  In this paper, we initiate the study of pro-Lie Polish abelian groups from
the perspective of homological algebra. We extend to this context the
type-decomposition of locally compact Polish abelian groups of Hoffmann and
Spitzweck, and prove that the category $\mathbf{proLiePAb}$ of pro-Lie Polish
abelian groups is a thick subcategory of the category of Polish abelian groups.
We completely characterize injective and projective objects in
$\mathbf{proLiePAb}$. We conclude that $\mathbf{proLiePAb}$ has enough
projectives but not enough injectives and homological dimension $1$. We also
completely characterize injective and projective objects in the category of
non-Archimedean Polish abelian groups, concluding that it has enough injectives
and projectives and homological dimension $1$. Injective objects are also
characterized for the categories of topological torsion Polish abelian groups
and for Polish abelian topological $p$-groups, showing that these categories
have enough injectives and homological dimension $1$.
