Localization for $THH(ku)$ and the Topological Hochschild and Cyclic Homology of Waldhausen Categories

The authors develop a theory of $THH$ and $TC$ of Waldhausen categories and prove the analogues of Waldhausen's theorems for $K$-theory. They resolve the longstanding confusion about localization sequences in $THH$ and $TC$, and establish a specialized dévissage theorem. As applications, the authors prove conjectures of Hesselholt and Ausoni-Rognes about localization cofiber sequences surrounding $THH(ku)$, and more generally establish a framework for advancing the Rognes program for studying Waldhausen's chromatic filtration on $A(*)$.