File: Mathlib/Topology/Order/Category/AlexDisc.lean

full_name: AlexandrovDiscreteSpace
code: /-- Auxiliary typeclass to define the category of Alexandrov-discrete spaces. Do not use this
directly. Use `AlexandrovDiscrete` instead. -/
class AlexandrovDiscreteSpace (α : Type*) extends TopologicalSpace α, AlexandrovDiscrete α
goal: Type u_1

full_name: AlexDisc
code: /-- The category of Alexandrov-discrete spaces. -/
def AlexDisc 
goal: Type (u_1 + 1)

full_name: AlexDisc.instCoeSort
code: instance instCoeSort : CoeSort AlexDisc Type* 
goal: CoeSort AlexDisc (Type u_1)

full_name: AlexDisc.instTopologicalSpace
code: instance instTopologicalSpace (α : AlexDisc) : TopologicalSpace α 
goal: TopologicalSpace ↑α

full_name: AlexDisc.instAlexandrovDiscrete
code: instance instAlexandrovDiscrete (α : AlexDisc) : AlexandrovDiscrete α 
goal: AlexandrovDiscrete ↑α

full_name: AlexDisc.instParentProjectionTopologicalSpaceAlexandrovDiscreteSpaceToTopologicalSpace
code: instance : BundledHom.ParentProjection @AlexandrovDiscreteSpace.toTopologicalSpace 
goal: CategoryTheory.BundledHom.ParentProjection @AlexandrovDiscreteSpace.toTopologicalSpace

full_name: AlexDisc.instConcreteCategory
code: instance instConcreteCategory : ConcreteCategory AlexDisc 
goal: CategoryTheory.ConcreteCategory AlexDisc

full_name: AlexDisc.instHasForgetToTop
code: instance instHasForgetToTop : HasForget₂ AlexDisc TopCat 
goal: CategoryTheory.HasForget₂ AlexDisc TopCat

full_name: AlexDisc.forgetToTop_full
code: instance forgetToTop_full : (forget₂ AlexDisc TopCat).Full 
goal: (CategoryTheory.forget₂ AlexDisc TopCat).Full

full_name: AlexDisc.forgetToTop_faithful
code: instance forgetToTop_faithful : (forget₂ AlexDisc TopCat).Faithful 
goal: (CategoryTheory.forget₂ AlexDisc TopCat).Faithful

full_name: AlexDisc.coe_forgetToTop
code: @[simp] lemma coe_forgetToTop (X : AlexDisc) : ↥((forget₂ _ TopCat).obj X) = X 
goal: ↑((CategoryTheory.forget₂ AlexDisc TopCat).obj X) = ↑X

full_name: AlexDisc.of
code: /-- Construct a bundled `AlexDisc` from the underlying topological space. -/
def of (α : Type*) [TopologicalSpace α] [AlexandrovDiscrete α] : AlexDisc 
goal: AlexDisc

full_name: AlexDisc.coe_of
code: @[simp] lemma coe_of (α : Type*) [TopologicalSpace α] [AlexandrovDiscrete α] : ↥(of α) = α 
goal: ↑(AlexDisc.of α) = α

full_name: AlexDisc.forgetToTop_of
code: @[simp] lemma forgetToTop_of (α : Type*) [TopologicalSpace α] [AlexandrovDiscrete α] :
  (forget₂ AlexDisc TopCat).obj (of α) = TopCat.of α 
goal: (CategoryTheory.forget₂ AlexDisc TopCat).obj (AlexDisc.of α) = TopCat.of α

full_name: AlexDisc.Iso.mk
code: /-- Constructs an equivalence between preorders from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : AlexDisc} (e : α ≃ₜ β) : α ≅ β 
goal: α ≅ β

full_name: AlexDisc.Iso.mk_inv
code: /-- Constructs an equivalence between preorders from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : AlexDisc} (e : α ≃ₜ β) : α ≅ β 
goal: (AlexDisc.Iso.mk e).inv = e.symm.toContinuousMap

full_name: AlexDisc.Iso.mk_hom
code: /-- Constructs an equivalence between preorders from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : AlexDisc} (e : α ≃ₜ β) : α ≅ β 
goal: (AlexDisc.Iso.mk e).hom = e.toContinuousMap

full_name: alexDiscEquivPreord_functor
code: /-- Sends a topological space to its specialisation order. -/
@[simps]
def alexDiscEquivPreord : AlexDisc ≌ Preord 
goal: alexDiscEquivPreord.functor = CategoryTheory.forget₂ AlexDisc TopCat ⋙ topToPreord

full_name: alexDiscEquivPreord_inverse_map
code: /-- Sends a topological space to its specialisation order. -/
@[simps]
def alexDiscEquivPreord : AlexDisc ≌ Preord 
goal: ∀ {X Y : Preord} (f : ↑X →o ↑Y), alexDiscEquivPreord.inverse.map f = Topology.WithUpperSet.map f

full_name: alexDiscEquivPreord_unitIso
code: /-- Sends a topological space to its specialisation order. -/
@[simps]
def alexDiscEquivPreord : AlexDisc ≌ Preord 
goal: alexDiscEquivPreord.unitIso =
  CategoryTheory.NatIso.ofComponents (fun X => AlexDisc.Iso.mk (id (homeoWithUpperSetTopologyorderIso ↑X)))
    @alexDiscEquivPreord.proof_4

full_name: alexDiscEquivPreord_counitIso
code: /-- Sends a topological space to its specialisation order. -/
@[simps]
def alexDiscEquivPreord : AlexDisc ≌ Preord 
goal: alexDiscEquivPreord.counitIso =
  CategoryTheory.NatIso.ofComponents (fun X => Preord.Iso.mk (id (orderIsoSpecializationWithUpperSetTopology ↑X).symm))
    @alexDiscEquivPreord.proof_5

full_name: alexDiscEquivPreord_inverse_obj
code: /-- Sends a topological space to its specialisation order. -/
@[simps]
def alexDiscEquivPreord : AlexDisc ≌ Preord 
goal: alexDiscEquivPreord.inverse.obj X = AlexDisc.of (Topology.WithUpperSet ↑X)

full_name: alexDiscEquivPreord
code: /-- Sends a topological space to its specialisation order. -/
@[simps]
def alexDiscEquivPreord : AlexDisc ≌ Preord 
goal: AlexDisc ≌ Preord