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Dold-Kan correspondence

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Contents

Idea

The Dold–Kan correspondence asserts there is an equivalence of categories between abelian simplicial groups and connective chain complexes of abelian groups.

Since every simplicial group is in particular a Kan complex with group structure, hence an ∞-groupoid with group structure, hence an ∞-group, the Dold-Kan correspondence says that connective chain complexes are a model for certain abelian ∞-groups. (This is part of the cosmic cube of higher category theory). The relevance of this is that chain complexes are typically easier to handle: all the tools of homological algebra apply. In fact, the functor that identifies simplicial abelian groups with their corresponding chain complexes – the normalized chains functor – does precisely this: it normalizes an abelian group by discarding irrelevant information and constructing a smaller and less redundant model for it.

There are various variants and generalizations of the Dold-Kan correspondence. These are discussed further below.

Statement (abelian case)

Let AA be an abelian category.

We say a chain complex in AA is connective if it is concentrated in non-negative degree. The full subcategory

Ch +(A)Ch (A) Ch^+_\bullet(A) \hookrightarrow Ch_\bullet(A)

of connective chain complexes is naturally identified with the category of \mathbb{N}-graded chain complexes.

Equivalence of categories

Theorem (Dold–Puppe)

For AA an abelian category there is an equivalence of categories

N:A Δ opCh +(A):Γ N \colon A^{\Delta^{op}} \stackrel{\leftarrow}{\to} Ch_\bullet^+(A) \colon \Gamma

between

where

Theorem (Kan)

For the case that AA is the category Ab of abelian groups, the functors NN and Γ\Gamma are nerve and realization with respect to the cosimplicial chain complex

[]:ΔCh +(Ab) \mathbb{Z}[-]: \Delta \to Ch_+(Ab)

that sends the standard nn-simplex to the normalized Moore complex of the free simplicial abelian group F (Δ n)F_{\mathbb{Z}}(\Delta^n) on the simplicial set Δ n\Delta^n, i.e.

Γ(V):[k]Hom Ch +(Ab)(N((Δ[k])),V). \Gamma(V) : [k] \mapsto Hom_{Ch_\bullet^+(Ab)}(N(\mathbb{Z}(\Delta[k])), V) \,.

More explicitly we have the following

Proposition
  • For VCh +V \in Ch_\bullet^+ the simplicial abelian group Γ(V)\Gamma(V) is in degree nn given by

    Γ(V) n= [n]surj[k]V k \Gamma(V)_n = \bigoplus_{[n] \underset{surj}{\to} [k]} V_k

    and for θ:[m][n]\theta : [m] \to [n] a morphism in Δ\Delta the corresponding map Γ(V) nΓ(V) m\Gamma(V)_n \to \Gamma(V)_m

    θ *: [n]surj[k]V k [m]surj[r]V r \theta^* : \bigoplus_{[n] \underset{surj}{\to} [k]} V_k \to \bigoplus_{[m] \underset{surj}{\to} [r]} V_r

    is given on the summand indexed by some σ:[n][k]\sigma : [n] \to [k] by the composite

    V kd *V s [m]surj[r]V r V_k \stackrel{d^*}{\to} V_s \hookrightarrow \bigoplus_{[m] \underset{surj}{\to} [r]} V_r

    where

    [m]t[s]d[k] [m] \stackrel{t}{\to} [s] \stackrel{d}{\to} [k]

    is the epi-mono factorization of the composite [m]θ[n]σ[k][m] \stackrel{\theta}{\to} [n] \stackrel{\sigma}{\to} [k].

  • The natural isomorphism ΓNId\Gamma N \to Id is given on AsAb Δ opA \in sAb^{\Delta^{op}} by the map

    [n]surj[k](NA) kA n \bigoplus_{[n] \underset{surj}{\to} [k]} (N A)_k \to A_n

    which on the direct summand indexed by σ:[n][k]\sigma : [n] \to [k] is the composite

    NA kA kσ *A n. N A_k \hookrightarrow A_k \stackrel{\sigma^*}{\to} A_n \,.
  • The natural isomorphism IdNΓId \to N \Gamma is on a chain complex VV given by the composite of the projection

    VC(Γ(V))C(Γ(C))/D(Γ(V)) V \to C(\Gamma(V)) \to C(\Gamma(C))/D(\Gamma(V))

    with the inverse

    C(Γ(V))/D(Γ(V))NΓ(V) C(\Gamma(V))/D(\Gamma(V)) \to N \Gamma(V)

    of

    NΓ(V)C(Γ(V))C(Γ(V))/D(Γ(V)) N \Gamma(V) \hookrightarrow C(\Gamma(V)) \to C(\Gamma(V))/D(\Gamma(V))

    (which is indeed an isomorphism, as discussed at Moore complex).

This is spelled out around prop. 2.2 in section III.2 of (GoerssJardine).

Proposition

With the explicit choice for ΓNId\Gamma N \stackrel{\simeq}{\to} Id is above we have that Γ\Gamma and NN form an adjoint equivalence (ΓN)(\Gamma \dashv N)

This is for instance exercise 8.4.2 in Weibel.

Remark

It follows that with the inverse structure maps, we also have an adjunction the other way round: (NΓ)(N \dashv \Gamma).

Quillen equivalence of model categories

Both Ch +(A)Ch_\bullet^+(A) and A Δ opA^{\Delta^{op}} are categories with weak equivalences in an standard way:

Proposition

These functors NN and Γ\Gamma both respect all weak equivalences with respect to the standard model structure on simplicial sets and on chain complexes in that they induce isomorphisms between simplicial homotopy groups and homology groups.

The structures of categories with weak equivalences have standard refinements to model category structures:

Proposition

Both

(NΓ):Ch +ΓNsAb (N \dashv \Gamma) : Ch_\bullet^+ \stackrel{\overset{N}{\leftarrow}}{\underset{\Gamma}{\to}} sAb

as well as

(ΓN):sAbNΓCh + (\Gamma \dashv N) : sAb \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{N}{\to}} Ch_\bullet^+

are Quillen equivalences with respect to these model structures.

This is discussed for instance in section 4.1 of SchwedeShipley

Remark

The category Ab Δ opAb^{\Delta^{op}} is – being a category of simplicial objects of a category with colimits – naturally an sSet-enriched category and with the model structure this makes it a simplicial model category.

Since the DK-correspondence is even an equivalence of categories, this induces accordingly the structure of a simplicial model category also on Ch +Ch_\bullet^+. Therefore the above Quillen equivalence is even a simplicial Quillen adjunction.

Remark

The free/forgetful adjunction (FU):AbSet(F \dashv U) : Ab \stackrel{\leftarrow}{\to} Set prolongs to simplicial objects

(FU):sAbUFsSet (F\dashv U) : sAb \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet

as an sSet-enriched adjunction. Moreover, by the above the right adjoint UU is a right Quillen functor to the standard model structure on simplicial sets.

This means we have a simplicial Quillen adjunction

(ΓFUN):Ch +UN. ( \Gamma F \dashv U N) : Ch_\bullet^+ \stackrel{\overset{}{\leftarrow}}{\underset{U N}{\to}} \,.

This manifestly presents connective chain complexes as models for certain ∞-groupoids.

Statement (general nonabelian case)

Globular and cubical version

There are versions of the Dold-Kan correspondence for other geometric shapes for higher structures than the simplex, also for the globe and the cube.

Theorem (globular Dold-Kan correspondence)

Write Ab for the category of abelian groups. (Could be any additive category with kernels for the following to be true). Then the following categories of structures internal to AbAb are equivalent.

  1. The category of chain complexes (in non-negative degree).

  2. The category of crossed complexes.

  3. The category of cubical sets with connection on a cubical set.

  4. The category of cubical strict ∞-groupoids.

  5. The category of globular strict ∞-groupoids.

A proof with references to the rich literature can be found for instance in

see the section Cubical Dold-Kan theorem.

This version of the Dold-Kan theorem reproduces the simplicial Dold-Kan theorem after application of the omega-nerve, i.e. the simplicial Dold-Kan correspondence factors through the globular one via the ω\omega-nerve.

Presentation of strict groupal \infty-groupoids

It was mentioned above that the standard simplicial Dold-Kan correspondence Ch (Ab)sAbCh_\bullet(Ab) \stackrel{\leftarrow}{\to} sAb may be understood as identifying strictly abelian strict ∞-groupoids among all ∞-groupoids. This statement is also surveyed and put into a larger context at cosmic cube of higher category theory.

We now give a formal version of this statement, following an observation by Richard Garner. A different but closely analogous sequence of arguments to the same extent is also in the book

see Dold-Kan map and omega-nerve.

Definition

Write

(LR):Ch (Ab) +StrCat(Ab)StrCat(Set) (L \dashv R) : Ch_\bullet(Ab)^+ \stackrel{\leftarrow}{\to} Str \infty Cat(Ab) \stackrel{\leftarrow}{\to} Str \infty Cat(Set)

for the adjunction obtained by composing the globular Dold-Kan correspondence with the forgetful functor which forgets the abelian group structure on a strict \infty-category in the image of the globular/cubical Dold-Kan map.

Proposition

The functor

C :ΔCh + C_\bullet : \Delta \to Ch_\bullet^+

which sends a simplex to its (normalized) chain complex factors as

C :Δ𝒪StrCatLCh +, C_\bullet : \Delta \stackrel{\mathcal{O}}{\to} Str \infty Cat \stackrel{L}{\to} Ch_\bullet^+ \,,

where the cosimplicial strict \infty-category 𝒪\mathcal{O} is the oriental functor.

This is a remark by Richard Garner posted here.

Proof

Use that 𝒪(n)\mathcal{O}(n) is the free strict \infty-category on a computad.

Observe that LL sends a strict ω\omega-category XX to the chain complex obtained from the abelian reflexive globular set X×mathbZX \times \mathb{Z}. In particular the value on the nn-globe is the chain complex

\mathbb{Z} \to \mathbb{Z}\oplus\mathbb{Z} \to \mathbb{Z}\oplus\mathbb{Z} \to \cdots \to \mathbb{Z}\oplus\mathbb{Z}

with (n+1)(n+1) terms and differential given by x(x,x)x \mapsto (x, -x) in each dimension.

Moreover, the value of LL on the boundary of the nn-globe is the chain complex obtained from this by removing the uppermost copy of \mathbb{Z}.

Given a computad CC, the associated abelian chain complex LCL C has for (LC) n(L C)_n the free abelian group on the set of generating nn-cells of CC, and differential given by x= jt j is i\partial x = \sum_j t_j - \sum_i s_i, where {s i}\{s_i\} and {t i}\{t_i\} are the sets of source- and or target-cells, respectively. A glance at Ross Street’s presentation of the orientals shows that L(𝒪(n))=C (Δ[n])L(\mathcal{O}(n)) = C_\bullet(\Delta[n]).

Corollary

The simplicial Dold-Kan map

Hom(C Δ[n],):Ch +sSet Hom(C_\bullet \Delta[n], -) : Ch_\bullet + \to sSet

factors as the identification of chain complexes with strictly abelian strict \infty-groupoids, followed by the functor that forgets the abelian structure and then followed by the omega-nerve operation that embeds strict \infty-groupoids into all \infty-groupoids.

Proof

Use the above adjunction and proposition to write for K K_\bullet a chain complex

Hom Ch (C Δ[n],K)=Hom Ch (L𝒪(n),K)=Hom StrCat(𝒪(n),RK)=N(RK) n. Hom_{Ch_\bullet}(C_\bullet \Delta[n], K) = Hom_{Ch_\bullet}(L \mathcal{O}(n), K) = Hom_{Str \infty Cat}(\mathcal{O}(n), R K) = N (R K)_n \,.

Remark The alternative construction in Nonabelian Algebraic Topology factors also versions of the nonabelian Dold-Kan correspondence through the ω\omega-nerve.

Non-abelian forms of the Dold-Kan correspondence.

Perhaps the ‘ultimate’ form of a ‘classical’ Dold–Kan result is by Pilar Carrasco, who identified the extra structure on chain complexes of groups in order that they be Moore complexes of simplicial groups. Dominique Bourn has a general form of this result for his semi-abelian categories. His results provide a neat categorical gloss on the theorem.

Dominique Bourn’s formulation is very pretty. The Moore complex functor is monadic when the basic category is semi-Abelian (Th. 1.4. p.113 in Bourn2007 below). Of course for simplicial groups, the monad on chain complexes of groups gives the hypercrossed complexes of Carrasco and Cegarra, but here they fall out from the theory. On the down side there is apparently no full analysis as yet of the actual form of this monad.

Stable Dold-Kan correspondence

The Dold-Kan correspondence stabilizes to identify unbounded chain complexes with the category of stably simplicial abelian groups. The latter are closely related to combinatorial spectra of Daniel Kan and can be defined as stably simplicial objects in the category of abelian groups. More precisely, we have the following definitions.

Definition

The category of stable simplices has integer numbers as objects. Given two objects kk and ll, the set of morphisms from kk to ll is the set of order-preserving maps hh from the set of natural numbers to itself such that h(n)=n+lkh(n)=n+l-k for all but a finite number of nn. Morphisms are composed by composing the corresponding maps.

Definition

A stably simplicial abelian group is a presheaf FF of abelian groups on the category of stable simplices such that for any integer kk every element xx of F(k)F(k) belongs to the kernels of all but a finite number of degeneracy maps. Morphisms of stably simplicial abelian groups are morphisms of presheaves.

The following theorem was established in 1963 by Daniel Kan in his paper “Semisimplicial spectra” (see Proposition 5.8):

Theorem

The category of unbounded chain complexes is equivalent to the category of stably simplicial abelian groups, with equivalences being given by the same functors

as in the unstable Dold-Kan correspondence, but appropriately extended to the above categories.

Similarly to the unstable case, the above categories, when interpreted as ∞-categories, are also equivalent to the ∞-category of module spectra over the Eilenberg-MacLane ring spectrum of the integers. For more see at stable Dold-Kan correspondence.

Generalizations and variants

There are various variants, generalizations and enhancements of the Dold–Kan correspondence.

Applications

Eilenberg-MacLane objects

The Dold-Kan correspondence gives a convenient construction of Eilenberg-MacLane objects in simplicial groups.

Proposition

For AA an abelian group write A[n]A[-n] for the chain complex concentrated on AA in degree nn.

The simplicial abelian group Γ(A[n])\Gamma (A[-n]) is an Eilenberg-MacLane object K(A,n)K(A,n).

And conversely, every such Eilenberg-MacLane object in simplicial abelian groups is related by an ∞-anafunctor-equivalence to a Γ(A[n])\Gamma(A[-n]).

Looping and delooping

The Dold-Kan correspondence provides a convenient way to describe formation of loop space objects and delooping for anything in the image of Ξ:Ch sSet\Xi : Ch_\bullet \to sSet:

by the basic fact that the homotopy groups of Ξ(V )\Xi(V_\bullet) are the homology groups of V V_\bullet, looping and delooping simply corresponds to shifting chain complexes up or down in degree.

But the relation is also strongly coherent: it respects the standard delooping functor W¯:sGrpsSet\bar W : sGrp \to sSet for simplicial groups (see there and at looping and delooping) (notice that restricted to simplicial abelian groups this produces simplicial abelian groups W¯:sAbGrpsAbGrp\bar W : sAbGrp \to sAbGrp):

Proposition

There is a natural isomorphism

NW¯G(NG)[1] N \bar W G \simeq (N G)[-1]

natural in GsAbGrpdG \in sAbGrpd.

This appears for instance as (GoerssJardine, remark III.5.6) or around (Jardine, theorem 4.57).

Abelian sheaf cohomology in nonabelian cohomology

Composed with the forgetful functor sAbsSetsAb \to sSet the Dold-Kan correspondence presents certain simplicial sets by chain complexes. Since this is entirely functorial, it prolongs to a functor from chain complexes of (pre)sheaves on any site SS, to simplicial presheaves

Γ:[S op,Ch +(ab)][S op,sSet]. \Gamma : [S^{op}, Ch_\bullet^+(ab)] \to [S^{op}, sSet] \,.

If [S op,sSet][S^{op}, sSet] is equipped with the projective model structure on simplicial presheaves it models the (∞,1)-sheaf (∞,1)-topos on SS. The derived hom-spaces compute general nonabelian cohomology.

If the coefficient objects come from sheaves of chain complexes along Γ\Gamma, this cohomology restricts to ordinary abelian sheaf cohomology. See there for more details.

Computational aspects

One may view the (monoidal) Dold-Kan correspondence as a relation between a well-behaved theory (simplicial/higher methods) that work in any characteristic but is very abstract and mainly suited to the proof of abstract theorems, and a more computational theory (strict structures in dg-modules) that are particularly well adapted to computations. The relation between these two (symmetric monoidal) theories may only be properly used with characteristic 0 coefficients. This remark is very naive and basic, but certainly at the center of computational implementations of abstract homotopical methods.

References

Historical references for the Dold–Kan correspondence are

which considers the correspondence for categories of modules, and

that generalizes the result to arbitrary abelian categories.

The expression of the correspondence in terms of nerve and realization is due to

  • Daniel Kan, Functors involving c.s.s complexes, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar., 1958), pp. 330–346 (jstor).

This remarkable article, which appeared shortly after the work by Dold and Puppe but was apparently not influenced by that, introduces not just the abstract nerve and realization form of the Dold-Kan correspondence, but introduces the general notion of nerve and realization and in fact the general notion of what is now called Kan extension.

A standard modern textbook reference for the ordinary Dold-Kan correspondence is chapter III.2 of

Similar material is in section 4.6 of

  • Rick Jardine, Generalized etale cohomology theories Modern Birkhäuser Classics, (1991)

Remarks about the interpretation in terms of model categories are in

Discussion in the generality of idempotent complete additive categories is in

The relation between strict ω-groupoids and crossed complexes is in

  • R. Brown and P.J. Higgins, The equivalence of \infty-groupoids and crossed complexes, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 22 no. 4 (1981), p. 371–386 (pdf)

P. 59 of

  • R. Brown, Groupoids and crossed objects in algebraic topology, Homology, Homotopy and Applications, 1 (1999) 1-78.

gives seven equivalent categories with the equivalences all expressing nonabelian versions of the Dold–Kan correspondence. One of these is given in

  • Ashley, N., Simplicial T-complexes and crossed complexes: a nonabelian version of a theorem of Dold and Kan. University of Wales PhD Thesis, (1978); Dissertationes Math. (Rozprawy Mat.) 265 (1988) 1–61.

The relation of these with the abelian version is given in

  • Brown, R. and Higgins, P. J., Cubical abelian groups with connections are equivalent to chain complexes. Homology Homotopy Appl. 5 (1) (2003) 49–52.

The discussion of Dold–Kan in the general context of semi-abelian categories is in

  • Dominique Bourn, Moore normalisation and Dold–Kan theorem for semi-Abelian categories, in Categories in algebra, geometry and mathematical physics , volume 431 of Contemp. Math., 105–124, Amer. Math. Soc., Providence, RI. (2007)

The classical Dold-Kan theorem occurs as a special case among others from combinatorics and representation theory here:

Revised on November 3, 2014 07:12:51 by Frédéric Paugam (78.194.68.97)