File: Mathlib/AlgebraicTopology/DoldKan/FunctorN.lean

full_name: AlgebraicTopology.DoldKan.N₁
code: /-- The functor `SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)` which maps
`X` to the formal direct factor of `K[X]` defined by `PInfty`. -/
@[simps]
def N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ) 
goal: CategoryTheory.SimplicialObject C ⥤ CategoryTheory.Idempotents.Karoubi (ChainComplex C ℕ)

full_name: AlgebraicTopology.DoldKan.N₁_map_f
code: /-- The functor `SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)` which maps
`X` to the formal direct factor of `K[X]` defined by `PInfty`. -/
@[simps]
def N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ) 
goal: ∀ {X Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y),
  (AlgebraicTopology.DoldKan.N₁.map f).f =
    AlgebraicTopology.DoldKan.PInfty ≫ AlgebraicTopology.AlternatingFaceMapComplex.map f

full_name: AlgebraicTopology.DoldKan.N₁_obj_X
code: /-- The functor `SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)` which maps
`X` to the formal direct factor of `K[X]` defined by `PInfty`. -/
@[simps]
def N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ) 
goal: (AlgebraicTopology.DoldKan.N₁.obj X).X = AlgebraicTopology.AlternatingFaceMapComplex.obj X

full_name: AlgebraicTopology.DoldKan.N₁_obj_p
code: /-- The functor `SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)` which maps
`X` to the formal direct factor of `K[X]` defined by `PInfty`. -/
@[simps]
def N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ) 
goal: (AlgebraicTopology.DoldKan.N₁.obj X).p = AlgebraicTopology.DoldKan.PInfty

full_name: AlgebraicTopology.DoldKan.N₂_map_f_f
code: /-- The extension of `N₁` to the Karoubi envelope of `SimplicialObject C`. -/
@[simps!]
def N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) 
goal: ∀ {X Y : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)} (f : X ⟶ Y) (i : ℕ),
  (AlgebraicTopology.DoldKan.N₂.map f).f.f i =
    AlgebraicTopology.DoldKan.PInfty.f i ≫ f.f.app (Opposite.op (SimplexCategory.mk i))

full_name: AlgebraicTopology.DoldKan.N₂_obj_p_f
code: /-- The extension of `N₁` to the Karoubi envelope of `SimplicialObject C`. -/
@[simps!]
def N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) 
goal: (AlgebraicTopology.DoldKan.N₂.obj P).p.f i =
  AlgebraicTopology.DoldKan.PInfty.f i ≫ P.p.app (Opposite.op (SimplexCategory.mk i))

full_name: AlgebraicTopology.DoldKan.N₂_obj_X_d
code: /-- The extension of `N₁` to the Karoubi envelope of `SimplicialObject C`. -/
@[simps!]
def N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) 
goal: (AlgebraicTopology.DoldKan.N₂.obj P).X.d i j =
  if h : i = j + 1 then CategoryTheory.eqToHom ⋯ ≫ ∑ i : Fin (j + 2), (-1) ^ ↑i • P.X.δ i else 0

full_name: AlgebraicTopology.DoldKan.N₂
code: /-- The extension of `N₁` to the Karoubi envelope of `SimplicialObject C`. -/
@[simps!]
def N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) 
goal: CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C) ⥤
  CategoryTheory.Idempotents.Karoubi (ChainComplex C ℕ)

full_name: AlgebraicTopology.DoldKan.N₂_obj_X_X
code: /-- The extension of `N₁` to the Karoubi envelope of `SimplicialObject C`. -/
@[simps!]
def N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) 
goal: (AlgebraicTopology.DoldKan.N₂.obj P).X.X n = P.X.obj (Opposite.op (SimplexCategory.mk n))

full_name: AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁
code: /-- The canonical isomorphism `toKaroubi (SimplicialObject C) ⋙ N₂ ≅ N₁`. -/
def toKaroubiCompN₂IsoN₁ : toKaroubi (SimplicialObject C) ⋙ N₂ ≅ N₁ 
goal: CategoryTheory.Idempotents.toKaroubi (CategoryTheory.SimplicialObject C) ⋙ AlgebraicTopology.DoldKan.N₂ ≅
  AlgebraicTopology.DoldKan.N₁

full_name: AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁_hom_app
code: @[simp]
lemma toKaroubiCompN₂IsoN₁_hom_app (X : SimplicialObject C) :
    (toKaroubiCompN₂IsoN₁.hom.app X).f = PInfty 
goal: (AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁.hom.app X).f = AlgebraicTopology.DoldKan.PInfty

full_name: AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁_inv_app
code: @[simp]
lemma toKaroubiCompN₂IsoN₁_inv_app (X : SimplicialObject C) :
    (toKaroubiCompN₂IsoN₁.inv.app X).f = PInfty 
goal: (AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁.inv.app X).f = AlgebraicTopology.DoldKan.PInfty