Example 10.2.3.21. Let $A_{\bullet }$ be a simplicial abelian group and let $\mathrm{N}_{\ast }(A)$ denote the normalized Moore complex of $A_{\bullet }$ (Construction 2.5.5.7). For every integer $n \geq 0$, let $\mathrm{N}_{\leq n}(A)$ denote the subcomplex of $\mathrm{N}_{\ast }(A)$ depicted in the diagram
\[ \cdots \rightarrow 0 \rightarrow \mathrm{N}_{n}(A) \xrightarrow {\partial } \mathrm{N}_{n-1}(A) \rightarrow \cdots \rightarrow \mathrm{N}_{1}(A) \xrightarrow {\partial } \mathrm{N}_{0}(A) \rightarrow 0 \rightarrow \cdots \]
Then the inclusion map
\[ \mathrm{K}( \mathrm{N}_{\leq n}(A) ) \hookrightarrow \mathrm{K}( \mathrm{N}_{\ast }(A) ) \simeq A_{\bullet } \]
exhibits the Eilenberg-MacLane space $\mathrm{K}( \mathrm{N}_{\leq n}(A) )$ as an $n$-skeleton of $A_{\bullet }$ in the category of simplicial abelian groups. Beware that the image of this inclusion is usually larger than the $n$-skeleton of $A_{\bullet }$ as a simplicial set (see Exercise 10.2.3.7).