Dold-Kan equivalence for simplicial modules with explicit inverse

The direct-sum decomposition. Put $K(n,i)={\bigcap }_{j<i}\ker (d_j\text{\hspace{0.17em}⁣}:U_n\to U_{n-1})$ for $0\le i\le n$. On $K(n,i)$ the map $d_i$ lands in $K(n-1,i)$: for $j<i$, $d_j{d}_i=d_{i-1}{d}_j$ vanishes. The map $s_i$ carries $K(n-1,i)$ into $K(n,i)$ by $d_j{s}_i=s_{i-1}{d}_j$ and is a section of $d_i$. Thus $x=(x-s_i{d}_{i}x)+s_i{d}_{i}x$ gives the unique splitting $K(n,i)=K(n,i+1)\oplus s_{i}K(n-1,i)$. Starting with $K(n,0)=U_n$, successively split for $i=0,\dots ,n-1$ and then recursively split each lower-dimensional $K(n-1,i)$. This produces exactly the summands $s_{i_1}\cdots s_{i_t}N(U{)}_{n-t}$ with $i_1\le \cdots \le i_t$, each with a unique coefficient; the sequences are precisely the canonical degeneracy factorizations of the order-preserving surjections $[n]\twoheadrightarrow [n-t]$, using $s_i{s}_j=s_{j+1}{s}_i$ to put any factorization in this form. The splitting formulas commute with homomorphisms of simplicial abelian groups, so the resulting direct-sum map with components $U(\alpha )$ is a natural isomorphism.

The inverse functor. For a nonnegative chain complex $C$ and $n\ge 0$ put $\mathrm{\Gamma }(C{)}_n={⨁}_{\alpha \text{\hspace{0.17em}⁣}:[n]\twoheadrightarrow [r]}{C}_r$. For $\phi \text{\hspace{0.17em}⁣}:[m]\to [n]$ and the $\alpha$-summand, put $\beta =\alpha \phi$ and write $[s]$ for the initial interval of $\mathrm{Im}\beta$ when it is one: if $\mathrm{Im}\beta =[r]$ map by the identity to the $\beta$-summand (with $\beta$ corestricted to its image), if $\mathrm{Im}\beta =[r-1]$ map by $(-1{)}^r{d}_C$, and otherwise map by zero; a gap in the image or the loss of at least two terminal vertices both fall under the zero case. These formulas respect composition: if an intermediate image has a gap, then any later initial-interval image lies below the gap and has lost at least two vertices, so the direct formula is zero; losing two or more terminal vertices stays zero under further restriction; if the first map loses none the second rule is the composite rule; if it loses exactly one index, the second map either loses none (same single signed differential), has a gap (zero), or loses at least one more, and the only possibly nonzero iterated case gives $d_C^2=0$. Identity operators act identically, so $\mathrm{\Gamma }(C)$ is a simplicial $R$-module, functorially in $C$.

The normalization differential. For $x\in N(U{)}_n$ and $j<n-1$ the identities give $d_j{d}_{n}x=d_{n-1}{d}_{j}x=0$, so $d_n$ maps $N(U{)}_n$ into $N(U{)}_{n-1}$; therefore $(-1{)}^n{d}_n$ defines a differential on $N(U)$ and $d_{n-1}{d}_n=d_{n-1}^2=0$ on normalized elements, as in [F2]. The passage to simplicial $R$-modules is $R$-linear throughout.

$N\mathrm{\Gamma }\cong \mathrm{i}\mathrm{d}$. The degenerate summands of $\mathrm{\Gamma }(C{)}_n$ are exactly those with $r<n$, since every nonidentity surjection factors through an elementary degeneracy and the rule of step 1.2 makes that factorization the identity on the corresponding coefficient. The ${\mathrm{i}\mathrm{d}}_{[n]}$-coefficient $C_n$ has all faces zero except the last, which is $(-1{)}^n{d}_C$. Hence the normalization of $\mathrm{\Gamma }(C)$ is exactly $C$ with its differential, so $N\mathrm{\Gamma }(C)\cong C$ naturally.

$\mathrm{\Gamma }N\cong \mathrm{i}\mathrm{d}$. The direct-sum map of step 1.1 is bijective in every degree, and its compatibility with a simplex operator $\phi$ can be checked on an $\alpha$-summand: if $\alpha \phi$ has a missing index $j<r$ it factors through the $j$-th face, which vanishes on $N(U{)}_r$; if its image is initial but has lost at least two terminal indices it factors through the face $r-1$, which also vanishes on $N(U{)}_r$; if no index is lost the composite is $U(\alpha \phi )$; and if only the last index is lost the restriction equals $(-1{)}^r{d}_N$. These are exactly the four rules defining $\mathrm{\Gamma }$, so the comparison $\mathrm{\Gamma }N(U)\to U$ is a natural isomorphism of simplicial $R$-modules.

Equivalence and scope. The two natural isomorphisms of steps 2.2 and 2.3 are inverse to each other on the nose by the uniqueness of the decomposition, so $N$ is an equivalence of categories; it replaces simplicial additive objects by nonnegative chain complexes as asserted. For exactness, let $M\twoheadrightarrow M^{\mathrm{\prime }}$ be degreewise surjective and let $y\in N(M^{\mathrm{\prime }}{)}_n$. Lift $y$ to $x\in M_n$ and project $x$ to its identity-surjection summand by the natural splitting of step 1.1; naturality makes this normalized projection a lift of $y$. Thus $N$ preserves epimorphisms; it preserves kernels because normalization is an intersection of face kernels. Applying these facts to a short exact sequence proves exactness. Since all constructions are $R$-linear formulas, the same proof applies to simplicial modules over a constant ring $R$; it does not identify a variable simplicial $A$-module with an ordinary chain complex over a fixed ring, for which a separate coefficient-base analysis is required.