I am fairly new to the concept of model categories, simplicial sets, etc. And so there is some questions, which may be obivuous, that I need to clarify.
Consider the cateogry of simplicial abelian groups $\textbf{S.Ab}$ and the category of non-negative chain complexes $\textbf{Ch}{+}$. The Dold-Kan correspondece tells us that theres is an equivalence between these categories, and between their homotopy categories.
Now consider the model structures on these two categories. Suppose we want to construct the eilenberg-maclane space $K(A,n)$ using this Dold-Kan correspondence. My naive intutive idea, would be to consider the chain complex which is just $A$ in the $n-$th position and $0$ otherwise. Then one would have that the homology of this chain complex is trivial for $k\neq n$ and $A$ at degree $n$.
Now I would like to understand if there is some kind of result that tells me that the homology of a chain complex $C$ is related to some homotopy groups on the model structure of $\textbf{Ch}_{+}$.
And then just take $\Gamma(A)\in \textbf{S.Ab}$ and finally take it's geometric realization to obtain the eilenberg-maclane space $K(A,n)$.
I guess my problem is understanding how the homology of a chain complex relates to the homotopy groups of the model structure , or if they related at all .
If anyone knows a reference where I can look into these things with more details, or as any insight, I appreciate it.
This is answered by Corollary III.2.7 in Goerss and Jardine's Simplicial Homotopy Theory, which establishes a natural isomorphism between homology groups of a chain complex and homotopy groups of the associated simplicial abelian group.