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Mathlib.AlgebraicTopology.DoldKan.NCompGamma

The unit isomorphism of the Dold-Kan equivalence

In order to construct the unit isomorphism of the Dold-Kan equivalence, we first construct natural transformations Γ₂N₁.natTrans : N₁ ⋙ Γ₂ ⟶ toKaroubi (SimplicialObject C) and Γ₂N₂.natTrans : N₂ ⋙ Γ₂ ⟶ 𝟭 (SimplicialObject C). It is then shown that Γ₂N₂.natTrans is an isomorphism by using that it becomes an isomorphism after the application of the functor N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) which reflects isomorphisms.

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

@[irreducible]

The natural transformation N₁ ⋙ Γ₂ ⟶ toKaroubi (SimplicialObject C).

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    @[irreducible]
    def AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] :
    (CategoryTheory.Idempotents.toKaroubi (CategoryTheory.SimplicialObject C)).comp (AlgebraicTopology.DoldKan.N₂.comp AlgebraicTopology.DoldKan.Γ₂) AlgebraicTopology.DoldKan.N₁.comp AlgebraicTopology.DoldKan.Γ₂

    The compatibility isomorphism relating N₂ ⋙ Γ₂ and N₁ ⋙ Γ₂.

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      @[simp]
      theorem AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso_hom_app {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : CategoryTheory.SimplicialObject C) :
      AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso.hom.app X = AlgebraicTopology.DoldKan.Γ₂.map (AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁.hom.app X)
      @[simp]
      theorem AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso_inv_app {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : CategoryTheory.SimplicialObject C) :
      AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso.inv.app X = AlgebraicTopology.DoldKan.Γ₂.map (AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁.inv.app X)
      @[irreducible]

      The natural transformation N₂ ⋙ Γ₂ ⟶ 𝟭 (SimplicialObject C).

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        theorem AlgebraicTopology.DoldKan.Γ₂N₂.natTrans_app_f_app {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)) :
        AlgebraicTopology.DoldKan.Γ₂N₂.natTrans.app P = CategoryTheory.CategoryStruct.comp ((AlgebraicTopology.DoldKan.N₂.comp AlgebraicTopology.DoldKan.Γ₂).map P.decompId_i) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.CategoryStruct.comp AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso.hom AlgebraicTopology.DoldKan.Γ₂N₁.natTrans).app P.X) P.decompId_p)
        theorem AlgebraicTopology.DoldKan.compatibility_Γ₂N₁_Γ₂N₂_natTrans {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : CategoryTheory.SimplicialObject C) :
        AlgebraicTopology.DoldKan.Γ₂N₁.natTrans.app X = CategoryTheory.CategoryStruct.comp (AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso.app X).inv (AlgebraicTopology.DoldKan.Γ₂N₂.natTrans.app ((CategoryTheory.Idempotents.toKaroubi (CategoryTheory.SimplicialObject C)).obj X))
        theorem AlgebraicTopology.DoldKan.identity_N₂_objectwise {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)) :
        CategoryTheory.CategoryStruct.comp (AlgebraicTopology.DoldKan.N₂Γ₂.inv.app (AlgebraicTopology.DoldKan.N₂.obj P)) (AlgebraicTopology.DoldKan.N₂.map (AlgebraicTopology.DoldKan.Γ₂N₂.natTrans.app P)) = CategoryTheory.CategoryStruct.id (AlgebraicTopology.DoldKan.N₂.obj P)
        theorem AlgebraicTopology.DoldKan.identity_N₂ {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] :
        CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id AlgebraicTopology.DoldKan.N₂ AlgebraicTopology.DoldKan.N₂Γ₂.inv) (CategoryTheory.CategoryStruct.comp (AlgebraicTopology.DoldKan.N₂.associator AlgebraicTopology.DoldKan.Γ₂ AlgebraicTopology.DoldKan.N₂).inv (AlgebraicTopology.DoldKan.Γ₂N₂.natTrans CategoryTheory.CategoryStruct.id AlgebraicTopology.DoldKan.N₂)) = CategoryTheory.CategoryStruct.id AlgebraicTopology.DoldKan.N₂
        @[simp]
        theorem AlgebraicTopology.DoldKan.Γ₂N₂_inv {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] :
        AlgebraicTopology.DoldKan.Γ₂N₂.inv = AlgebraicTopology.DoldKan.Γ₂N₂.natTrans

        The unit isomorphism of the Dold-Kan equivalence.

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        • AlgebraicTopology.DoldKan.Γ₂N₂ = (CategoryTheory.asIso AlgebraicTopology.DoldKan.Γ₂N₂.natTrans).symm
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          @[simp]
          theorem AlgebraicTopology.DoldKan.Γ₂N₁_inv {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] :
          AlgebraicTopology.DoldKan.Γ₂N₁.inv = AlgebraicTopology.DoldKan.Γ₂N₁.natTrans

          The natural isomorphism toKaroubi (SimplicialObject C) ≅ N₁ ⋙ Γ₂.

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          • AlgebraicTopology.DoldKan.Γ₂N₁ = (CategoryTheory.asIso AlgebraicTopology.DoldKan.Γ₂N₁.natTrans).symm
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