(Looking for) the heart of abelian Polish groups

We prove that the category $M$ 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 $A$ of abelian Polish groups in the sense of Beilinson–Bernstein–Deligne and Schneiders. Thus, $M$ is an abelian category containing $A$ as a full subcategory such that the inclusion functor $A\to M$ is exact and finitely continuous. Furthermore, $M$ is uniquely characterized up to equivalence by the following universal property: for every abelian category $B$, a functor $A\to B$ is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor $M\to B$. In particular, this provides a description of the left heart of $A$ 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 Fréchet spaces over a separable complete non-Archimedean valued field.