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.AxwS.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)K(B)K(E(A,C,B)). This ...
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.AxwS.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)K(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.
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