2022/02/27 by Martino Lupini, Lupini, Martino · 3 citations
Mathematics · #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.2202.13439
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→ 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→ B is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor M→ 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.