Spectral Waldhausen categories, the S•-construction, and the Dennis trace
Jonathan A. Campbell, John A. Lind, Cary Malkiewich, Kate Ponto and Inna Zakharevich – GJM, Volume 9, Issue 2 (2024), 27-60.
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- décembre 31, 2024
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We give an explicit point-set construction of the Dennis trace map from the K-theory of endomorphisms \textnormal{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.
Milestones:
Received: August 22, 2024
Accepted: December 17, 2024
Revised: December 31, 2024
Published: January 2025
Authors:
Jonathan A. Campbell4320 Westerra Ct, San Diego, CA 92121, USA.
John A. LindDepartment of Mathematics and Statistics,
California State University, Chico, CA USA.
Cary MalkiewichDepartment of Mathematics and Statistics, Binghamton University,
PO Box 6000, Binghamton, NY 13902, USA.
Kate PontoDepartment of Mathematics, University of Kentucky,
719 Patterson Office Tower, Lexington, KY, USA.
Inna Zakharevich587 Malott, Ithaca, NY 14853, USA.