Abstract
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.
Original language | English |
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Pages (from-to) | 151-200 |
Number of pages | 50 |
Journal | Journal of Homotopy and Related Structures |
Volume | 8 |
Issue number | 2 |
DOIs | |
State | Published - Oct 2013 |
Bibliographical note
Funding Information:K. Ponto and M. Shulman were supported by National Science Foundation postdoctoral fellowships during the writing of this paper.
Keywords
- Bicategory
- Fixed-point theory
- Trace
ASJC Scopus subject areas
- Algebra and Number Theory
- Geometry and Topology