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

Autor:	Campbell, Jonathan A., Lind, John A., Malkiewich, Cary, Ponto, Kate, Zakharevich, Inna
Rok vydání:	2020
Předmět:	Mathematics - Algebraic Topology Mathematics - Category Theory Mathematics - K-Theory and Homology 55N15 55P42 18D20 16E40
Druh dokumentu:	Working Paper
Popis:	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. Comment: This paper is a companion to arxiv:2005.04334
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2006.04006 Zobrazit plný text záznamu View this record from Arxiv