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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.)

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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.

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    @[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) :
    (N₁.obj X).p = PInfty
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    @[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✝) :
    (N₁.map f).f = CategoryTheory.CategoryStruct.comp PInfty (AlternatingFaceMapComplex.map f)
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    @[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) :
    (N₁.obj X).X = AlternatingFaceMapComplex.obj X
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    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.

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      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 : ℕ) :
      (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.δ i) else 0
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      @[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) :
      (N₂.obj P).p.f i = CategoryTheory.CategoryStruct.comp (PInfty.f i) (P.p.app (Opposite.op (SimplexCategory.mk i)))
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      @[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) :
      (N₂.obj P).X.X n = P.X.obj (Opposite.op (SimplexCategory.mk n))
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      @[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) :
      (N₂.map f).f.f i = CategoryTheory.CategoryStruct.comp (PInfty.f i) (f.f.app (Opposite.op (SimplexCategory.mk i)))
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      def AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁ {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] :
      (CategoryTheory.Idempotents.toKaroubi (CategoryTheory.SimplicialObject C)).comp N₂ ≅ N₁

      The canonical isomorphism toKaroubi (SimplicialObject C) ⋙ N₂ ≅ N₁.

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