(Looking For) The Heart of Abelian Polish Groups

We prove that the category $\mathcal{M}$ of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category $\mathcal{A}$ of abelian Polish groups in the sense of Beilinson--Bernstein--Deligne and Schneiders. Thus, $\mathcal{M}$ is an abelian category containing $\mathcal{A}$ as a full subcategory such that the inclusion functor $\mathcal{A}\rightarrow \mathcal{M}$ is exact and finitely continuous. Furthermore, $\mathcal{M}$ is uniquely characterized up to equivalence by the following universal property: for every abelian category $\mathcal{B}$, a functor $\mathcal{A}\rightarrow \mathcal{B}$ is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor $\mathcal{M}\rightarrow \mathcal{B}$. In particular, this provides a description of the left heart of $\mathcal{A}$ as a concrete category. We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish $R$-modules, for a given Polish group or Polish ring $R$; and separable Banach spaces and separable Fr\'{e}chet spaces over a separable complete non-Archimedean valued field.