Spectral Waldhausen categories, the $S_\bullet$-construction, and the Dennis trace
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2020
- DOI:
- 10.48550/arXiv.2006.04006
- arXiv:
- arXiv:2006.04006
- Bibcode:
- 2020arXiv200604006C
- Keywords:
-
- Mathematics - Algebraic Topology;
- Mathematics - Category Theory;
- Mathematics - K-Theory and Homology;
- 55N15;
- 55P42;
- 18D20;
- 16E40
- E-Print:
- This paper is a companion to arxiv:2005.04334