Theorem of the Heart in Negative K-Theory for Weight Structures

Abstract

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