Trace map, Cayley transform and LS category of Lie groups

正则时钟映射与迹空间

Regular clock maps and trace spaces

本文研究正则时钟映射，证明其构成时钟映射范畴的小正交类，从而局部可表现，并证明正则时钟映射下从有向路径到迹的典范商映射是同伦等价。

Analogy between the cyclotomic trace map $K \rightarrow TC$ and the Grothendieck trace formula via noncommutative geometry

In this article, we suggest a categorification procedure in order to capture an analogy between Crystalline Grothendieck-Lefschetz trace formula and the cyclotomic trace map $K\rightarrow TC$ from the algebraic $K$-theory to the topological cyclic homology $TC$. First, we categorify the category of schemes to the $(2, \infty)$-category of noncommuatative schemes a la Kontsevich. This gives a categorification of the set of rational points of a scheme. Then, we categorify the Crystalline Grothendieck-Lefschetz trace formula and find an analogue to the Crystalline cohomology in the setting of non…

The additivity of traces in monoidal derivators

Motivated by traces of matrices and Euler characteristics of topological spaces, we expect abstract traces in a symmetric monoidal category to be "additive". When the category is "stable" in some sense, additivity along cofiber sequences is a question about the interaction of stability and the monoidal structure. May proved such an additivity theorem when the stable structure is a triangulation, based on new axioms for monoidal triangulated categories. In this paper we use stable derivators instead, which are a different model for "stable homotopy theories". We define and study monoidal struct…

追踪方法用于稳定范畴 I：代数K-理论的线性近似

Trace methods for stable categories I: The linear approximation of algebraic K-theory

本文提出在稳定范畴的双模设定下，通过追踪方法构建K-理论与THH之间的映射，并扩展了Dundas-McCarthy的结果。

The Homology Groups of a Partial Trace Monoid Action

The aim of this paper is to investigate the homology groups of mathematical models of concurrency. We study the Baues-Wirsching homology groups of a small category associated with a partial monoid action on a set. We prove that these groups can be reduced to the Leech homology groups of the monoid. For a trace monoid with an action on a set, we will build a cubical complex of free Abelian groups with homology groups isomorphic to the integral homology groups of the action category. It allows us to solve the problem posed by the author in 2004 of the constructing an algorithm for computing homo…

Uniqueness of the multiplicative cyclotomic trace

Making use of the theory of noncommutative motives, we characterize the topological Dennis trace map as the unique multiplicative natural transformation from algebraic K-theory to topological Hochschild homology (THH) and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology (TC). Moreover, we prove that the space of all multiplicative structures on algebraic K-theory is contractible. We also show that the algebraic K-theory functor from small stable infinity categories to spectra is lax symmetric monoidal, which in particular implies that E_n ring spec…

一维因子化同调的迹

Traces for factorization homology in dimension 1

本研究构造了对称幺半∞-范畴中与可对偶对象相关的圆不变因子化同调迹，证明了Toën–Vezzosi猜想，还通过仿循环范畴计算步行伴随的圆因子化同调，展现了一维因子化同调的庞加莱对偶形式。

Traces in monoidal derivators, and homotopy colimits

A variant of the trace in a monoidal category is given in the setting of closed monoidal derivators, which is applicable to endomorphisms of fiberwise dualizable objects. Functoriality of this trace is established. As an application, an explicit formula for the trace of the homotopy colimit of endomorphisms over finite EI-categories is deduced. This result can be seen as a generalization of the additivity of traces in monoidal categories with a compatible triangulation.