Example 3.5.3.24. 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 $n \geq 0$, the coskeleton $\operatorname{cosk}_{n}( X)$ inherits the structure of a simplicial abelian group. It follows from Theorem 2.5.6.1 that $\operatorname{cosk}_{n}(X)$ 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 conditions $(a)$ and $(b)$ of Proposition 3.5.3.9 and are equipped with a chain map $A_{\ast } \rightarrow A'_{\ast }$. More concretely, we can identify $A'_{\ast }$ with the chain complex
where $Z_{n}$ is the group of $n$-cycles of $A_{\ast }$.