Homological algebra of pro-Lie Polish abelian groups

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 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 proLiePAb. We conclude that 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.