Theorem of the heart in negative K-theory for weight structures

Abstract

We construct the strong weight complex functor (in the sense of Bondarko) for a stable infinity-category $\underline{C}$ equipped with a bounded weight structure $w$. Along the way we prove that $\underline{C}$ is determined by the infinity-categorical heart of $w$. This allows us to compare the K-theory of $\underline{C}$ and the K-theory of $\underline{Hw}$, the classical heart of $w$. In particular, we prove that $\operatorname{K}_{n}(\underline{C}) \to \operatorname{K}_{n}(\underline{Hw})$ are isomorphisms for $n \le 0$.