The localisation theorem for the K-theory of stable ∞-categories

Abstract

We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic $\operatorname K$ -theory of stable $\infty$ -categories. It is based on a general formula for the evaluation of an additive functor on a Verdier quotient closely following work of Waldhausen. We also include a new proof of the additivity theorem of $\operatorname K$ -theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and a second proof of the cofinality theorem which is based on the universal property of $\operatorname K$ -theory.