Shadows and traces in bicategories
Abstract
Traces in symmetric monoidal categories are wellknown 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 HattoriStallings 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 2functoriality, which are essential to its applications in fixedpoint theory. Throughout we make use of an appropriate "cylindrical" type of string diagram, which we justify formally in an appendix.
 Publication:

arXiv eprints
 Pub Date:
 October 2009
 arXiv:
 arXiv:0910.1306
 Bibcode:
 2009arXiv0910.1306P
 Keywords:

 Mathematics  Category Theory;
 18D05 (Primary) 18D10 (Secondary)
 EPrint:
 46 pages