File: Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean

full_name: AlgebraicTopology.DoldKan.Γ₀NondegComplexIso_inv_f
code: /-- The isomorphism `(Γ₀.splitting K).nondegComplex ≅ K` for all `K : ChainComplex C ℕ`. -/
@[simps!]
def Γ₀NondegComplexIso (K : ChainComplex C ℕ) : (Γ₀.splitting K).nondegComplex ≅ K 
goal: (AlgebraicTopology.DoldKan.Γ₀NondegComplexIso K).inv.f i = 𝟙 ((AlgebraicTopology.DoldKan.Γ₀.splitting K).N i)

full_name: AlgebraicTopology.DoldKan.Γ₀NondegComplexIso_hom_f
code: /-- The isomorphism `(Γ₀.splitting K).nondegComplex ≅ K` for all `K : ChainComplex C ℕ`. -/
@[simps!]
def Γ₀NondegComplexIso (K : ChainComplex C ℕ) : (Γ₀.splitting K).nondegComplex ≅ K 
goal: (AlgebraicTopology.DoldKan.Γ₀NondegComplexIso K).hom.f i = 𝟙 ((AlgebraicTopology.DoldKan.Γ₀.splitting K).N i)

full_name: AlgebraicTopology.DoldKan.Γ₀NondegComplexIso
code: /-- The isomorphism `(Γ₀.splitting K).nondegComplex ≅ K` for all `K : ChainComplex C ℕ`. -/
@[simps!]
def Γ₀NondegComplexIso (K : ChainComplex C ℕ) : (Γ₀.splitting K).nondegComplex ≅ K 
goal: (AlgebraicTopology.DoldKan.Γ₀.splitting K).nondegComplex ≅ K

full_name: AlgebraicTopology.DoldKan.Γ₀'CompNondegComplexFunctor
code: /-- The natural isomorphism `(Γ₀.splitting K).nondegComplex ≅ K` for `K : ChainComplex C ℕ`. -/
def Γ₀'CompNondegComplexFunctor : Γ₀' ⋙ Split.nondegComplexFunctor ≅ 𝟭 (ChainComplex C ℕ) 
goal: AlgebraicTopology.DoldKan.Γ₀' ⋙ SimplicialObject.Split.nondegComplexFunctor ≅ 𝟭 (ChainComplex C ℕ)

full_name: AlgebraicTopology.DoldKan.N₁Γ₀
code: /-- The natural isomorphism `Γ₀ ⋙ N₁ ≅ toKaroubi (ChainComplex C ℕ)`. -/
def N₁Γ₀ : Γ₀ ⋙ N₁ ≅ toKaroubi (ChainComplex C ℕ) 
goal: AlgebraicTopology.DoldKan.Γ₀ ⋙ AlgebraicTopology.DoldKan.N₁ ≅ CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ)

full_name: AlgebraicTopology.DoldKan.N₁Γ₀_app
code: theorem N₁Γ₀_app (K : ChainComplex C ℕ) :
    N₁Γ₀.app K = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.symm ≪≫
      (toKaroubi _).mapIso (Γ₀NondegComplexIso K) 
goal: AlgebraicTopology.DoldKan.N₁Γ₀.app K =
  (AlgebraicTopology.DoldKan.Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.symm ≪≫
    (CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ)).mapIso (AlgebraicTopology.DoldKan.Γ₀NondegComplexIso K)

full_name: AlgebraicTopology.DoldKan.N₁Γ₀_hom_app
code: theorem N₁Γ₀_hom_app (K : ChainComplex C ℕ) :
    N₁Γ₀.hom.app K = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv ≫
        (toKaroubi _).map (Γ₀NondegComplexIso K).hom 
goal: AlgebraicTopology.DoldKan.N₁Γ₀.hom.app K =
  (AlgebraicTopology.DoldKan.Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv ≫
    (CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ)).map (AlgebraicTopology.DoldKan.Γ₀NondegComplexIso K).hom

full_name: AlgebraicTopology.DoldKan.N₁Γ₀_inv_app
code: theorem N₁Γ₀_inv_app (K : ChainComplex C ℕ) :
    N₁Γ₀.inv.app K = (toKaroubi _).map (Γ₀NondegComplexIso K).inv ≫
        (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom 
goal: AlgebraicTopology.DoldKan.N₁Γ₀.inv.app K =
  (CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ)).map (AlgebraicTopology.DoldKan.Γ₀NondegComplexIso K).inv ≫
    (AlgebraicTopology.DoldKan.Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom

full_name: AlgebraicTopology.DoldKan.N₁Γ₀_hom_app_f_f
code: @[simp]
theorem N₁Γ₀_hom_app_f_f (K : ChainComplex C ℕ) (n : ℕ) :
    (N₁Γ₀.hom.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv.f.f n 
goal: (AlgebraicTopology.DoldKan.N₁Γ₀.hom.app K).f.f n =
  (AlgebraicTopology.DoldKan.Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv.f.f n

full_name: AlgebraicTopology.DoldKan.N₁Γ₀_inv_app_f_f
code: @[simp]
theorem N₁Γ₀_inv_app_f_f (K : ChainComplex C ℕ) (n : ℕ) :
    (N₁Γ₀.inv.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom.f.f n 
goal: (AlgebraicTopology.DoldKan.N₁Γ₀.inv.app K).f.f n =
  (AlgebraicTopology.DoldKan.Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom.f.f n

full_name: AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso
code: /-- Compatibility isomorphism between `toKaroubi _ ⋙ Γ₂ ⋙ N₂` and `Γ₀ ⋙ N₁` which
are functors `ChainComplex C ℕ ⥤ Karoubi (ChainComplex C ℕ)`. -/
def N₂Γ₂ToKaroubiIso : toKaroubi (ChainComplex C ℕ) ⋙ Γ₂ ⋙ N₂ ≅ Γ₀ ⋙ N₁ 
goal: CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ) ⋙ AlgebraicTopology.DoldKan.Γ₂ ⋙ AlgebraicTopology.DoldKan.N₂ ≅
  AlgebraicTopology.DoldKan.Γ₀ ⋙ AlgebraicTopology.DoldKan.N₁

full_name: AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso_hom_app
code: @[simp]
lemma N₂Γ₂ToKaroubiIso_hom_app (X : ChainComplex C ℕ) :
    (N₂Γ₂ToKaroubiIso.hom.app X).f = PInfty 
goal: (AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso.hom.app X).f = AlgebraicTopology.DoldKan.PInfty

full_name: AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso_inv_app
code: @[simp]
lemma N₂Γ₂ToKaroubiIso_inv_app (X : ChainComplex C ℕ) :
    (N₂Γ₂ToKaroubiIso.inv.app X).f = PInfty 
goal: (AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso.inv.app X).f = AlgebraicTopology.DoldKan.PInfty

full_name: AlgebraicTopology.DoldKan.N₂Γ₂
code: /-- The counit isomorphism of the Dold-Kan equivalence for additive categories. -/
def N₂Γ₂ : Γ₂ ⋙ N₂ ≅ 𝟭 (Karoubi (ChainComplex C ℕ)) 
goal: AlgebraicTopology.DoldKan.Γ₂ ⋙ AlgebraicTopology.DoldKan.N₂ ≅ 𝟭 (CategoryTheory.Idempotents.Karoubi (ChainComplex C ℕ))

full_name: AlgebraicTopology.DoldKan.N₂Γ₂_inv_app_f_f
code: @[simp]
theorem N₂Γ₂_inv_app_f_f (X : Karoubi (ChainComplex C ℕ)) (n : ℕ) :
    (N₂Γ₂.inv.app X).f.f n =
      X.p.f n ≫ ((Γ₀.splitting X.X).cofan _).inj (Splitting.IndexSet.id (op [n])) 
goal: (AlgebraicTopology.DoldKan.N₂Γ₂.inv.app X).f.f n =
  X.p.f n ≫
    ((AlgebraicTopology.DoldKan.Γ₀.splitting X.X).cofan (Opposite.op [n])).inj
      (SimplicialObject.Splitting.IndexSet.id (Opposite.op [n]))

full_name: AlgebraicTopology.DoldKan.whiskerLeft_toKaroubi_N₂Γ₂_hom
code: lemma whiskerLeft_toKaroubi_N₂Γ₂_hom :
    whiskerLeft (toKaroubi (ChainComplex C ℕ)) N₂Γ₂.hom = N₂Γ₂ToKaroubiIso.hom ≫ N₁Γ₀.hom 
goal: CategoryTheory.whiskerLeft (CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ))
    AlgebraicTopology.DoldKan.N₂Γ₂.hom =
  AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso.hom ≫ AlgebraicTopology.DoldKan.N₁Γ₀.hom

full_name: AlgebraicTopology.DoldKan.N₂Γ₂_compatible_with_N₁Γ₀
code: theorem N₂Γ₂_compatible_with_N₁Γ₀ (K : ChainComplex C ℕ) :
    N₂Γ₂.hom.app ((toKaroubi _).obj K) = N₂Γ₂ToKaroubiIso.hom.app K ≫ N₁Γ₀.hom.app K 
goal: AlgebraicTopology.DoldKan.N₂Γ₂.hom.app ((CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ)).obj K) =
  AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso.hom.app K ≫ AlgebraicTopology.DoldKan.N₁Γ₀.hom.app K