Condensed Learning Seminar

Introduce the notion of Verdier sequences of presentable stable $\infty$-categories and then discuss the behavior of dualizable $\infty$-categories in Verdier sequence; in particular, discuss Thomason's trick. Introduce the notion of localizing invariants of small stable $\infty$-categories and then illustrate it by non-connective $K$-theory. In particular, define negative $K$-groups for commutative rings by classical means and then explain its relation to the definition of non-connective $K$-theory as the universal localizing $\mathrm{Sp}$-valued invariant under $\mathrm{core}$. Following Efimov, prove that localizing invariants extend uniquely to all dualizable stable $\infty$-categories.