谱Waldhausen范畴、$S_\bullet$ -构造和Dennis迹
Spectral Waldhausen categories, the $S_\bullet$-construction, and the Dennis trace
摘要：
对于任意谱Waldhausen范畴$\mathcal{C}$，我们给出了从自同态$K\mathrm{End}(\mathcal{C})$到拓扑霍赫希尔德同调调$\mathrm{THH}(\mathcal{C})$的$K$ -理论的Dennis迹映射的显式点集构造.我们描述了必要的技术基础，最值得注意的是一个良好的表现模型的光谱类别的图表在$\mathcal{C}$索引的普通类别通过摩尔结束。这被应用于定义一个版本的Waldhausen的$S_{\bullet}$ -构造光谱Waldhausen类别，这是中央的丹尼斯跟踪地图的这个帐户。我们的目标是方便和透明-我们提供了所有的细节，除了$\mathrm{THH}$的可加性定理的证明，这是理所当然的-和博览会关注的不是原创性的想法，而是旨在提供一个有用的资源，学习丹尼斯跟踪和它的基本机制。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
