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Mathlib.AlgebraicTopology.DoldKan.FunctorN

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_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

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

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Instances For
    @[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 : ), (AlgebraicTopology.DoldKan.N₂.map f).f.f i = CategoryTheory.CategoryStruct.comp (AlgebraicTopology.DoldKan.PInfty.f i) (f.f.app (Opposite.op (SimplexCategory.mk i)))
    @[simp]
    @[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 : ) (j : ) :
    (AlgebraicTopology.DoldKan.N₂.obj P).X.d i j = if h : i = j + 1 then CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom ) (i : Fin (j + 2), (-1) ^ i P.X i) else 0

    The canonical isomorphism toKaroubi (SimplicialObject C) ⋙ N₂N₁.

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      @[simp]
      theorem AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁_hom_app {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) :
      (AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁.hom.app X).f = AlgebraicTopology.DoldKan.PInfty
      @[simp]
      theorem AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁_inv_app {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) :
      (AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁.inv.app X).f = AlgebraicTopology.DoldKan.PInfty