Construction of functors N for the Dold-Kan correspondence

In this file, we construct functors N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ) and N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) for any preadditive category C. (The indices of these functors are the number of occurrences of Karoubi at the source or the target.)

In the case C is additive, the functor N₂ shall be the functor of the equivalence CategoryTheory.Preadditive.DoldKan.equivalence defined in EquivalenceAdditive.lean.

In the case the category C is pseudoabelian, the composition of N₁ with the inverse of the equivalence ChainComplex C ℕ ⥤ Karoubi (ChainComplex C ℕ) will be the functor CategoryTheory.Idempotents.DoldKan.N of the equivalence of categories CategoryTheory.Idempotents.DoldKan.equivalence : SimplicialObject C ≌ ChainComplex C ℕ defined in EquivalencePseudoabelian.lean.

When the category C is abelian, a relation between N₁ and the normalized Moore complex functor shall be obtained in Normalized.lean.

(See Equivalence.lean for the general strategy of proof of the Dold-Kan equivalence.)

@[simp]

theorem AlgebraicTopology.DoldKan.N₁_map_f {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] : ∀ {X Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y), (AlgebraicTopology.DoldKan.N₁.map f).f = CategoryTheory.CategoryStruct.comp AlgebraicTopology.DoldKan.PInfty (AlgebraicTopology.AlternatingFaceMapComplex.map f)

@[simp]

theorem AlgebraicTopology.DoldKan.N₁_obj_p {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) : (AlgebraicTopology.DoldKan.N₁.obj X).p = AlgebraicTopology.DoldKan.PInfty

@[simp]

theorem AlgebraicTopology.DoldKan.N₁_obj_X {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) : (AlgebraicTopology.DoldKan.N₁.obj X).X = AlgebraicTopology.AlternatingFaceMapComplex.obj X

def AlgebraicTopology.DoldKan.N₁ {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] : CategoryTheory.Functor (CategoryTheory.SimplicialObject C) (CategoryTheory.Idempotents.Karoubi (ChainComplex C ℕ))

The functor SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ) which maps X to the formal direct factor of K[X] defined by PInfty.

@[simp]

theorem AlgebraicTopology.DoldKan.N₂_map_f_f {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] : ∀ {X Y : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)} (f : X ⟶ Y) (i : ℕ), HomologicalComplex.Hom.f (AlgebraicTopology.DoldKan.N₂.map f).f i = CategoryTheory.CategoryStruct.comp (HomologicalComplex.Hom.f AlgebraicTopology.DoldKan.PInfty i) (f.f.app (Opposite.op (SimplexCategory.mk i)))

@[simp]

theorem AlgebraicTopology.DoldKan.N₂_obj_X_d {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)) (i : ℕ) (j : ℕ) : HomologicalComplex.d (AlgebraicTopology.DoldKan.N₂.obj P).X i j = if h : i = j + 1 then CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom (_ : P.X.obj (Opposite.op (SimplexCategory.mk i)) = P.X.obj (Opposite.op (SimplexCategory.mk (j + 1))))) (Finset.sum Finset.univ fun i => (-1) ^ ↑i • CategoryTheory.SimplicialObject.δ P.X i) else 0

@[simp]

theorem AlgebraicTopology.DoldKan.N₂_obj_X_X {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)) (n : ℕ) : HomologicalComplex.X (AlgebraicTopology.DoldKan.N₂.obj P).X n = P.X.obj (Opposite.op (SimplexCategory.mk n))

@[simp]

theorem AlgebraicTopology.DoldKan.N₂_obj_p_f {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)) (i : ℕ) : HomologicalComplex.Hom.f (AlgebraicTopology.DoldKan.N₂.obj P).p i = CategoryTheory.CategoryStruct.comp (HomologicalComplex.Hom.f AlgebraicTopology.DoldKan.PInfty i) (P.p.app (Opposite.op (SimplexCategory.mk i)))

def AlgebraicTopology.DoldKan.N₂ {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] : CategoryTheory.Functor (CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)) (CategoryTheory.Idempotents.Karoubi (ChainComplex C ℕ))

The extension of N₁ to the Karoubi envelope of SimplicialObject C.

theorem AlgebraicTopology.DoldKan.compatibility_N₁_N₂ {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] : CategoryTheory.Functor.comp (CategoryTheory.Idempotents.toKaroubi (CategoryTheory.SimplicialObject C)) AlgebraicTopology.DoldKan.N₂ = AlgebraicTopology.DoldKan.N₁