File: Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean

full_name: AlgebraicTopology.DoldKan.PInfty_comp_map_mono_eq_zero
code: theorem PInfty_comp_map_mono_eq_zero (X : SimplicialObject C) {n : ℕ} {Δ' : SimplexCategory}
    (i : Δ' ⟶ [n]) [hi : Mono i] (h₁ : Δ'.len ≠ n) (h₂ : ¬Isδ₀ i) :
    PInfty.f n ≫ X.map i.op = 0 
goal: AlgebraicTopology.DoldKan.PInfty.f n ≫ X.map i.op = 0

full_name: AlgebraicTopology.DoldKan.Γ₀_obj_termwise_mapMono_comp_PInfty
code: @[reassoc]
theorem Γ₀_obj_termwise_mapMono_comp_PInfty (X : SimplicialObject C) {Δ Δ' : SimplexCategory}
    (i : Δ ⟶ Δ') [Mono i] :
    Γ₀.Obj.Termwise.mapMono (AlternatingFaceMapComplex.obj X) i ≫ PInfty.f Δ.len =
      PInfty.f Δ'.len ≫ X.map i.op 
goal: AlgebraicTopology.DoldKan.Γ₀.Obj.Termwise.mapMono K[X] i ≫ AlgebraicTopology.DoldKan.PInfty.f Δ.len =
  AlgebraicTopology.DoldKan.PInfty.f Δ'.len ≫ X.map i.op

full_name: AlgebraicTopology.DoldKan.Γ₂N₁.natTrans_app_f_app
code: /-- The natural transformation `N₁ ⋙ Γ₂ ⟶ toKaroubi (SimplicialObject C)`. -/
@[simps]
def natTrans : (N₁ : SimplicialObject C ⥤ _) ⋙ Γ₂ ⟶ toKaroubi _ 
goal: (AlgebraicTopology.DoldKan.Γ₂N₁.natTrans.app X).f.app Δ =
  (AlgebraicTopology.DoldKan.Γ₀.splitting K[X]).desc Δ fun A =>
    AlgebraicTopology.DoldKan.PInfty.f (Opposite.unop A.fst).len ≫ X.map A.e.op

full_name: AlgebraicTopology.DoldKan.Γ₂N₁.natTrans
code: /-- The natural transformation `N₁ ⋙ Γ₂ ⟶ toKaroubi (SimplicialObject C)`. -/
@[simps]
def natTrans : (N₁ : SimplicialObject C ⥤ _) ⋙ Γ₂ ⟶ toKaroubi _ 
goal: AlgebraicTopology.DoldKan.N₁ ⋙ AlgebraicTopology.DoldKan.Γ₂ ⟶
  CategoryTheory.Idempotents.toKaroubi (CategoryTheory.SimplicialObject C)

full_name: AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso
code: /-- The compatibility isomorphism relating `N₂ ⋙ Γ₂` and `N₁ ⋙ Γ₂`. -/
def Γ₂N₂ToKaroubiIso : toKaroubi (SimplicialObject C) ⋙ N₂ ⋙ Γ₂ ≅ N₁ ⋙ Γ₂ 
goal: CategoryTheory.Idempotents.toKaroubi (CategoryTheory.SimplicialObject C) ⋙
    AlgebraicTopology.DoldKan.N₂ ⋙ AlgebraicTopology.DoldKan.Γ₂ ≅
  AlgebraicTopology.DoldKan.N₁ ⋙ AlgebraicTopology.DoldKan.Γ₂

full_name: AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso_hom_app
code: @[simp]
lemma Γ₂N₂ToKaroubiIso_hom_app (X : SimplicialObject C) :
    Γ₂N₂ToKaroubiIso.hom.app X = Γ₂.map (toKaroubiCompN₂IsoN₁.hom.app X) 
goal: AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso.hom.app X =
  AlgebraicTopology.DoldKan.Γ₂.map (AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁.hom.app X)

full_name: AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso_inv_app
code: @[simp]
lemma Γ₂N₂ToKaroubiIso_inv_app (X : SimplicialObject C) :
    Γ₂N₂ToKaroubiIso.inv.app X = Γ₂.map (toKaroubiCompN₂IsoN₁.inv.app X) 
goal: AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso.inv.app X =
  AlgebraicTopology.DoldKan.Γ₂.map (AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁.inv.app X)

full_name: AlgebraicTopology.DoldKan.Γ₂N₂.natTrans
code: /-- The natural transformation `N₂ ⋙ Γ₂ ⟶ 𝟭 (SimplicialObject C)`. -/
def natTrans : (N₂ : Karoubi (SimplicialObject C) ⥤ _) ⋙ Γ₂ ⟶ 𝟭 _ 
goal: AlgebraicTopology.DoldKan.N₂ ⋙ AlgebraicTopology.DoldKan.Γ₂ ⟶
  𝟭 (CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C))

full_name: AlgebraicTopology.DoldKan.Γ₂N₂.natTrans_app_f_app
code: theorem natTrans_app_f_app (P : Karoubi (SimplicialObject C)) :
    Γ₂N₂.natTrans.app P =
      (N₂ ⋙ Γ₂).map P.decompId_i ≫
        (Γ₂N₂ToKaroubiIso.hom ≫ Γ₂N₁.natTrans).app P.X ≫ P.decompId_p 
goal: AlgebraicTopology.DoldKan.Γ₂N₂.natTrans.app P =
  (AlgebraicTopology.DoldKan.N₂ ⋙ AlgebraicTopology.DoldKan.Γ₂).map P.decompId_i ≫
    (AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso.hom ≫ AlgebraicTopology.DoldKan.Γ₂N₁.natTrans).app P.X ≫ P.decompId_p

full_name: AlgebraicTopology.DoldKan.compatibility_Γ₂N₁_Γ₂N₂_natTrans
code: theorem compatibility_Γ₂N₁_Γ₂N₂_natTrans (X : SimplicialObject C) :
    Γ₂N₁.natTrans.app X =
      (Γ₂N₂ToKaroubiIso.app X).inv ≫
        Γ₂N₂.natTrans.app ((toKaroubi (SimplicialObject C)).obj X) 
goal: AlgebraicTopology.DoldKan.Γ₂N₁.natTrans.app X =
  (AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso.app X).inv ≫
    AlgebraicTopology.DoldKan.Γ₂N₂.natTrans.app
      ((CategoryTheory.Idempotents.toKaroubi (CategoryTheory.SimplicialObject C)).obj X)

full_name: AlgebraicTopology.DoldKan.identity_N₂_objectwise
code: theorem identity_N₂_objectwise (P : Karoubi (SimplicialObject C)) :
    (N₂Γ₂.inv.app (N₂.obj P) : N₂.obj P ⟶ N₂.obj (Γ₂.obj (N₂.obj P))) ≫
    N₂.map (Γ₂N₂.natTrans.app P) = 𝟙 (N₂.obj P) 
goal: AlgebraicTopology.DoldKan.N₂Γ₂.inv.app (AlgebraicTopology.DoldKan.N₂.obj P) ≫
    AlgebraicTopology.DoldKan.N₂.map (AlgebraicTopology.DoldKan.Γ₂N₂.natTrans.app P) =
  𝟙 (AlgebraicTopology.DoldKan.N₂.obj P)

full_name: AlgebraicTopology.DoldKan.identity_N₂
code: theorem identity_N₂ :
    (𝟙 (N₂ : Karoubi (SimplicialObject C) ⥤ _) ◫ N₂Γ₂.inv) ≫
    (Functor.associator _ _ _).inv ≫ Γ₂N₂.natTrans ◫ 𝟙 (@N₂ C _ _) = 𝟙 N₂ 
goal: (𝟙 AlgebraicTopology.DoldKan.N₂ ◫ AlgebraicTopology.DoldKan.N₂Γ₂.inv) ≫
    (AlgebraicTopology.DoldKan.N₂.associator AlgebraicTopology.DoldKan.Γ₂ AlgebraicTopology.DoldKan.N₂).inv ≫
      AlgebraicTopology.DoldKan.Γ₂N₂.natTrans ◫ 𝟙 AlgebraicTopology.DoldKan.N₂ =
  𝟙 AlgebraicTopology.DoldKan.N₂

full_name: AlgebraicTopology.DoldKan.instIsIsoFunctorKaroubiSimplicialObjectNatTrans
code: instance : IsIso (Γ₂N₂.natTrans : (N₂ : Karoubi (SimplicialObject C) ⥤ _) ⋙ _ ⟶ _) 
goal: CategoryTheory.IsIso AlgebraicTopology.DoldKan.Γ₂N₂.natTrans

full_name: AlgebraicTopology.DoldKan.instIsIsoFunctorSimplicialObjectKaroubiNatTrans
code: instance : IsIso (Γ₂N₁.natTrans : (N₁ : SimplicialObject C ⥤ _) ⋙ _ ⟶ _) 
goal: CategoryTheory.IsIso AlgebraicTopology.DoldKan.Γ₂N₁.natTrans

full_name: AlgebraicTopology.DoldKan.Γ₂N₂
code: /-- The unit isomorphism of the Dold-Kan equivalence. -/
@[simps! inv]
def Γ₂N₂ : 𝟭 _ ≅ (N₂ : Karoubi (SimplicialObject C) ⥤ _) ⋙ Γ₂ 
goal: 𝟭 (CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)) ≅
  AlgebraicTopology.DoldKan.N₂ ⋙ AlgebraicTopology.DoldKan.Γ₂

full_name: AlgebraicTopology.DoldKan.Γ₂N₂_inv
code: /-- The unit isomorphism of the Dold-Kan equivalence. -/
@[simps! inv]
def Γ₂N₂ : 𝟭 _ ≅ (N₂ : Karoubi (SimplicialObject C) ⥤ _) ⋙ Γ₂ 
goal: AlgebraicTopology.DoldKan.Γ₂N₂.inv = AlgebraicTopology.DoldKan.Γ₂N₂.natTrans

full_name: AlgebraicTopology.DoldKan.Γ₂N₁
code: /-- The natural isomorphism `toKaroubi (SimplicialObject C) ≅ N₁ ⋙ Γ₂`. -/
@[simps! inv]
def Γ₂N₁ : toKaroubi _ ≅ (N₁ : SimplicialObject C ⥤ _) ⋙ Γ₂ 
goal: CategoryTheory.Idempotents.toKaroubi (CategoryTheory.SimplicialObject C) ≅
  AlgebraicTopology.DoldKan.N₁ ⋙ AlgebraicTopology.DoldKan.Γ₂

full_name: AlgebraicTopology.DoldKan.Γ₂N₁_inv
code: /-- The natural isomorphism `toKaroubi (SimplicialObject C) ≅ N₁ ⋙ Γ₂`. -/
@[simps! inv]
def Γ₂N₁ : toKaroubi _ ≅ (N₁ : SimplicialObject C ⥤ _) ⋙ Γ₂ 
goal: AlgebraicTopology.DoldKan.Γ₂N₁.inv = AlgebraicTopology.DoldKan.Γ₂N₁.natTrans