Spectral Waldhausen categories, the S_•-construction, and the Dennis trace

We give an explicit point-set construction of the Dennis trace map from the K-theory of endomorphisms K End(C) to topological Hochschild homology THH(C) for any spectral Waldhausen category C. We describe the necessary technical foundations, most notably a well-behaved model for the spectral category of diagrams in C indexed by an ordinary category via the Moore end. This is applied to define a version of Waldhausen’s S•-construction for spectral Waldhausen categories, which is central to this account of the Dennis trace map. Our goals are both convenience and transparency—we provide all details except for a proof of the additivity theorem for THH, which is taken for granted—and the exposition is concerned not with originality of ideas, but rather aims to provide a useful resource for learning about the Dennis trace and its underlying machinery.