Documentation

Mathlib.AlgebraicTopology.AlternatingFaceMapComplex

The alternating face map complex of a simplicial object in a preadditive category #

We construct the alternating face map complex, as a functor alternatingFaceMapComplex : SimplicialObject C ⥤ ChainComplex C ℕ for any preadditive category C. For any simplicial object X in C, this is the homological complex ... → X_2 → X_1 → X_0 where the differentials are alternating sums of faces.

The dual version alternatingCofaceMapComplex : CosimplicialObject C ⥤ CochainComplex C ℕ is also constructed.

We also construct the natural transformation inclusionOfMooreComplex : normalizedMooreComplex A ⟶ alternatingFaceMapComplex A when A is an abelian category.

References #

Construction of the alternating face map complex #

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def AlgebraicTopology.AlternatingFaceMapComplex.objD {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) :
X.obj (Opposite.op (SimplexCategory.mk (n + 1))) ⟶ X.obj (Opposite.op (SimplexCategory.mk n))

The differential on the alternating face map complex is the alternate sum of the face maps

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    theorem AlgebraicTopology.AlternatingFaceMapComplex.d_squared {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) :
    CategoryTheory.CategoryStruct.comp (AlgebraicTopology.AlternatingFaceMapComplex.objD X (n + 1)) (AlgebraicTopology.AlternatingFaceMapComplex.objD X n) = 0

    The chain complex relation d ≫ d #

    Construction of the alternating face map complex functor #

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    def AlgebraicTopology.AlternatingFaceMapComplex.obj {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) :
    ChainComplex C ℕ

    The alternating face map complex, on objects

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      theorem AlgebraicTopology.AlternatingFaceMapComplex.obj_X {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) :
      (AlgebraicTopology.AlternatingFaceMapComplex.obj X).X n = X.obj (Opposite.op (SimplexCategory.mk n))
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      theorem AlgebraicTopology.AlternatingFaceMapComplex.obj_d_eq {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) :
      (AlgebraicTopology.AlternatingFaceMapComplex.obj X).d (n + 1) n = ∑ i : Fin (n + 2), (-1) ^ ↑i • X.δ i
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      def AlgebraicTopology.AlternatingFaceMapComplex.map {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} {Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y) :
      AlgebraicTopology.AlternatingFaceMapComplex.obj X ⟶ AlgebraicTopology.AlternatingFaceMapComplex.obj Y

      The alternating face map complex, on morphisms

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        theorem AlgebraicTopology.AlternatingFaceMapComplex.map_f {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} {Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y) (n : ℕ) :
        (AlgebraicTopology.AlternatingFaceMapComplex.map f).f n = f.app (Opposite.op (SimplexCategory.mk n))
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        def AlgebraicTopology.alternatingFaceMapComplex (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] :
        CategoryTheory.Functor (CategoryTheory.SimplicialObject C) (ChainComplex C ℕ)

        The alternating face map complex, as a functor

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          theorem AlgebraicTopology.alternatingFaceMapComplex_obj_X {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) :
          ((AlgebraicTopology.alternatingFaceMapComplex C).obj X).X n = X.obj (Opposite.op (SimplexCategory.mk n))
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          theorem AlgebraicTopology.alternatingFaceMapComplex_obj_d {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) :
          ((AlgebraicTopology.alternatingFaceMapComplex C).obj X).d (n + 1) n = AlgebraicTopology.AlternatingFaceMapComplex.objD X n
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          theorem AlgebraicTopology.alternatingFaceMapComplex_map_f {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} {Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y) (n : ℕ) :
          ((AlgebraicTopology.alternatingFaceMapComplex C).map f).f n = f.app (Opposite.op (SimplexCategory.mk n))
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          theorem AlgebraicTopology.map_alternatingFaceMapComplex {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] {D : Type u_2} [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [F.Additive] :
          (AlgebraicTopology.alternatingFaceMapComplex C).comp (F.mapHomologicalComplex (ComplexShape.down ℕ)) = ((CategoryTheory.SimplicialObject.whiskering C D).obj F).comp (AlgebraicTopology.alternatingFaceMapComplex D)
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          theorem AlgebraicTopology.karoubi_alternatingFaceMapComplex_d {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)) (n : ℕ) :
          ((AlgebraicTopology.AlternatingFaceMapComplex.obj (CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj P)).d (n + 1) n).f = CategoryTheory.CategoryStruct.comp (P.p.app (Opposite.op (SimplexCategory.mk (n + 1)))) ((AlgebraicTopology.AlternatingFaceMapComplex.obj P.X).d (n + 1) n)
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          def AlgebraicTopology.AlternatingFaceMapComplex.ε {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] :
          CategoryTheory.SimplicialObject.Augmented.drop.comp (AlgebraicTopology.alternatingFaceMapComplex C) ⟶ CategoryTheory.SimplicialObject.Augmented.point.comp (ChainComplex.single₀ C)

          The natural transformation which gives the augmentation of the alternating face map complex attached to an augmented simplicial object.

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            theorem AlgebraicTopology.AlternatingFaceMapComplex.ε_app_f_zero {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] (X : CategoryTheory.SimplicialObject.Augmented C) :
            (AlgebraicTopology.AlternatingFaceMapComplex.ε.app X).f 0 = X.hom.app (Opposite.op (SimplexCategory.mk 0))
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            theorem AlgebraicTopology.AlternatingFaceMapComplex.ε_app_f_succ {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] (X : CategoryTheory.SimplicialObject.Augmented C) (n : ℕ) :
            (AlgebraicTopology.AlternatingFaceMapComplex.ε.app X).f (n + 1) = 0

            Construction of the natural inclusion of the normalized Moore complex #

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            def AlgebraicTopology.inclusionOfMooreComplexMap {A : Type u_2} [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] (X : CategoryTheory.SimplicialObject A) :
            (AlgebraicTopology.normalizedMooreComplex A).obj X ⟶ (AlgebraicTopology.alternatingFaceMapComplex A).obj X

            The inclusion map of the Moore complex in the alternating face map complex

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              theorem AlgebraicTopology.inclusionOfMooreComplexMap_f {A : Type u_2} [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] (X : CategoryTheory.SimplicialObject A) (n : ℕ) :
              (AlgebraicTopology.inclusionOfMooreComplexMap X).f n = (AlgebraicTopology.NormalizedMooreComplex.objX X n).arrow
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              def AlgebraicTopology.inclusionOfMooreComplex (A : Type u_2) [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] :
              AlgebraicTopology.normalizedMooreComplex A ⟶ AlgebraicTopology.alternatingFaceMapComplex A

              The inclusion map of the Moore complex in the alternating face map complex, as a natural transformation

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                theorem AlgebraicTopology.inclusionOfMooreComplex_app (A : Type u_2) [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] (X : CategoryTheory.SimplicialObject A) :
                (AlgebraicTopology.inclusionOfMooreComplex A).app X = AlgebraicTopology.inclusionOfMooreComplexMap X
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                def AlgebraicTopology.AlternatingCofaceMapComplex.objD {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.CosimplicialObject C) (n : ℕ) :
                X.obj (SimplexCategory.mk n) ⟶ X.obj (SimplexCategory.mk (n + 1))

                The differential on the alternating coface map complex is the alternate sum of the coface maps

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                  theorem AlgebraicTopology.AlternatingCofaceMapComplex.d_eq_unop_d {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.CosimplicialObject C) (n : ℕ) :
                  AlgebraicTopology.AlternatingCofaceMapComplex.objD X n = (AlgebraicTopology.AlternatingFaceMapComplex.objD ((CategoryTheory.cosimplicialSimplicialEquiv C).functor.obj (Opposite.op X)) n).unop
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                  theorem AlgebraicTopology.AlternatingCofaceMapComplex.d_squared {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.CosimplicialObject C) (n : ℕ) :
                  CategoryTheory.CategoryStruct.comp (AlgebraicTopology.AlternatingCofaceMapComplex.objD X n) (AlgebraicTopology.AlternatingCofaceMapComplex.objD X (n + 1)) = 0
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                  def AlgebraicTopology.AlternatingCofaceMapComplex.obj {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.CosimplicialObject C) :
                  CochainComplex C ℕ

                  The alternating coface map complex, on objects

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                    def AlgebraicTopology.AlternatingCofaceMapComplex.map {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.CosimplicialObject C} {Y : CategoryTheory.CosimplicialObject C} (f : X ⟶ Y) :
                    AlgebraicTopology.AlternatingCofaceMapComplex.obj X ⟶ AlgebraicTopology.AlternatingCofaceMapComplex.obj Y

                    The alternating face map complex, on morphisms

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                      def AlgebraicTopology.alternatingCofaceMapComplex (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] :
                      CategoryTheory.Functor (CategoryTheory.CosimplicialObject C) (CochainComplex C ℕ)

                      The alternating coface map complex, as a functor

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                        theorem AlgebraicTopology.alternatingCofaceMapComplex_obj (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.CosimplicialObject C) :
                        (AlgebraicTopology.alternatingCofaceMapComplex C).obj X = AlgebraicTopology.AlternatingCofaceMapComplex.obj X
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                        theorem AlgebraicTopology.alternatingCofaceMapComplex_map (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] :
                        ∀ {X Y : CategoryTheory.CosimplicialObject C} (f : X ⟶ Y), (AlgebraicTopology.alternatingCofaceMapComplex C).map f = AlgebraicTopology.AlternatingCofaceMapComplex.map f