Example 3.5.4.25. Let $A_{\ast }$ be a nonnegatively graded chain complex of abelian groups and let $X = \mathrm{K}( A_{\ast } )$ denote the associated simplicial abelian group. For every integer $n \geq -1$, the weak $n$-coskeleton $\operatorname{cosk}^{\circ }_{n}( X)$ inherits the structure of a simplicial abelian group. It follows from Theorem 2.5.6.1 that $\operatorname{cosk}^{\circ }_{n}(X)$ can be identified with the Eilenberg-MacLane space $\mathrm{K}( A'_{\ast } )$, for some nonnegatively graded chain complex $A'_{\ast }$. Here $A'_{\ast }$ is universal among chain complexes which satisfy the criterion of Exercise 3.5.4.8 and are equipped with a chain map $A_{\ast } \rightarrow A'_{\ast }$. More concretely, we can identify $A'_{\ast }$ with the chain complex
where $Z_{n+1} \subseteq A_{n+1}$ denotes the subgroup of $(n+1)$-cycles of $A_{\ast }$.