Waldhausen Additivity: Classical and Quasicategorical

TL;DR

This paper extends Waldhausen's Additivity theorem to quasicategories, establishing a framework for Waldhausen K-theory in the quasicategorical setting and proving related stability and equivalence results.

Contribution

It generalizes Waldhausen Additivity to quasicategories, develops Waldhausen quasicategories, and provides conditions for split exact sequences to be standard, applicable to stable quasicategories.

Findings

Proves Waldhausen Additivity using simplicial product version of Quillen's Theorem A.

Establishes stable equivalence of spectra for split exact sequences in quasicategories.

Shows Waldhausen K-theory preserves split cofiber sequences in stable quasicategories.

Abstract

We use a simplicial product version of Quillen's Theorem A to prove classical Waldhausen Additivity of wS., which says that the "subobject" and "quotient" functors of cofiber sequences induce a weak equivalence wS.E(A,C,B)--> wS.A x wS.B . A consequence is Additivity for the Waldhausen K-theory spectrum of the associated split exact sequence, namely a stable equivalence of spectra K(A)vK(B)--> K(E(A,C,B)). This paper is dedicated to transferring these proofs to the quasicategorical setting and developing Waldhausen quasicategories and their sequences. We also give sufficient conditions for a split exact sequence to be equivalent to a standard one. These conditions are always satisfied by stable quasicategories, so Waldhausen K-theory sends any split exact sequence of pointed stable quasicategories to a split cofiber sequence. Presentability is not needed. In an effort to make the article self-contained, we recall all the necessary results from the theory of quasicategories, and prove a few quasicategorical results that are not in the literature.