Weight structures and the algebraic $K$-theory of stable $\infty$-categories
We introduce the notion of a bounded weight structure on a stable $\infty$-category and use this to prove the natural generalization of Waldhausen's sphere theorem: We show that the algebraic $K$-theory of a stable $\infty$-category with a bounded non-degenerate weight structure is equivalent to the algebraic $K$-theory of the heart of the weight structure.