Applications of Borel-definable homological algebra to locally compact groups

We show that the $\mathrm{H}\mathrm{o}\mathrm{m}$ functor from the category $\mathbf{L}\mathbf{C}\mathbf{P}\mathbf{A}\mathbf{b}$ of locally compact Polish abelian groups to the category $\mathbf{P}\mathbf{A}\mathbf{b}$ of Polish abelian groups has a total right derived functor, improving on Hoffmann and Spitzweck’s construction of its cohomological right derived functor. We also apply the description of the left heart of subcategories of $\mathbf{P}\mathbf{A}\mathbf{b}$ in terms of groups with a Polish cover and Borel-definable group homomorphisms to completely characterize the injective and projective objects in the left heart of $\mathbf{L}\mathbf{C}\mathbf{P}\mathbf{A}\mathbf{b}$, as well as in the left heart of its full subcategories spanned by: compactly generated groups, Lie groups, totally disconnected groups, topological torsion groups, topological $p$-groups, locally compact groups of finite ranks, topological torsion groups of finite ranks, topological $p$-groups of finite ranks, and type $\mathbb{A}$ groups.