Injective and projective objects in categories of topological torsion Polish abelian groups

We give complete proofs of the injective and projective classifications in four categories of non-Archimedean Polish abelian groups, according to whether the groups are locally compact and whether their discrete quotients are torsion or p-primary. The new results characterize projectives without a local compactness assumption: they are precisely the compact torsion-free groups, namely products over primes p of powers of Z_p in the torsion case and powers of Z_p in the p-primary case, with all multiplicities at most countable. These results answer two questions posed by Casarosa, Codenotti, and Lupini in 2024. We also prove their known injective classifications: the locally compact injectives are the countable discrete divisible groups of the relevant prime class, and the injectives in the larger categories are countable products of such groups. None of the four categories has enough projectives, and precisely the compact objects admit strict projective resolutions. The two larger categories have enough injectives; in the locally compact categories, precisely the discrete objects embed strictly into injectives. We also characterize injectives and projectives in all four left hearts: only zero is injective, and the projectives remain exactly the compact torsion-free groups. The locally compact left-heart results are known; for the two larger hearts, the absence of nonzero injectives follows from an argument using Solecki ranks.

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