The hearts of weight structures are the weakly idempotent complete categories

TL;DR

This paper characterizes the categories that serve as hearts of weight structures as exactly the weakly idempotent complete additive categories, providing new conditions and insights into their properties and completions.

Contribution

It establishes a precise characterization of hearts of weight structures and introduces new equivalent conditions for weak idempotent completeness.

Findings

Hearts of weight structures are exactly weakly idempotent complete categories.

Several new conditions equivalent to weak idempotent completeness are identified.

Discussion on weak idempotent completions of additive categories.

Abstract

In this note we prove that additive categories that occur as hearts of weight structures are precisely the weakly idempotent completecategories, that is, the categories where all split monomorphisms give direct sum decompositions. We also give several other conditions equivalent to weak idempotent completeness (some of them are completely new), and discuss weak idempotent completions of additive categories.