File: Mathlib/Order/Category/CompleteLat.lean

full_name: CompleteLat
code: /-- The category of complete lattices. -/
def CompleteLat 
goal: Type (u_1 + 1)

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

full_name: CompleteLat.instCompleteLatticeα
code: instance (X : CompleteLat) : CompleteLattice X 
goal: CompleteLattice ↑X

full_name: CompleteLat.of
code: /-- Construct a bundled `CompleteLat` from a `CompleteLattice`. -/
def of (α : Type*) [CompleteLattice α] : CompleteLat 
goal: CompleteLat

full_name: CompleteLat.coe_of
code: @[simp]
theorem coe_of (α : Type*) [CompleteLattice α] : ↥(of α) = α 
goal: ↑(CompleteLat.of α) = α

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

full_name: CompleteLat.instBundledHomCompleteLatticeCompleteLatticeHom
code: instance : BundledHom @CompleteLatticeHom 
goal: CategoryTheory.BundledHom CompleteLatticeHom

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

full_name: CompleteLat.hasForgetToBddLat
code: instance hasForgetToBddLat : HasForget₂ CompleteLat BddLat 
goal: CategoryTheory.HasForget₂ CompleteLat BddLat

full_name: CompleteLat.Iso.mk
code: /-- Constructs an isomorphism of complete lattices from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : CompleteLat.{u}} (e : α ≃o β) : α ≅ β 
goal: α ≅ β

full_name: CompleteLat.Iso.mk_hom_toFun
code: /-- Constructs an isomorphism of complete lattices from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : CompleteLat.{u}} (e : α ≃o β) : α ≅ β 
goal: (CompleteLat.Iso.mk e).hom.toFun a = e a

full_name: CompleteLat.Iso.mk_inv_toFun
code: /-- Constructs an isomorphism of complete lattices from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : CompleteLat.{u}} (e : α ≃o β) : α ≅ β 
goal: (CompleteLat.Iso.mk e).inv.toFun a = e.symm a

full_name: CompleteLat.dual_obj
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : CompleteLat ⥤ CompleteLat 
goal: CompleteLat.dual.obj X = CompleteLat.of (↑X)ᵒᵈ

full_name: CompleteLat.dual
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : CompleteLat ⥤ CompleteLat 
goal: CompleteLat ⥤ CompleteLat

full_name: CompleteLat.dual_map
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : CompleteLat ⥤ CompleteLat 
goal: CompleteLat.dual.map a = CompleteLatticeHom.dual a

full_name: CompleteLat.dualEquiv_inverse
code: /-- The equivalence between `CompleteLat` and itself induced by `OrderDual` both ways. -/
@[simps functor inverse]
def dualEquiv : CompleteLat ≌ CompleteLat 
goal: CompleteLat.dualEquiv.inverse = CompleteLat.dual

full_name: CompleteLat.dualEquiv_functor
code: /-- The equivalence between `CompleteLat` and itself induced by `OrderDual` both ways. -/
@[simps functor inverse]
def dualEquiv : CompleteLat ≌ CompleteLat 
goal: CompleteLat.dualEquiv.functor = CompleteLat.dual

full_name: CompleteLat.dualEquiv
code: /-- The equivalence between `CompleteLat` and itself induced by `OrderDual` both ways. -/
@[simps functor inverse]
def dualEquiv : CompleteLat ≌ CompleteLat 
goal: CompleteLat ≌ CompleteLat

full_name: completeLat_dual_comp_forget_to_bddLat
code: theorem completeLat_dual_comp_forget_to_bddLat :
    CompleteLat.dual ⋙ forget₂ CompleteLat BddLat =
    forget₂ CompleteLat BddLat ⋙ BddLat.dual 
goal: CompleteLat.dual ⋙ CategoryTheory.forget₂ CompleteLat BddLat = CategoryTheory.forget₂ CompleteLat BddLat ⋙ BddLat.dual