Simplicial abelian group

A simplicial abelian group is a simplicial object in abelian groups. Its underlying simplicial set is always a Kan complex.

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- Normalized chain complex of a simplicial abelian group Simplicial abelian group
  - Dold–Kan correspondence Normalized chain complex of a simplicial abelian group

Normalized chain complex of a simplicial abelian group ( $N_{*} A$ )

For a simplicial abelian group

A

, one convention for normalized chains is

N_{n} A = ⋂_{i = 1}^{n} ker (d_{i} : A_{n} \to A_{n - 1}), \partial = d_{0} .

Equivalently, it is the quotient of the unnormalized chain complex by its degenerate subcomplex.

Dold–Kan correspondence

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

K

.

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