  Let $\text{Ch}$ be the category of (possibly unbounded) chain complexes of
abelian groups. In this note we construct the standard Quillen model structure
on $\text{Ch}$, by a method that is somewhat different from the standard one.
Essentially, we use a functorial two-stage projective resolution for abelian
groups, and build everything directly from that. This has the advantage of
being very concrete, explicit and functorial. It does not rely on the small
object argument, or make any explicit use of transfinite induction. On the
other hand, it is not so conceptual, and it does use the fact that subgroups of
free abelian groups are free, so it does not generalise to many rings other
than $\mathbb{Z}$. We do not claim any great technical benefit for this
approach, but it seems like an interesting alternative, and may be
pedagogically useful.
