Shadows and traces in bicategories
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.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2009
- DOI:
- 10.48550/arXiv.0910.1306
- arXiv:
- arXiv:0910.1306
- Bibcode:
- 2009arXiv0910.1306P
- Keywords:
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- Mathematics - Category Theory;
- 18D05 (Primary) 18D10 (Secondary)
- E-Print:
- 46 pages