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  "body": "\n## Statement\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]].\nLet $s\\mathcal{A}$ be the category of [[simplicial objects|WK0A32TZHB]] in $\\mathcal{A}$, and let $\\mathrm{Ch}_{\\ge 0}(\\mathcal{A})$ be the category of [[chain complexes|WQ82C0M3QB]] in $\\mathcal{A}$ concentrated in degrees $n \\ge 0$.\nFor $A \\in s\\mathcal{A}$ with face maps $d_i \\colon A_n \\to A_{n-1}$, let the normalized chain complex $N(A)$ be\n$$N(A)_n = \\bigcap_{i=1}^{n} \\ker\\left(d_i \\colon A_n \\to A_{n-1}\\right), \\qquad \\partial_n = d_0 \\colon N(A)_n \\to N(A)_{n-1},$$\nwith $N(A)_0 = A_0$, where $\\ker$ denotes the [[kernel|S417YNEJ6M]].\nThen $N \\colon s\\mathcal{A} \\to \\mathrm{Ch}_{\\ge 0}(\\mathcal{A})$ is an [[equivalence of categories|K7A04ZBD92]].\n\n## Remarks\n\nA quasi-inverse $\\Gamma$ is given by\n$$\\Gamma(C)_n = \\bigoplus_{[n] \\twoheadrightarrow [k]} C_k.$$\nThe sum runs over all surjective order-preserving maps from $\\set{0, \\ldots, n}$ onto $\\set{0, \\ldots, k}$.\nThe face and degeneracy maps are defined from the simplicial identities and the differential of $C$.\nTo 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.\n\nThe equivalence carries more structure.\nFor a simplicial abelian group $A$, the homotopy groups $\\pi_n(A)$ are the [[homology|2J4JJJYP22]] groups $H_n(N(A))$.\nSimplicial homotopies correspond to [[chain homotopies|4EBHAWJJ8C]].\nThe 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.\n\nExample: 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 $K(M, n) = \\Gamma(M[n])$.\nIts geometric realization is an Eilenberg–MacLane space for $M$ in degree $n$.\n\n## Explanations\n\n### L1 human\n\nSimplicial abelian groups are combinatorial models of spaces that carry an addition.\nChain complexes are the basic objects of homological algebra.\nThe Dold–Kan correspondence says that the two are the same thing, once one discards the redundant degenerate simplices.\nThis lets one move freely between homotopy theory and homological algebra.\n\n### L2 human\n\nThe correspondence turns homotopy groups of simplicial abelian groups into homology of chain complexes.\nIt 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.\nThe functors $N$ and $\\Gamma$ are not monoidal equivalences; the Eilenberg–Zilber and Alexander–Whitney maps relate the tensor products only up to homotopy.\nThe statement extends to any idempotent complete additive category, and it has an $\\infty$-categorical version for stable $\\infty$-categories.\n",
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