Applications of Borel-definable homological algebra to locally compact groups

Abstract

We apply the description of the left heart of the category $\mathbf{LCPAb}$ of locally compact Polish abelian groups in terms of groups with a Polish cover and Borel-definable group homomorphisms to study injective and projective objects in the left heart of $\mathbf{LCPAb}$, as well as in the left hearts of its thick subcategories. In particular, we prove that the left hearts of the following categories have no nonzero injective objects: locally compact Polish abelian groups, compactly generated locally compact Polish abelian groups, locally compact Polish abelian groups of finite ranks, abelian Lie groups, totally disconnected locally compact Polish abelian groups, topological torsion locally compact Polish abelian groups, locally compact Polish abelian topological $p$-groups for any prime $p$.