Dold–Kan correspondence

Remarks

A quasi-inverse $\Gamma$ is given by $$\Gamma(C)_n = \bigoplus_{[n] \twoheadrightarrow [k]} C_k.$$ The sum runs over all surjective order-preserving maps from $\set{0, \ldots, n}$ onto $\set{0, \ldots, k}$. The face and degeneracy maps are defined from the simplicial identities and the differential of $C$. To see that $N$ is an equivalence, one shows that every $A_n$ splits as the direct sum of the images of $N(A)_k$ under the iterated degeneracies, indexed by these surjections.

The equivalence carries more structure. For a simplicial abelian group $A$, the homotopy groups $\pi_n(A)$ are the homology groups $H_n(N(A))$. Simplicial homotopies correspond to chain homotopies. The normalized complex $N(A)$ is a direct summand of the unnormalized complex with differential $\sum_i (-1)^i d_i$, and the inclusion is a chain homotopy equivalence.

Example: for an abelian group $M$ and $n \ge 0$, the chain complex $M[n]$ with $M$ in degree $n$ and $0$ elsewhere corresponds to a simplicial abelian group $K(M, n) = \Gamma(M[n])$. Its geometric realization is an Eilenberg–MacLane space for $M$ in degree $n$.

Going deeper

For a student

The correspondence turns homotopy groups of simplicial abelian groups into homology of chain complexes. It is used to build Eilenberg–MacLane spaces, to define derived functors of non-additive functors, as Dold and Puppe did, and to compare simplicial and differential graded algebra. The functors $N$ and $\Gamma$ are not monoidal equivalences; the Eilenberg–Zilber and Alexander–Whitney maps relate the tensor products only up to homotopy. The statement extends to any idempotent complete additive category, and it has an $\infty$-categorical version for stable $\infty$-categories.