nLab
Dold-Kan correspondence

Idea

The Dold–Kan correspondence asserts that the nerve and realization adjunction between abelian simplicial groups and chain complexes encodes $\mathrm{\infty }$-groupoids with abelian group structure equivalently in terms of chain complexes of abelian groups.

Analogous statement hold for the category of abelian groups replaces by any other abelian category.

This allows one

• on the one hand to understand the true conceptual home of many constructions with chain complexes in homological algebra;

• on the other to efficiently compute with certain higher categorical structures.

Compare this to the role played by rational homotopy theory as a tool for handling certain topological spaces.

The classical Dold–Kan correspondence states the equivalence of

simplicial abelian groups $\simeq$ non-negatively graded chain complexes of abelian groups.

Since every simplicial group is a Kan complex, one can equivalently read this more suggestively as

$\mathrm{\infty }$-groupoids with abelian group structure $\simeq$ chain complexes${}_{+}$ internal to abelian groups.

The fact that this involves abelian groups is essential for the correspondence. However, a similar correspondence still holds in a slightly more nonabelian context. This and various other generalizations are described further below.

Details

We now spell out technical details of the classical Dold–Kan correspondence.

Theorem (Dold–Puppe)

For $A$ an abelian category there is an adjoint equivalence

$$N:[{\Delta }^{\mathrm{op}},A]\overleftarrow{\to }{\mathrm{Ch}}_{+}(A):\Xi$$N : [\Delta^{op},A] \stackrel{\leftarrow}{\to} Ch_+(A) : \Xi

between

• the category $[{\Delta }^{\mathrm{op}},A]$ of simplicial objects in $A$;

• the category of chain complexes in $A$ that are 0 in negative degree,

where

• $N$ is the normalized Moore complex functor, given by

  $$(NG{)}_n=\bigcap _{i=1}^n\mathrm{Ker}{d}_i^n$$(N G)_n=\bigcap_{i=1}^{n}Ker\,d_i^n

  with differential ${\partial }_n:N{G}_n\to N{G}_{n-1}$ given by $d_0^n$.

These functors respect the standard 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 group?s.

Note: Given a simplicial object $G$ in an abelian category, the normalized Moore complex $N_{•}(G)$ defined above is naturally isomorphic to another chain complex $C_{•}(G)/D_{•}(G)$ formed as follows. First define the alternating face map complex, denoted $C_{•}(G)$, by setting $C_n(G)=G_n$ and letting the differential $\partial$ be given by

$$\partial =\sum _{i=0}^n(-1{)}^i{d}_i$$\partial = \sum_{i = 0}^n (-1)^i d_i

Then define $C_{•}(G)/D_{•}(G)$ to be the quotient of $C_{•}(G)$ by the subcomplex $D_{•}(G)$ where $D_n(G)$ consists of simplices in the image of a degeneracy map. $C_{•}(G)/D_{•}(G)$ is sometimes called the normalized chain complex of the simplicial object $G$, and it is probably more familiar to many people. One advantage of the normalized Moore complex is that it also plays an important role in the nonabelian context. For more details, see the entry Moore complex.

Theorem (Kan)

For the case that $A$ is the category of abelian groups, the functors $N$ and $\Xi$ are nerve and realization with respect to the cosimplicial chain complex

$${\mathrm{Ch}}_{•}:=N{F}_{\mathbb{Z}}(-):\Delta \to {\mathrm{Ch}}_{+}(\mathrm{Ab})$$Ch_\bullet := N F_\mathbb{Z}(-): \Delta \to Ch_+(Ab)

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

$$\Xi (V_{•}{)}_n={\mathrm{Hom}}_{{\mathrm{Ch}}_{•}(\mathrm{Ab})}(C_{•}({\Delta }^n),V_{•}).$$\Xi(V_\bullet)_n= Hom_{Ch_\bullet(Ab)}(C_\bullet(\Delta^n), V_\bullet) \,.

Before proving this, first briefly recall an alternate, but naturally isomorphic, construction of the normalized Moore complex of a simplicial abelian group.

Definition

For $A=(A_n)$ a simplicial abelian group, its alternating face complex is

$$C_{•}(A)=(\cdots \to A_n\stackrel{{\partial }_n:=\sum _{i=0}^n(-1{)}^i{d}_i}{\to }{A}_{n-1}\to \cdots ),$$C_\bullet(A) = \left( \cdots \to A_{n} \stackrel{\partial_n := \sum_{i = 0}^{n} (-1)^i d_i} {\to} A_{n-1} \to \cdots \right) \,,

where the $d_i$ are the face maps of $A$.

The degenerate complex $D_{•}(A)$ of $A$ is the full subcomplex generated from degenerate elements

$$D_n(A):=s(A_{n-1})$$D_n(A) := s(A_{n-1})

Lemma

This is indeed well defined in that the boundary map satisfies $\partial \circ \partial =0$ in $C_{•}(A)$ and restricts to a boundary map on the degenerate subcomplex $\partial :A_n{\mid }_{s(A_{n-1})}\to A_{n-1}{\mid }_{s(A_{n-2})}$.

Proof

For the first statement one checks

$$\begin{array}{rl}{\partial }_n{\partial }_{n+1}& =\sum _{i,j}(-1{)}^{i+j}{d}_i\circ d_j\\ & =\sum _{i\ge j}(-1{)}^{i+j}{d}_i\circ d_j+\sum _{i<j}(-1{)}^{i+j}{d}_i\circ d_j\\ & =\sum _{i\ge j}(-1{)}^{i+j}{d}_i\circ d_j+\sum _{i<j}(-1{)}^{i+j}{d}_{j-1}\circ d_i\\ & =\sum _{i\ge j}(-1{)}^{i+j}{d}_i\circ d_j-\sum _{i\le k}(-1{)}^{i+k}{d}_k\circ d_i\\ & =0\end{array}$$\begin{aligned} \partial_n \partial_{n+1} & = \sum_{i, j} (-1)^{i+j} d_i \circ d_{j} \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_i \circ d_j \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_{j-1} \circ d_i \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j - \sum_{i \leq k} (-1)^{i+k} d_{k} \circ d_i \\ &= 0 \end{aligned}

using the simplicial identity $d_i\circ d_j=d_{j-1}\circ d_i$ for $i<j$.

Similarly, using the mixed simplicial identities we find that for $s_j(a)\in A_n$ a degenerate element, its boundary is

$$\begin{array}{rl}\sum _i(-1{)}^i{d}_i{s}_j(a)& =\sum _{i<j}(-1{)}^i{s}_{j-1}{d}_i(a)+\sum _{i=j,j+1}(-1{)}^{i}a+\sum _{i>j+1}(-1{)}^i{s}_j{d}_{i-1}(a)\\ & =\sum _{i<j}(-1{)}^i{s}_{j-1}{d}_i(a)+\sum _{i>j+1}(-1{)}^i{s}_j{d}_{i-1}(a)\end{array}$$\begin{aligned} \sum_i (-1)^i d_i s_j(a) &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i = j, j+1} (-1)^i a + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \\ &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \end{aligned}

which is again a combination of elements in the image of the degeneracy maps.

Lemma

There is a splitting

$$C_{•}(A)\simeq N_{•}(A)\oplus D_{•}(A)$$C_\bullet(A) \simeq N_\bullet(A) \oplus D_\bullet(A)

where the first summand is naturally isomorphic to the normalized Moore complex.

Theorem (Eilenberg–Mac Lane)

The inclusion

$$N_{•}(A)\hookrightarrow C_{•}(A)$$N_\bullet(A) \hookrightarrow C_\bullet(A)

is a homotopy equivalence, i.e. the complex $D_{•}(X)$ is null-homotopic.

Proof

Following the proof of what is Theorem 2.1 in Goerss–Jardine we look for each $n\in \mathbb{N}$ and each $j<n$ at the groups

$$N_n(A{)}_j:={\cap }_{i=0}^j\ker (d_i)\subset A_n$$N_n(A)_j := \cap_{i=0}^j ker (d_i) \subset A_n

and similarly at

$$D_n(A{)}_j=\{s_i{\}}_{i\le j}(A_{n-1})\subset A_n,$$D_n(A)_j = \{s_{i}\}_{i \leq j}(A_{n-1}) \subset A_n \,,

the subgroup generated by the first $j$ degeneracies.

For $j=n-1$ these coincide with $N_n(A)$ and with $D_n(A)$, respectively. We show by induction on $j$ that the composite

$$N_n(A{)}_j\hookrightarrow A_n\stackrel{}{\to }{A}_n/D_n(A{)}_j$$N_n(A)_j \hookrightarrow A_n \stackrel{}{\to} A_n/D_n(A)_j

is an isomorphism of all $j<n$. For $j=n-1$ this is then the desired result.

…

Now consider the free chain complex on ${\Delta }^n$…

Generalizations and enhancements

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

Monoidal version

Both the category of simplicial abelian groups as well as the category of nonnegatively graded chain complexes of abelian groups carry a standard structure of a monoidal category. For simplicial abelian groups this is the levelwise or ‘pointwise’ tensor product

$$(A\otimes B{)}_n=A_n\otimes B_n$$(A \otimes B)_n = A_n \otimes B_n

For chain complexes of abelian groups the tensor product is given by

$$(C\otimes D{)}_n=\sum _{i,j:i+j=n}{C}_i\otimes D_j$$(C \otimes D)_n = \sum_{i,j : i+j = n} C_i \otimes D_j

The same is true if we replace abelian groups by $R$-modules for any commutative ring $R$.

This means that one may ask whether the normalized Moore complex functor

$$N:[{\Delta }^{\mathrm{op}},R-\mathrm{Mod}]\to {\mathrm{Ch}}_{+}(R\mathrm{Mod})$$N : [\Delta^{op},R-Mod] \to Ch_+(R Mod)

and its adjoint

$$\Xi :{\mathrm{Ch}}_{+}(R\mathrm{Mod})\to [{\Delta }^{\mathrm{op}},R\mathrm{Mod}]$$\Xi: Ch_+(R Mod) \to [\Delta^{op},R Mod]

respect these monoidal structures. A partial answer is:

Theorem

If $R$ is any commutative ring, the functor

$$N:[{\Delta }^{\mathrm{op}},R\mathrm{Mod}]\to {\mathrm{Ch}}_{+}(R\mathrm{Mod})$$N : [\Delta^{op},R Mod] \to Ch_+(R Mod)

is lax monoidal with respect to the Eilenberg–Zilber map?

$$\mathrm{EZ}:N(A)\otimes N(B)\to N(A\otimes B)$$EZ : N(A) \otimes N(B) \to N(A \otimes B)

and oplax monoidal with respect to the Alexander–Whitney map?

$$\mathrm{AW}:N(A\otimes B)\to N(A)\otimes N(B)$$AW : N(A \otimes B) \to N(A) \otimes N(B)

The Alexander–Whitney map is left inverse to the Eilenberg–Zilber map:

$$\mathrm{AW}\circ \mathrm{EZ}=1$$AW \circ EZ = 1

However, the Alexander–Whitney map is not right inverse to the Eilenberg–Zilber map:

$$\mathrm{EZ}\circ \mathrm{AW}\ne 1$$EZ \circ AW \ne 1

Instead there is a chain homotopy

$$\mathrm{EZ}\circ \mathrm{AW}\simeq 1$$EZ \circ AW \simeq 1

Similarly, the adjoint to the normalized Moore complex functor

$$\Xi :{\mathrm{Ch}}_{+}(\mathrm{Ab})\to [{\Delta }^{\mathrm{op}},\mathrm{Ab}]$$\Xi : Ch_+(Ab) \to [\Delta^{op},Ab]

is lax monoidal with respect to the Alexander–Whitney map and oplax monoidal with respect to the Eilenberg–Zilber map.

Proof

Apparently this result appears in Mac Lane’s Homology. It was proved by Eilenberg and Mac Lane in their 1954 Annals paper “On the groups $H(\pi ,n)$. II”.

As a consequence, both $N$ and $\Xi$ send monoids to monoids and comonoids to comonoids. In other words:

• if $A$ is a monoid internal to simplicial $R$-modules, $N(A)$ is a differential graded algebra over $R$;

• if $A$ is a comonoid internal to simplicial $R$-modules, $N(A)$ is a differential graded coalgebra over $R$;

• if $C$ is a differential graded algebra over $R$, $\Xi (C)$ is a monoid internal to simplicial $R$-modules.

• if $C$ is a differential graded coalgebra over $R$, $\Xi (C)$ is a comonoid internal to simplicial $R$-modules.

For more details, see:

• monoidal Dold–Kan correspondence

These remarks by Kathryn Hess are also very useful, and should be integrated into the story here. She uses “lax comonoidal functor” to mean what above is called an “oplax monoidal functor”.

The normalized chains functor from simplicial sets to chain complexes (with any coefficients) is both lax monoidal and lax comonoidal. The Eilenberg-Zilber equivalence, from the tensor product of the chains on X and on Y to the chains on the cartesian product of X and Y, provides the natural transformation that shows that the chain functor is lax monoidal. The Alexander-Whitney equivalence goes in the opposite direction and shows that the chain functor is lax comonoidal.

Since the chain functor is lax comonoidal, the normalized chains on any simplicial set is a dg coalgebra, where the comultiplication is given by the composite of the chain functor applied to the diagonal map, followed be the Alexadnder-Whitney transformation. It turns out that the Eilenberg-Zilber equivalence is actually itself a morphism of coalgebras with respect to this comultiplication. On the other hand, the Alexander-Whitney map is a morphism of coalgebras up to strong homotopy.

The A-W/E-Z equivalences for the normalized chains functor are a special case of the strong deformation retract of chain complexes that was constructed by Eilenberg and MacLane in their 1954 Annals paper “On the groups H(pi, n). II”. For any commutative ring R, they defined chain equivalences between the tensor product of the normalized chains on two simplicial R-modules and the normalized chains on their levelwise tensor product.

Steve Lack and I observed recently that the normalized chains functor is actually even Frobenius monoidal. We then discovered that Aguiar and Mahajan already had a proof of this fact in their recent monograph… :-)

As to the Hopf algebra/bialgebra question, I admit I’ve often been guilty of referring to loop space homology as a “graded Hopf algebra”, without any mention of antipode. I was happy to learn a few years ago that connected graded bialgebras admit a unique antipode, as Mike has already pointed out. On the chain level, if the simplicial group you’re considering is reduced (i.e., has a unique 0-simplex), then its normalized chain complex is a connected dg bialgebra and therefore a dg Hopf algebra.

Dual Dold–Kan correspondence

There is a version relating cochain complexes in non-negative degree to cosimplicial abelian groups. Indeed, replacing the abelian category $A$ by its opposite category $A^{\mathrm{op}}$ in the Dold–Puppe theorem above, we instantly see:

Theorem (dual Dold–Puppe)

For $A$ an abelian category there is an adjoint equivalence between

• the category $[\Delta ,A]$ of cosimplicial objects in $A$;

and

• the category of cochain complexes? in $A$ that are 0 in negative degree.

More concretely, let us take the case where $A$ is the category of abelian groups, and construct an explicit equivalence.

If $G$ is a cosimplicial abelian group, the dual Moore complex $C^{•}$ is formally defined just as the ordinary Moore complex with

$$C^k:=G^k$$C^k := G^k

by taking the the alternating sum of the face maps ${\partial }_i$.

$$d:\sum _{i=0}^k(-1){\partial }_i:C^k\to C^{k+1}.$$d : \sum_{i = 0}^{k} (-1) \partial_i : C^k \to C^{k+1} \,.

Notice that since we have now a cosimplicial object this differential indeed increases degree, as befits a cochain complex.

It is in the definition of the normalized dual Moore complex $N^{•}G$ that the role of face maps $d_i$ and boundary maps $s_i$ is interchanged as compared to the ordinary normalized complex. We have

$$N^{k}G:=G^n/\sum {i=0}^n{d}_i{G}^n\simeq {\cap }_{i=0}^{n-1}\ker (s_i).$$N^k G := G^n/\sum{i=0}^n d_i G^n \simeq \cap_{i=0}^{n-1} ker(s_i) \,.

This statement is reviewed and various further references are given in section 4 of

• J.L. Castiglioni, G. Cortiñas, Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence, J. Pure Appl. Algebra 191 (2004), no. 1-2, 119–142, arXiv:math.KT/0306289.

Nonabelian versions

In a series of articles by Brown and Higgins, based on old work by Whitehead, the above was generalized to

strict $\mathrm{\infty }$-groupoids $\simeq$ crossed complexes.

Here on the left we have strict ω-groupoids and on the right crossed complexes, a non-abelian generalization of chain complexes of groups.

crossed complexes $\supset$ chain complexes${}_{+}$ internal of abelian groups

There is a discussion of some of these and extensions to them to the non-abelian case, in the entry on Moore complex.

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.

Version for Lie algebras

In rational homotopy theory, Quillen proved and used an analogous statement for Lie algebras: Quillen equivalence between the reduced rational dg Lie algebras and reduced rational simplicial Lie algebras:

• D. G. Quillen, Rational homotopy theory, Ann. Math. 90 (1969), 204–265.

Parameterized version

The statement of the Dold–Kan correspondence generalizes to sheaves with values in the respective categories and this way from ∞ Grpd to more general $(\mathrm{\infty }\mathrm{,1})$-topoi:

For $X$ be a site, let $\mathrm{Sh}(X,\mathrm{sAb})$ be the category of simplicial abelian sheaves – i.e. simplicial sheaves which take values in simplicial abelian groups – and let $\mathrm{Sh}(X,{\mathrm{Ch}}_{+}(\mathrm{Ab}))$ be the category of sheaves on $S$ with values in non-negatively graded chain complexes of abelian groups. The normalized chain complex extends objectwise to a functor

$$\mathrm{Sh}(X,\mathrm{sAb})\stackrel{\simeq }{\to }\mathrm{Sh}(X,{\mathrm{Ch}}_{+}(\mathrm{Ab}))$$Sh(X,sAb) \stackrel{\simeq}{\to} Sh(X, Ch_+(Ab))

which is an equivalence of categories. Moreover, both these categories are naturally categories with weak equivalences: the weak equivalences in $\mathrm{Sh}(X,\mathrm{sAb})$ are the stalkwise weak equivalences of simplicial sets and the weak equivalences in $\mathrm{Sh}(X,{\mathrm{Ch}}_{+}(\mathrm{Ab}))$ are the quasi-isomorphisms. The normalized chain complex functor preserves these weak equivalences. This sheaf version of the Dold–Kan correspondence allows to understand abelian sheaf cohomology as a special case of nonabelian cohomology.

See page 9,10 of

• K. Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology

$(\mathrm{\infty }\mathrm{,1})$-Version

There is a version of the Dold–Kan correspondence in the context of $(\mathrm{\infty }\mathrm{,1})$-categories:

let $C$ be a stable (∞,1)-category. Then the $(\mathrm{\infty }\mathrm{,1})$-categories of complexes in $C$ is equivalent to the $(\mathrm{\infty }\mathrm{,1})$-category of simplicial objects in $C$

$$\mathrm{Fun}(N({\mathbb{Z}}_{\ge 0}),C)\simeq \mathrm{Fun}(N(\Delta {)}^{\mathrm{op}},C).$$Fun(N(\mathbb{Z}_{\geq 0}), C) \simeq Fun(N(\Delta)^{op}, C) \,.

This is theorem 12.8, p. 50 of

• Jacob Lurie, Stable ∞-Categories

Dendroidal version

There is a version of the Dold–Kan correspondence with simplicial sets generalized to dendroidal sets. This is described in

• Javier Gutiérrez?, Andor Lukacs?, Ittay Weiss, Dold-Kan correspondence for dendroidal abelian groups (arXiv)

Related nerve constructions

This gives a pattern for constructing simplicial structures, often called the simplicial nerve, from algebraic structures.

For example the nerve of a groupoid $G$ may be defined as the simplicial set which is $\mathrm{Ob}(G)$ in dimension 0 and is given in dimension $n>0$ by

$$\mathrm{Nerve}(G{)}_n=\mathrm{Gpd}({\pi }_1({\Delta }^n,{\Delta }_0^n),G).$$Nerve(G)_n= Gpd(\pi_1(\Delta^n,\Delta^n_0) , G).

where ${\Delta }_0^n$ denotes the set of vertices of ${\Delta }^n$.

A filtered space $X_{*}$ has a fundamental crossed complex $\Pi X_{*}$, and a geometric simplex ${\Delta }^n$ has a filtration ${\Delta }_{*}^n$ by skeleta. The nerve of a crossed complex $C$ is then the simplicial set given in dimension $n$ by

$$\mathrm{Nerve}(C{)}_n=\mathrm{Crs}(\Pi ({\Delta }_{*}^n,C).$$Nerve(C)_n= Crs(\Pi(\Delta^n_*, C).

This includes the case when $C$ is a crossed module (of groupoids) regarded as a crossed complex of length 2. The geometric realisation of this simplicial set gives the classifying space $\mathrm{BC}$ of the crossed complex $C$. This space is filtered by the length of truncations of $C$ to give a filtered space $(\mathrm{BC}{)}_{*}$ and it is a theorem that $\Pi (\mathrm{BC}{)}_{*}\cong C$.

An obvious analogue gives cubical or globular nerves.

References

Historical references for the Dold–Kan correspondence are

• A. Dold, Homology of symmetric products and other functors of complexes (jstor)

which considers the correspondence for categories of modules, and

• A. Dold, D. Puppe, Homologie nicht-additiver Funktoren. Anwendugen (numdam)

that generalizes the result to arbitrary abelian categories.

The expression of the correspondence in terms of nerves 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).

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

• Goerss, Jardine, Simplicial Homotopy Theory (web)

The monoidal structure on the Dold–Kan correspondence functors is discussed in

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

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

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

• Bourn2007 D. 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)