbrev compHausToTop : CategoryTheory.Functor CompHaus TopCat

The fully faithful embedding of CompHaus in TopCat.

Equations

• compHausToTop = CompHausLike.compHausLikeToTop fun (x : TopCat) => True

@[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

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

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

@[reducible, inline]

abbrev compHausLikeToCompHaus (P : TopCat → Prop) : CategoryTheory.Functor (CompHausLike P) CompHaus

Every CompHausLike admits a functor to CompHaus.

Equations

• compHausLikeToCompHaus P = CompHausLike.toCompHausLike ⋯