Waldhausen Additivity: Classical and Quasicategorical
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.
Keywords
Cite
@article{arxiv.1207.6613,
title = {Waldhausen Additivity: Classical and Quasicategorical},
author = {Thomas M. Fiore and Malte Pieper},
journal= {arXiv preprint arXiv:1207.6613},
year = {2018}
}
Comments
83 pages. The revised and copy-edited version of this obsolete arXiv version will be soon published in the Journal of Homotopy and Related Structures. https://link.springer.com/journal/volumesAndIssues/40062 . The final, pre-copy-edit, pre-publication accepted version is at http://www-personal.umd.umich.edu/~tmfiore/1/research.html and will replace this arXiv version after the embargo period