The heart of the Banach spaces

TL;DR

This paper constructs a heart-like category in exact categories, including Banach spaces, by localizing the monomorphism category, extending the concept beyond traditional exact structures in functional analysis.

Contribution

It provides a direct, rigorous construction of a heart-like category for Banach spaces and similar categories lacking an exact structure, broadening the scope of categorical analysis.

Findings

Constructs a heart-like category for Banach spaces.

Extends the concept to categories without an exact structure.

Provides a new approach using localization of monomorphism categories.

Abstract

Consider an exact category in the sense of Quillen. Assume that in this category every morphism has a kernel and that every kernel is an inflation. In their seminal 1982 paper, Beilinson, Bernstein and Deligne consider in this setting a t-structure on the derived category and remark that its heart can be described as a category of formal quotients. They further point out that the category of Banach spaces is an example, and that here a similar category of formal quotients was studied by Waelbroeck already in 1962. In the current article, we give a direct and rigorous construction of the latter category by considering first the monomorphism category. Then we localize with respect to a multiplicative system. Our approach gives rise to a heart-like category not only for the Banach spaces. In particular, the main results apply to categories in which the set of all kernel-cokernel pairs does not form an exact structure. Such categories arise frequently in functional analysis.