(Looking For) The Heart of Abelian Polish Groups

TL;DR

This paper characterizes the category of abelian groups with a Polish cover as the left heart of the derived category of abelian Polish groups, providing a concrete description and universal property.

Contribution

It establishes that the category of abelian groups with a Polish cover is the left heart of the derived category of abelian Polish groups, with a universal property and concrete description.

Findings

Identifies the category as the left heart of the derived category of abelian Polish groups.

Provides a universal property characterizing this category.

Extends the description to various algebraic categories with topology.

Abstract

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.