The linearity of traces in monoidal categories and bicategories

Kate Ponto, Michael Shulman

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


We show that in any symmetric monoidal category, if a weight for colim-its is absolute, then the resulting colimit of any diagram of dualizable objects is again dualizable. Moreover, in this case, if an endomorphism of the colimit is induced by an endomorphism of the diagram, then its trace can be calculated as a linear combination of traces on the objects in the diagram. The formal nature of this result makes it easy to generalize to traces in homotopical contexts (using derivators) and traces in bicate-gories. These generalizations include the familiar additivity of the Euler characteristic and Lefschetz number along cofiber sequences, as well as an analogous result for the Reidemeister trace, but also the orbit-counting theorem for sets with a group action, and a general formula for homotopy colimits over EI-categories.

Original languageEnglish
Pages (from-to)594-689
Number of pages96
JournalTheory and Applications of Categories
StatePublished - Jun 27 2016

Bibliographical note

Publisher Copyright:
© Kate Ponto and Michael Shulman, 2016.


  • Absolute colimit
  • Derivator
  • Duality
  • Trace

ASJC Scopus subject areas

  • Mathematics (miscellaneous)


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