For context, I am reading chapter III.2 of Goerss and Jardine's book on simplicial homotopy theory, but I am not an expert in model categories. In that chapter, they introduce the model structure on simplicial abelian groups $sAb$ with weak equivalences the weak equivalences of the underlying simplicial sets and fibrations the Kan fibrations. With this definition the cofibrations are those morphisms with the left lifting property with respect to trivial fibrations. They later remark that
Remark 2.9 [...] All cofibrations of $sAb$ are levelwise monomorphisms.
For simplicial sets, the converse is also true. As a consequence, all simplicial sets are cofibrant.
Are cofibrations in $sAb$ exactly the levelwise monomorphisms, as well?
Are all simplicial abelian groups cofibrant? If not, is there a characterization of a cofibrant object in this model category?
From the way they talk about cofibrancy of an object $A$ as an assumption throughout the chapter, it seems that this is not true. They also mention without proof that the free simplicial abelian group $\mathbb{Z} X$ is cofibrant for any simplicial set $X$, is there an easy way to see this? Is cofibrancy related to freeness of an object?
Finally, I am more generally interested in the same constructions for a general abelian category $\mathcal{A}$. It is my understanding that the same simplicial model structure can be defined on $s \mathcal{A}$. What happens to questions 1. and 2. when replacing abelian groups with objects in a different abelian category $\mathcal{A}$?
I would appreciate clearer references on this topic, if there are any.
