The Dold-Kan correspondence

The Dold-Kan correspondence states that for any abelian category A, there is an equivalence between the category of simplicial objects in A and the category of chain complexes in A (with degrees indexed by ℕ and the homological convention that the degree is decreased by the differentials).

In this file, we finish the construction of this equivalence by providing CategoryTheory.Abelian.DoldKan.equivalence which is of type SimplicialObject A ≌ ChainComplex A ℕ for any abelian category A. The functor SimplicialObject A ⥤ ChainComplex A ℕ of this equivalence is definitionally equal to normalizedMooreComplex A.

Overall strategy of the proof of the correspondence

Before starting the implementation of the proof in Lean, the author noticed that the Dold-Kan equivalence not only applies to abelian categories, but should also hold generally for any pseudoabelian category C (i.e. a category with instances [Preadditive C] [HasFiniteCoproducts C] and [IsIdempotentComplete C]): this is CategoryTheory.Idempotents.DoldKan.equivalence.

When the alternating face map complex K[X] of a simplicial object X in an abelian is studied, it is shown that it decomposes as a direct sum of the normalized subcomplex and of the degenerate subcomplex. The crucial observation is that in this decomposition, the projection on the normalized subcomplex can be defined in each degree using simplicial operators. Then, the definition of this projection PInfty : K[X] ⟶ K[X] can be carried out for any X : SimplicialObject C when C is a preadditive category.

The construction of the endomorphism PInfty is done in the files Homotopies.lean, Faces.lean, Projections.lean and PInfty.lean. Eventually, as we would also like to show that the inclusion of the normalized Moore complex is a homotopy equivalence (cf. file HomotopyEquivalence.lean), this projection PInfty needs to be homotopic to the identity. In our construction, we get this for free because PInfty is obtained by altering the identity endomorphism by null homotopic maps. More details about this aspect of the proof are in the file Homotopies.lean.

When the alternating face map complex K[X] is equipped with the idempotent endomorphism PInfty, it becomes an object in Karoubi (ChainComplex C ℕ) which is the idempotent completion of the category ChainComplex C ℕ. In FunctorN.lean, we obtain this functor N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ), which is formally extended as N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ). (Here, some functors have an index which is the number of occurrences of Karoubi at the source or the target.)

In FunctorGamma.lean, assuming that the category C is additive, we define the functor in the other direction Γ₂ : Karoubi (ChainComplex C ℕ) ⥤ Karoubi (SimplicialObject C) as the formal extension of a functor Γ₀ : ChainComplex C ℕ ⥤ SimplicialObject C which is defined similarly as in [Simplicial Homotopy Theory by Goerss-Jardine][goerss-jardine-2009]. In Degeneracies.lean, we show that PInfty vanishes on the image of degeneracy operators, which is one of the key properties that makes it possible to contruct the isomorphism N₂Γ₂ : Γ₂ ⋙ N₂ ≅ 𝟭 (Karoubi (ChainComplex C ℕ)).

The rest of the proof follows the strategy in the [original paper by Dold][dold1958]. We show that the functor N₂ reflects isomorphisms in NReflectsIso.lean: this relies on a decomposition of the identity of X _[n] using PInfty.f n and degeneracies obtained in Decomposition.lean. Then, in NCompGamma.lean, we construct a natural transformation Γ₂N₂.trans : N₂ ⋙ Γ₂ ⟶ 𝟭 (Karoubi (SimplicialObject C)). It is shown that it is an isomorphism using the fact that N₂ reflects isomorphisms, and because we can show that the composition N₂ ⟶ N₂ ⋙ Γ₂ ⋙ N₂ ⟶ N₂ is the identity (see identity_N₂). The fact that N₂ is defined as a formal direct factor makes the proof easier because we only have to compare endomorphisms of an alternating face map complex K[X] and we do not have to worry with inclusions of kernel subobjects.

In EquivalenceAdditive.lean, we obtain the equivalence equivalence : Karoubi (SimplicialObject C) ≌ Karoubi (ChainComplex C ℕ). It is in the namespace CategoryTheory.Preadditive.DoldKan. The functors in this equivalence are named N and Γ: by definition, they are N₂ and Γ₂.

In EquivalencePseudoabelian.lean, assuming C is idempotent complete, we obtain equivalence : SimplicialObject C ≌ ChainComplex C ℕ in the namespace CategoryTheory.Idempotents.DoldKan. This could be roughly obtained by composing the previous equivalence with the equivalences SimplicialObject C ≌ Karoubi (SimplicialObject C) and Karoubi (ChainComplex C ℕ) ≌ ChainComplex C ℕ. Instead, we polish this construction in Compatibility.lean by ensuring good definitional properties of the equivalence (e.g. the inverse functor is definitionallly equal to Γ₀' : ChainComplex C ℕ ⥤ SimplicialObject C) and showing compatibilities for the unit and counit isomorphisms.

In this file Equivalence.lean, assuming the category A is abelian, we obtain equivalence : SimplicialObject A ≌ ChainComplex A ℕ in the namespace CategoryTheory.Abelian.DoldKan. This is obtained by replacing the functor CategoryTheory.Idempotents.DoldKan.N of the equivalence in the pseudoabelian case with the isomorphic functor normalizedMooreComplex A thanks to the isomorphism obtained in Normalized.lean.

TODO: Show functoriality properties of the three equivalences above. More precisely, for example in the case of abelian categories A and B, if F : A ⥤ B is an additive functor, we can show that the functors N for A and B are compatible with the functors SimplicialObject A ⥤ SimplicialObject B and ChainComplex A ℕ ⥤ ChainComplex B ℕ induced by F. (Note that this does not require that F is an exact functor!)

TODO: Introduce the degenerate subcomplex D[X] which is generated by degenerate simplices, show that the projector PInfty corresponds to a decomposition K[X] ≅ N[X] ⊞ D[X].

TODO: dualise all of this as CosimplicialObject A ⥤ CochainComplex A ℕ. (It is unclear what is the best way to do this. The exact design may be decided when it is needed.)

References

• [Albrecht Dold, Homology of Symmetric Products and Other Functors of Complexes][dold1958]
• [Paul G. Goerss, John F. Jardine, Simplicial Homotopy Theory][goerss-jardine-2009]

def CategoryTheory.Abelian.DoldKan.N {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] : CategoryTheory.Functor (CategoryTheory.SimplicialObject A) (ChainComplex A ℕ)

The functor N for the equivalence is normalizedMooreComplex A

Equations

• CategoryTheory.Abelian.DoldKan.N = AlgebraicTopology.normalizedMooreComplex A

def CategoryTheory.Abelian.DoldKan.Γ {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] : CategoryTheory.Functor (ChainComplex A ℕ) (CategoryTheory.SimplicialObject A)

The functor Γ for the equivalence is the same as in the pseudoabelian case.

Equations

• CategoryTheory.Abelian.DoldKan.Γ = CategoryTheory.Idempotents.DoldKan.Γ

@[simp]

theorem CategoryTheory.Abelian.DoldKan.comparisonN_hom_app_f {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] (X : CategoryTheory.SimplicialObject A) (i : ℕ) : (CategoryTheory.Abelian.DoldKan.comparisonN.hom.app X).f i = CategoryTheory.CategoryStruct.comp (((CategoryTheory.Idempotents.toKaroubiEquivalence (ChainComplex A ℕ)).unitIso.hom.app (CategoryTheory.Abelian.DoldKan.N.toPrefunctor.obj X)).f i) (((CategoryTheory.Idempotents.toKaroubiEquivalence (ChainComplex A ℕ)).inverse.toPrefunctor.map ((AlgebraicTopology.DoldKan.N₁_iso_normalizedMooreComplex_comp_toKaroubi A).inv.app X)).f i)

@[simp]

theorem CategoryTheory.Abelian.DoldKan.comparisonN_inv_app_f {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] (X : CategoryTheory.SimplicialObject A) (i : ℕ) : (CategoryTheory.Abelian.DoldKan.comparisonN.inv.app X).f i = CategoryTheory.CategoryStruct.comp (((CategoryTheory.Idempotents.toKaroubiEquivalence (ChainComplex A ℕ)).inverse.toPrefunctor.map ((AlgebraicTopology.DoldKan.N₁_iso_normalizedMooreComplex_comp_toKaroubi A).hom.app X)).f i) (((CategoryTheory.Idempotents.toKaroubiEquivalence (ChainComplex A ℕ)).unitIso.inv.app (CategoryTheory.Abelian.DoldKan.N.toPrefunctor.obj X)).f i)

def CategoryTheory.Abelian.DoldKan.comparisonN {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] : CategoryTheory.Abelian.DoldKan.N ≅ CategoryTheory.Idempotents.DoldKan.N

The comparison isomorphism between normalizedMooreComplex A and the functor Idempotents.DoldKan.N from the pseudoabelian case

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.Abelian.DoldKan.equivalence_functor {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] : CategoryTheory.Abelian.DoldKan.equivalence.functor = CategoryTheory.Abelian.DoldKan.N

def CategoryTheory.Abelian.DoldKan.equivalence {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] : CategoryTheory.SimplicialObject A ≌ ChainComplex A ℕ

The Dold-Kan equivalence for abelian categories

Equations

• One or more equations did not get rendered due to their size.

theorem CategoryTheory.Abelian.DoldKan.equivalence_inverse {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] : CategoryTheory.Abelian.DoldKan.equivalence.inverse = CategoryTheory.Abelian.DoldKan.Γ