The hearts of weight structures are the weakly idempotent complete 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.