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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 $K$.

The Dold–Kan correspondence says thatis an equivalence from simplicial abelian groups to nonnegatively graded chain complexes of abelian groups. The restriction of the right adjoint $K$ to ${\mathbf{Ch}}_{\ge 0}(\mathbf{Ab})$ is a quasi-inverse: both the unit $C_{\ast }\to N_{\ast }K(C_{\ast })$ and counit $K(N_{\ast }A)\to A$ are natural isomorphisms.

For a simplicial abelian group $A$, its normalized chain complex of a simplicial abelian group isThe simplicial identities give ${\partial }^2=0$. Equivalently, $N_{n}A$ is the quotient of $A_n$ by the subgroup generated by degenerate simplices, with differential induced by ${\sum }_i(-1{)}^i{d}_i$.

Regard the given short exact sequence as a degreewise short exact sequence of chain complexes concentrated in degree $n$. The inverse functor in the Dold–Kan correspondence is exact, so it produces a degreewise short exact sequence of simplicial abelian groupsThe last map is degreewise surjective and hence a Kan fibration. Its strict fiber is $K(A_1,n)$, and a strict fiber of a fibration computes the homotopy fiber. This proves the asserted homotopy fiber sequence of pointed Kan complexes.