扭曲双范畴影子与迹

中文摘要 AI 辅助

双范畴影子提供了一个范畴框架，涵盖Hochschild同调与拓扑Hochschild同调（THH），编码它们的Morita不变性，并将迹的概念从对称幺半范畴扩展到双范畴。某些等变变体，如$C_n$-扭曲THH，无法纳入普通影子框架。我们引入带有$G$-扭曲数据的双范畴，证明此类双范畴上的每个普通影子会对每个$g\in G$诱导出关联$G$-扭曲双范畴上的一个$g$-扭曲影子。这些$g$-扭曲影子在$G$-Morita等价下不变；例子包括$C_n$-扭曲THH和$C_n$-Green函子的扭曲Hochschild同调。我们进一步定义$g$-扭曲双范畴迹，其可恢复$g$-扭曲Hattori–Stallings迹。对于$C_n$-Green函子，逐轨道的$g$-扭曲Hattori–Stallings迹组装为$C_n$-Mackey函子的态射，在轨道$C_n/C_n$处可恢复零次扭曲Dennis迹。

英文摘要

Bicategorical shadows provide a categorical framework that encompasses Hochschild homology and topological Hochschild homology (THH), encodes their Morita invariance, and extends the notion of trace from symmetric monoidal categories to bicategories. Certain equivariant variants, such as $C_n$-twisted THH, do not fit into the ordinary shadow framework. We introduce bicategories with $G$-twisting data and show that every ordinary shadow on such a bicategory induces, for each $g\in G$, a $g$-twisted shadow on the associated bicategory of $G$-twists. These $g$-twisted shadows are invariant under $G$-Morita equivalence; examples include $C_n$-twisted THH and twisted Hochschild homology of $C_n$-Green functors. We further define a $g$-twisted bicategorical trace that recovers the $g$-twisted Hattori--Stallings trace. For $C_n$-Green functors, the orbitwise $g$-twisted Hattori--Stallings traces assemble into a morphism of $C_n$-Mackey functors that, at the orbit $C_n/C_n$, recovers the degree-zero twisted Dennis trace.

发表机构

• Michigan State University（密歇根州立大学）

机构由 AI 辅助整理，请以论文原文为准。