Shadows and traces in bicategories
Shadows and traces in bicategories
复制标题
双类别中的阴影和痕迹
DOI：
10.1007/s40062-012-0017-0
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发表时间：
2009
影响因子：
0.5
通讯作者：
Michael Shulman
中科院分区：
文献类型：
--
作者：
K. Ponto;Michael Shulman
对称么半群范畴中的迹是众所周知的，并且有许多应用；例如，它们的函数性直接蕴含着Lefschetz不动点定理。然而，对于一些应用，如Lefschetz定理的推广，需要“非对易”迹，例如非对易环上的模的Hattori-Stallings迹。本文研究了对称么半群的一个推广，它适用于非对易情形；它的背景是一个带有额外结构的双范畴，称为“影子”。特别地，我们证明了它的函数性和2-函数性，这对于它在不动点理论中的应用是必不可少的。在整个过程中，我们使用了适当的“圆柱形”类型的串图，我们在附录中正式证明了这一点。
Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs “noncommutative” traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a “shadow”. In particular, we prove its functoriality and 2-functoriality, which are essential to its applications in fixed-point theory. Throughout we make use of an appropriate “cylindrical” type of string diagram, which we justify formally in an appendix.