File: Mathlib/Topology/Category/CompHaus/Basic.lean

full_name: CompHaus
code: /-- The category of compact Hausdorff spaces. -/
abbrev CompHaus 
goal: Type (u_1 + 1)

full_name: CompHaus.instInhabited
code: instance : Inhabited CompHaus 
goal: Inhabited CompHaus

full_name: CompHaus.instCoeSortType
code: instance : CoeSort CompHaus Type* 
goal: CoeSort CompHaus (Type u_1)

full_name: CompHaus.instCompactSpaceαTopologicalSpaceToTopTrue
code: instance {X : CompHaus} : CompactSpace X 
goal: CompactSpace ↑X.toTop

full_name: CompHaus.instT2SpaceαTopologicalSpaceToTopTrue
code: instance {X : CompHaus} : T2Space X 
goal: T2Space ↑X.toTop

full_name: CompHaus.instHasPropTrue
code: instance : HasProp (fun _ ↦ True) X 
goal: CompHausLike.HasProp (fun x => True) X

full_name: CompHaus.of
code: /-- A constructor for objects of the category `CompHaus`,
taking a type, and bundling the compact Hausdorff topology
found by typeclass inference. -/
abbrev of : CompHaus 
goal: CompHaus

full_name: compHausToTop
code: /-- The fully faithful embedding of `CompHaus` in `TopCat`. -/
-- Porting note: `semireducible` -> `.default`.
abbrev compHausToTop : CompHaus.{u} ⥤ TopCat.{u} 
goal: CompHaus ⥤ TopCat

full_name: stoneCechObj_toTop_str_IsOpen
code: /-- (Implementation) The object part of the compactification functor from topological spaces to
compact Hausdorff spaces.
-/
@[simps!]
def stoneCechObj (X : TopCat) : CompHaus 
goal: TopologicalSpace.IsOpen s = IsOpen (Quotient.mk' ⁻¹' s)

full_name: stoneCechObj_toTop_α
code: /-- (Implementation) The object part of the compactification functor from topological spaces to
compact Hausdorff spaces.
-/
@[simps!]
def stoneCechObj (X : TopCat) : CompHaus 
goal: ↑(stoneCechObj X).toTop = StoneCech ↑X

full_name: stoneCechObj
code: /-- (Implementation) The object part of the compactification functor from topological spaces to
compact Hausdorff spaces.
-/
@[simps!]
def stoneCechObj (X : TopCat) : CompHaus 
goal: CompHaus

full_name: stoneCechEquivalence
code: /-- (Implementation) The bijection of homsets to establish the reflective adjunction of compact
Hausdorff spaces in topological spaces.
-/
noncomputable def stoneCechEquivalence (X : TopCat.{u}) (Y : CompHaus.{u}) :
    (stoneCechObj X ⟶ Y) ≃ (X ⟶ compHausToTop.obj Y) 
goal: (stoneCechObj X ⟶ Y) ≃ (X ⟶ compHausToTop.obj Y)

full_name: topToCompHaus
code: /-- The Stone-Cech compactification functor from topological spaces to compact Hausdorff spaces,
left adjoint to the inclusion functor.
-/
noncomputable def topToCompHaus : TopCat.{u} ⥤ CompHaus.{u} 
goal: TopCat ⥤ CompHaus

full_name: topToCompHaus_obj
code: theorem topToCompHaus_obj (X : TopCat) : ↥(topToCompHaus.obj X) = StoneCech X 
goal: ↑(topToCompHaus.obj X).toTop = StoneCech ↑X

full_name: compHausToTop.reflective
code: /-- The category of compact Hausdorff spaces is reflective in the category of topological spaces.
-/
noncomputable instance compHausToTop.reflective : Reflective compHausToTop 
goal: CategoryTheory.Reflective compHausToTop

full_name: compHausToTop.createsLimits
code: noncomputable instance compHausToTop.createsLimits : CreatesLimits compHausToTop 
goal: CategoryTheory.CreatesLimits compHausToTop

full_name: CompHaus.hasLimits
code: instance CompHaus.hasLimits : Limits.HasLimits CompHaus 
goal: CategoryTheory.Limits.HasLimits CompHaus

full_name: CompHaus.hasColimits
code: instance CompHaus.hasColimits : Limits.HasColimits CompHaus 
goal: CategoryTheory.Limits.HasColimits CompHaus

full_name: CompHaus.limitCone
code: /-- An explicit limit cone for a functor `F : J ⥤ CompHaus`, defined in terms of
`TopCat.limitCone`. -/
def limitCone {J : Type v} [SmallCategory J] (F : J ⥤ CompHaus.{max v u}) : Limits.Cone F 
goal: CategoryTheory.Limits.Cone F

full_name: CompHaus.limitConeIsLimit
code: /-- The limit cone `CompHaus.limitCone F` is indeed a limit cone. -/
def limitConeIsLimit {J : Type v} [SmallCategory J] (F : J ⥤ CompHaus.{max v u}) :
    Limits.IsLimit.{v} (limitCone.{v,u} F) 
goal: CategoryTheory.Limits.IsLimit (CompHaus.limitCone F)

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

full_name: compHausLikeToCompHaus
code: /-- Every `CompHausLike` admits a functor to `CompHaus`. -/
abbrev compHausLikeToCompHaus (P : TopCat → Prop) : CompHausLike P ⥤ CompHaus 
goal: CompHausLike P ⥤ CompHaus