The localisation theorem for the $\mathrm{K}$-theory of stable $\infty$-categories

TL;DR

This paper presents a comprehensive account of key theorems in algebraic K-theory for stable ∞-categories, including new proofs and derivations based on universal properties and classical constructions.

Contribution

It offers a self-contained presentation of localisation and cofinality theorems, introduces a new proof of the additivity theorem, and derives the cofinality theorem from universal properties.

Findings

New proof of the additivity theorem inspired by Ranicki's algebraic Thom construction

Short proof of the universality theorem of Blumberg, Gepner and Tabuada

Cofinality theorem derived from universal property alone

Abstract

We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic $\mathrm{K}$-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 $\mathrm{K}$-theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and demonstrate that the cofinality theorem can be derived from the universal property alone.