TITLE: Is there any generalization of the Dold-Kan correspondence?
QUESTION [13 upvotes]: 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?)

REPLY [5 votes]: 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.)