The normalized-chain functor gives an equivalence between simplicial abelian groups and nonnegatively graded chain complexes of abelian groups. Its inverse is the Eilenberg–MacLane denormalization functor .
For an abelian group , an Eilenberg–MacLane space has and all other homotopy groups trivial. In the simplicial model it is , where is the chain complex concentrated in degree .
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The Dold–Kan correspondence is a fundamental theorem in algebraic topology and homological algebra that establishes a relationship between two important categories: the category of simplicial sets and the category of chain complexes of abelian groups (or modules). It is named after mathematicians Alfred Dold and D. K. Kan, who formulated it in the context of homotopy theory.