Algebraic K-theory and abstract homotopy theory

TL;DR

This paper develops a new approach to algebraic K-theory using homotopy theory, providing criteria for when functors induce equivalences of K-theory spectra, thus unifying and generalizing previous results.

Contribution

It introduces a decomposition of K-theory spaces via Dwyer-Kan localization and establishes criteria for functors to induce K-theory equivalences, broadening understanding in the field.

Findings

Decomposition of K-theory space using Dwyer-Kan localization

Criteria for functors to induce K-theory spectrum equivalences

Weakly exact functors induce K-theory equivalences under mild conditions

Abstract

We decompose the K-theory space of a Waldhausen category in terms of its Dwyer-Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria appearing in the literature. We show that under mild hypotheses, a weakly exact functor that induces an equivalence of homotopy categories induces an equivalence of K-theory spectra.