On the algebraic K-theory of higher categories

TL;DR

This paper extends Waldhausen K-theory to higher categories, showing it can be described as a Goodwillie differential, which leads to new proofs of classical theorems and applications to ring spectra and stacks.

Contribution

It introduces a higher categorical framework for Waldhausen K-theory, demonstrating its description as a Goodwillie differential and establishing its universal property.

Findings

K-theory spaces admit canonical deloopings

The K-theory functor satisfies a universal property

New proofs of additivity and fibration theorems

Abstract

We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective) deloopings, and the K-theory functor enjoys a universal property. Using this, we give new, higher categorical proofs of both the additivity and fibration theorems of Waldhausen. As applications of this technology, we study the algebraic K-theory of associative ring spectra and spectral Deligne-Mumford stacks.