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The Dold-Kan correspondence gives an equivalence between simplicial abelian groups and chain complexes of abelian groups supported on negative degrees. It actually works for any abelian category. I'm wondering if there is any generalization of this corrensponces, more precisely, for other category of algebraic objects, like groups, rings, monoids, A-algebras (where A is a ring), should simplicial objects in them correspond to some more familiar objects? (hopefully some graded thing and maybe we can still take cohomologies of them?)

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It may be worth reading the nLab entries on the Moore complex: ncatlab.org/nlab/show/Moore+complex and of course the Dold-Kan correspondence: ncatlab.org/nlab/show/… I however am somewhat out of my depth with the terminology and would appreciate an easier to understand answer to your question. –  James Griffin Jul 21 '10 at 15:58
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4 Answers 4

Associated to a simplicial group $G_n$, there is indeed a "chain complex". Namely, for each n one has the group $C_n$ of all elements in $G_n$ that map to zero under the boundary maps $d_1, \ldots, d_n$ (but not necessarily $d_0$) that are part of the simplicial structure. The map $d_0$ is then a group homomorphism $C_n \to C_{n-1}$ with $Im(d_0) \subset ker(d_0)$, and the image is a normal subgroup. One could view this as a "chain complex" of nonabelian groups, and the homology groups have a nice interpretation: they're the homotopy groups of the geometric realization |G|.

However, this method of passing to a graded object isn't in any sense an equivalence of categories. Moreover, it is known that simplicial groups mod weak equivalence are a model for the homotopy category of pointed CW-complexes. In some ways, then, we don't expect much in the way of purely algebraic characterizations of simplicial groups.

For simplicial rings there is the slight issue that the normalized chain functor plays somewhat poorly together with the tensor product, and so the Dold-Kan correspondence doesn't preserve the symmetric monoidal structure. This is not as serious an issue for associative algebras or commutative algebras in characteristic zero if one is only working up to weak equivalence/quasi-isomorphism, but if you are interested in an actual equivalence of categories it means that a simplicial commutative $\mathbb{Q}$-algebra is not quite the same as a nonnegatively graded commutative differential graded algebra.

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As was pointed out by James Griffin, one can generalize the characterization of simplicial groups by certain chain complexes, quite a bit.

There is also a DK-correspondence for simplicial rings i.e. for simplicial objects in the category of rings. This is the

monoidal Dold-Kan correspondence.

Notice that this is no longer an equivalence of categories, but of (oo,1)-categories.

Generally, the Dold-Kan correspondence may be thought of as identifying certain very "strict" oo-groupoids inside all oo-groupoids. For instance a simplicial abelian group is necessarily a Kan complex with unique fillers and with an extra abelian grup structure on it, so simplicial abelian groups model strict oo-groupoids with strict abelian group structure on them. The DK-correspondence observes that their information is all encoded in the chain complex of homotopy groups of the oo-groupoid.

The monoidal DK-correspondence identifies something like strict $E_\infty$-rings.

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There are nice generalisations with a certain amount of non-abelian aspects. See Nan~Tie, G. {A {D}old-{K}an theorem for crossed complexes}. {J. Pure Appl. Algebra} \textbf{56}~(2) (1989) 177--194.

and the references to work of Nick Ashley. See also Brown, R. and Higgins, P.~J.{Cubical abelian groups with connections are equivalent to chain complexes}.{Homology Homotopy Appl.} \textbf{5}~(1) (2003) 49--52

and the references there. A search on MathSciNet for Dold-Kan in title will reveal other aspects.

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For simplicial groups, there is the generalised Dold-Kan-Puppe theorem by Carrasco-Cegarra: "Group-theoretic algebraic models for homotopy types", J. Pure Appl. Alg. 75 (1991), no. 3, 195--235 (see also MathSciNet MR1137837). (The additive notation in the article is to be read non-commutatively.)

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