File: Mathlib/CategoryTheory/Limits/IsConnected.lean

full_name: CategoryTheory.Limits.Types.constPUnitFunctor
code: /-- The functor mapping every object to `PUnit`. -/
def constPUnitFunctor : C ⥤ Type w 
goal: CategoryTheory.Functor C (Type w)

full_name: CategoryTheory.Limits.Types.pUnitCocone_ι_app
code: /-- The cocone on `constPUnitFunctor` with cone point `PUnit`. -/
@[simps]
def pUnitCocone : Cocone (constPUnitFunctor.{w} C) 
goal: (CategoryTheory.Limits.Types.pUnitCocone C).ι.app X a = id a

full_name: CategoryTheory.Limits.Types.pUnitCocone
code: /-- The cocone on `constPUnitFunctor` with cone point `PUnit`. -/
@[simps]
def pUnitCocone : Cocone (constPUnitFunctor.{w} C) 
goal: CategoryTheory.Limits.Cocone (CategoryTheory.Limits.Types.constPUnitFunctor C)

full_name: CategoryTheory.Limits.Types.pUnitCocone_pt
code: /-- The cocone on `constPUnitFunctor` with cone point `PUnit`. -/
@[simps]
def pUnitCocone : Cocone (constPUnitFunctor.{w} C) 
goal: (CategoryTheory.Limits.Types.pUnitCocone C).pt = PUnit.{w + 1}

full_name: CategoryTheory.Limits.Types.isColimitPUnitCocone
code: /-- If `C` is connected, the cocone on `constPUnitFunctor` with cone point `PUnit` is a colimit
    cocone. -/
noncomputable def isColimitPUnitCocone [IsConnected C] : IsColimit (pUnitCocone.{w} C) 
goal: CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Types.pUnitCocone C)

full_name: CategoryTheory.Limits.Types.instHasColimitConstPUnitFunctor
code: instance instHasColimitConstPUnitFunctor [IsConnected C] : HasColimit (constPUnitFunctor.{w} C) 
goal: CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.Types.constPUnitFunctor C)

full_name: CategoryTheory.Limits.Types.instSubsingletonColimitPUnit
code: instance instSubsingletonColimitPUnit
    [IsPreconnected C] [HasColimit (constPUnitFunctor.{w} C)] :
    Subsingleton (colimit (constPUnitFunctor.{w} C)) 
goal: Subsingleton (CategoryTheory.Limits.colimit (CategoryTheory.Limits.Types.constPUnitFunctor C))

full_name: CategoryTheory.Limits.Types.colimitConstPUnitIsoPUnit
code: /-- Given a connected index category, the colimit of the constant unit-valued functor is `PUnit`. -/
noncomputable def colimitConstPUnitIsoPUnit [IsConnected C] :
    colimit (constPUnitFunctor.{w} C) ≅ PUnit.{w + 1} 
goal: CategoryTheory.Limits.colimit (CategoryTheory.Limits.Types.constPUnitFunctor C) ≅ PUnit.{w + 1}

full_name: CategoryTheory.Limits.Types.zigzag_of_eqvGen_quot_rel
code: /-- Let `F` be a `Type`-valued functor. If two elements `a : F c` and `b : F d` represent the same
element of `colimit F`, then `c` and `d` are related by a `Zigzag`. -/
theorem zigzag_of_eqvGen_quot_rel (F : C ⥤ Type w) (c d : Σ j, F.obj j)
    (h : EqvGen (Quot.Rel F) c d) : Zigzag c.1 d.1 
goal: CategoryTheory.Zigzag c.fst d.fst

full_name: CategoryTheory.Limits.Types.isConnected_iff_colimit_constPUnitFunctor_iso_pUnit
code: /-- An index category is connected iff the colimit of the constant singleton-valued functor is a
singleton. -/
theorem isConnected_iff_colimit_constPUnitFunctor_iso_pUnit
    [HasColimit (constPUnitFunctor.{w} C)] :
    IsConnected C ↔ Nonempty (colimit (constPUnitFunctor.{w} C) ≅ PUnit) 
goal: CategoryTheory.IsConnected C ↔
  Nonempty (CategoryTheory.Limits.colimit (CategoryTheory.Limits.Types.constPUnitFunctor C) ≅ PUnit.{w + 1})

full_name: CategoryTheory.Limits.Types.isConnected_iff_isColimit_pUnitCocone
code: theorem isConnected_iff_isColimit_pUnitCocone :
    IsConnected C ↔ Nonempty (IsColimit (pUnitCocone.{w} C)) 
goal: CategoryTheory.IsConnected C ↔ Nonempty (CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Types.pUnitCocone C))

full_name: CategoryTheory.Limits.Types.isConnected_iff_of_final
code: /-- The domain of a final functor is connected if and only if its codomain is connected. -/
theorem isConnected_iff_of_final (F : C ⥤ D) [CategoryTheory.Functor.Final F] :
    IsConnected C ↔ IsConnected D 
goal: CategoryTheory.IsConnected C ↔ CategoryTheory.IsConnected D