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Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Split simplicial objects in preadditive categories #

In this file we define a functor nondegComplex : SimplicialObject.Split C ⥤ ChainComplex C ℕ when C is a preadditive category with finite coproducts, and get an isomorphism toKaroubiNondegComplexFunctorIsoN₁ : nondegComplex ⋙ toKaroubi _ ≅ forget C ⋙ DoldKan.N₁.

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

The projection on a summand of the coproduct decomposition given by a splitting of a simplicial object.

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    If a simplicial object X in an additive category is split, then PInfty vanishes on all the summands of X _[n] which do not correspond to the identity of [n].

    The differentials s.d i j : s.N i ⟶ s.N j on nondegenerate simplices of a split simplicial object are induced by the differentials on the alternating face map complex.

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      If s is a splitting of a simplicial object X in a preadditive category, s.nondegComplex is a chain complex which is given in degree n by the nondegenerate n-simplices of X.

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      • s.nondegComplex = { X := s.N, d := s.d, shape := , d_comp_d' := }
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        The chain complex s.nondegComplex attached to a splitting of a simplicial object X becomes isomorphic to the normalized Moore complex N₁.obj X defined as a formal direct factor in the category Karoubi (ChainComplex C ℕ).

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          @[simp]
          theorem SimplicialObject.Split.nondegComplexFunctor_map_f {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ : SimplicialObject.Split C} {S₂ : SimplicialObject.Split C} (Φ : S₁ S₂) (n : ) :
          (SimplicialObject.Split.nondegComplexFunctor.map Φ).f n = Φ.f n
          @[simp]
          theorem SimplicialObject.Split.nondegComplexFunctor_obj {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : SimplicialObject.Split C) :
          SimplicialObject.Split.nondegComplexFunctor.obj S = S.s.nondegComplex

          The functor which sends a split simplicial object in a preadditive category to the chain complex which consists of nondegenerate simplices.

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            @[simp]
            theorem SimplicialObject.Split.toKaroubiNondegComplexFunctorIsoN₁_hom_app_f_f {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : SimplicialObject.Split C) (n : ) :
            (SimplicialObject.Split.toKaroubiNondegComplexFunctorIsoN₁.hom.app X).f.f n = CategoryTheory.CategoryStruct.comp ((X.s.cofan { unop := SimplexCategory.mk n }).inj (SimplicialObject.Splitting.IndexSet.id { unop := SimplexCategory.mk n })) (AlgebraicTopology.DoldKan.PInfty.f n)
            @[simp]
            theorem SimplicialObject.Split.toKaroubiNondegComplexFunctorIsoN₁_inv_app_f_f {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : SimplicialObject.Split C) (n : ) :
            (SimplicialObject.Split.toKaroubiNondegComplexFunctorIsoN₁.inv.app X).f.f n = X.s.πSummand (SimplicialObject.Splitting.IndexSet.id { unop := SimplexCategory.mk n })
            noncomputable def SimplicialObject.Split.toKaroubiNondegComplexFunctorIsoN₁ {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] :
            SimplicialObject.Split.nondegComplexFunctor.comp (CategoryTheory.Idempotents.toKaroubi (ChainComplex C )) (SimplicialObject.Split.forget C).comp AlgebraicTopology.DoldKan.N₁

            The natural isomorphism (in Karoubi (ChainComplex C ℕ)) between the chain complex of nondegenerate simplices of a split simplicial object and the normalized Moore complex defined as a formal direct factor of the alternating face map complex.

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