Simplicial abelian group

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

For a simplicial abelian group $A$, one convention for normalized chains isEquivalently, it is the quotient of the unnormalized chain complex by its degenerate subcomplex.

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 $K$.

For an abelian group $A$, an Eilenberg–MacLane space $K(A,n)$ has ${\pi }_n\cong A$ and all other homotopy groups trivial. In the simplicial model it is $K(A[n])$, where $A[n]$ is the chain complex concentrated in degree $n$.