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Mathlib.AlgebraicTopology.DoldKan.HomotopyEquivalence

The normalized Moore complex and the alternating face map complex are homotopy equivalent #

In this file, when the category A is abelian, we obtain the homotopy equivalence homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex between the normalized Moore complex and the alternating face map complex of a simplicial object in A.

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noncomputable def AlgebraicTopology.DoldKan.homotopyPToId {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (q : ℕ) :
Homotopy (AlgebraicTopology.DoldKan.P q) (CategoryTheory.CategoryStruct.id (AlgebraicTopology.AlternatingFaceMapComplex.obj X))

Inductive construction of homotopies from P q to 𝟙 _

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    def AlgebraicTopology.DoldKan.homotopyQToZero {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (q : ℕ) :
    Homotopy (AlgebraicTopology.DoldKan.Q q) 0

    The complement projection Q q to P q is homotopic to zero.

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      theorem AlgebraicTopology.DoldKan.homotopyPToId_eventually_constant {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) {q n : ℕ} (hqn : n < q) :
      (AlgebraicTopology.DoldKan.homotopyPToId X (q + 1)).hom n (n + 1) = (AlgebraicTopology.DoldKan.homotopyPToId X q).hom n (n + 1)
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      def AlgebraicTopology.DoldKan.homotopyPInftyToId {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) :
      Homotopy AlgebraicTopology.DoldKan.PInfty (CategoryTheory.CategoryStruct.id (AlgebraicTopology.AlternatingFaceMapComplex.obj X))

      Construction of the homotopy from PInfty to the identity using eventually (termwise) constant homotopies from P q to the identity for all q

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        @[simp]
        theorem AlgebraicTopology.DoldKan.homotopyPInftyToId_hom {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (i j : ℕ) :
        (AlgebraicTopology.DoldKan.homotopyPInftyToId X).hom i j = (AlgebraicTopology.DoldKan.homotopyPToId X (j + 1)).hom i j
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        def AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex {A : Type u_2} [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] {Y : CategoryTheory.SimplicialObject A} :
        HomotopyEquiv ((AlgebraicTopology.normalizedMooreComplex A).obj Y) ((AlgebraicTopology.alternatingFaceMapComplex A).obj Y)

        The inclusion of the Moore complex in the alternating face map complex is a homotopy equivalence

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          @[simp]
          theorem AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_hom {A : Type u_2} [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] {Y : CategoryTheory.SimplicialObject A} :
          AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.hom = AlgebraicTopology.inclusionOfMooreComplexMap Y
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          @[simp]
          theorem AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_inv {A : Type u_2} [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] {Y : CategoryTheory.SimplicialObject A} :
          AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.inv = AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex Y
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          @[simp]
          theorem AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_homotopyHomInvId {A : Type u_2} [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] {Y : CategoryTheory.SimplicialObject A} :
          AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.homotopyHomInvId = Homotopy.ofEq ⋯
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          @[simp]
          theorem AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_homotopyInvHomId {A : Type u_2} [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] {Y : CategoryTheory.SimplicialObject A} :
          AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.homotopyInvHomId = (Homotopy.ofEq ⋯).trans (AlgebraicTopology.DoldKan.homotopyPInftyToId Y)