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

TL;DR

This paper develops a functorial framework for weight structures in stable infinity-categories, establishing isomorphisms between their K-theories in non-positive degrees, thus advancing the understanding of negative K-theory in this context.

Contribution

It constructs the strong weight complex functor for stable infinity-categories with bounded weight structures and proves the equivalence of their K-theories with those of their hearts in non-positive degrees.

Findings

Constructed the strong weight complex functor for stable infinity-categories.

Proved that the infinity-category is determined by its heart.

Established isomorphisms between K-theories of the category and its heart for n ≤ 0.

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 $n \le 0$.