Waldhausen Additivity: classical and quasicategorical

Abstract

We use a simplicial product version of Quillen’s Theorem A to prove classical Waldhausen Additivity of \(wS_\bullet \), which says that the “subobject” and “quotient” functors of cofiber sequences induce a weak equivalence \(wS_\bullet {\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}) \rightarrow wS_\bullet {\mathcal {A}}\times wS_\bullet {\mathcal {B}}\). A consequence is Additivity for the Waldhausen K-theory spectrum of the associated split exact sequence, namely a stable equivalence of spectra \({\mathbf {K}}({\mathcal {A}}) \vee {\mathbf {K}}({\mathcal {B}}) \rightarrow {\mathbf {K}}({\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}))\). 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.