Spectral Waldhausen categories, the $S_\bullet$-construction, and the Dennis trace

TL;DR

This paper constructs an explicit Dennis trace map from the K-theory of endomorphisms to topological Hochschild homology for spectral Waldhausen categories, providing detailed foundations and a spectral $S_ullet$-construction.

Contribution

It offers a detailed, explicit point-set construction of the Dennis trace for spectral Waldhausen categories, including a well-behaved model for diagram categories and a spectral $S_ullet$-construction.

Findings

Explicit Dennis trace map constructed

Well-behaved model for diagram categories established

Spectral $S_ullet$-construction defined

Abstract

We give an explicit point-set construction of the Dennis trace map from the $K$-theory of endomorphisms $K\mathrm{End}(\mathcal{C})$ to topological Hochschild homology $\mathrm{THH}(\mathcal{C})$ for any spectral Waldhausen category $\mathcal{C}$. We describe the necessary technical foundations, most notably a well-behaved model for the spectral category of diagrams in $\mathcal{C}$ indexed by an ordinary category via the Moore end. This is applied to define a version of Waldhausen's $S_{\bullet}$-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 $\mathrm{THH}$, 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.