Shadows and traces in bicategories

Kate Ponto, Michael Shulman

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


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 languageEnglish
Pages (from-to)151-200
Number of pages50
JournalJournal of Homotopy and Related Structures
Issue number2
StatePublished - 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.


  • Bicategory
  • Fixed-point theory
  • Trace

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology


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