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Mathlib.Algebra.Homology.AlternatingConst

The alternating constant complex #

In this file we define the chain complex X ←0- X β†πŸ™- X ←0- X β†πŸ™- X β‹―, calculate its homology, and show that it is homotopy equivalent to the single complex where X is in degree 0.

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def ChainComplex.alternatingConst {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] :
CategoryTheory.Functor C (ChainComplex C β„•)

The chain complex X ←0- X β†πŸ™- X ←0- X β†πŸ™- X β‹―. It is exact away from 0 and has homology X at 0.

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    @[simp]
    theorem ChainComplex.alternatingConst_obj_X {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (X : C) (x✝ : β„•) :
    (alternatingConst.obj X).X x✝ = X
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    @[simp]
    theorem ChainComplex.alternatingConst_obj_d {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (X : C) (i j : β„•) :
    (alternatingConst.obj X).d i j = if Even i ∧ j + 1 = i then CategoryTheory.CategoryStruct.id X else 0
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    @[simp]
    theorem ChainComplex.alternatingConst_map_f {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} (f : X ⟢ Y) (x✝ : β„•) :
    (alternatingConst.map f).f x✝ = f
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    noncomputable def ChainComplex.alternatingConstHomologyDataEvenNEZero {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] (X : C) (n : β„•) (hn : Even n) (hβ‚€ : n β‰  0) :
    (HomologicalComplex.sc (alternatingConst.obj X) n).HomologyData

    The n-th homology of the alternating constant complex is zero for non-zero even n.

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      noncomputable def ChainComplex.alternatingConstHomologyDataOdd {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] (X : C) (n : β„•) (hn : Odd n) :
      (HomologicalComplex.sc (alternatingConst.obj X) n).HomologyData

      The n-th homology of the alternating constant complex is zero for odd n.

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        noncomputable def ChainComplex.alternatingConstHomologyDataZero {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (X : C) (n : β„•) (hn : n = 0) :
        (HomologicalComplex.sc (alternatingConst.obj X) n).HomologyData

        The n-th homology of the alternating constant complex is X for n = 0.

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          instance ChainComplex.instHasHomologyNatObjAlternatingConst {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] (X : C) (n : β„•) :
          HomologicalComplex.HasHomology (alternatingConst.obj X) n
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          theorem ChainComplex.alternatingConst_exactAt {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] (X : C) (n : β„•) (hn : n β‰  0) :
          HomologicalComplex.ExactAt (alternatingConst.obj X) n

          The n-th homology of the alternating constant complex is X for n β‰  0.

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          noncomputable def ChainComplex.alternatingConstHomologyZero {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] (X : C) :
          HomologicalComplex.homology (alternatingConst.obj X) 0 β‰… X

          The n-th homology of the alternating constant complex is X for n = 0.

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            def AlgebraicTopology.alternatingFaceMapComplexConst {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] :
            (CategoryTheory.Functor.const SimplexCategoryα΅’α΅–).comp (alternatingFaceMapComplex C) β‰… ChainComplex.alternatingConst

            The alternating face complex of the constant complex is the alternating constant complex.

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              noncomputable def ChainComplex.alternatingConstHomotopyEquiv {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] (X : C) :
              HomotopyEquiv (alternatingConst.obj X) ((singleβ‚€ C).obj X)

              alternatingConst.obj X is homotopy equivalent to the chain complex (singleβ‚€ C).obj X.

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