Waldhausen Additivity: Classical and Quasicategorical

We give a short proof of classical Waldhausen Additivity, and then prove Waldhausen Additivity for an ∞-version of Waldhausen K-theory. Namely, we prove that Waldhausen K-theory sends a split-exact sequence of Waldausen quasicategories A → E → B to a stable equivalence of spectra K(E) → K(A) ∨ K(B) under a few mild hypotheses. For example, each cofiber sequence in A of the form A0 → A1 → ∗ is required to have the first map an equivalence. Model structures, presentability, and stability are not needed. In an effort to make the article self-contained, we provide many details in our proofs, recall all the prerequisites from the theory of quasicategories, and prove some of those as well. For instance, we develop the expected facts about (weak) adjunctions between quasicategories and (weak) adjunctions between simplicial categories.