The additivity of traces in monoidal derivators

Moritz Groth, Kate Ponto, Michael Shulman

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

Motivated by traces of matrices and Euler characteristics of topological spaces, we expect abstract traces in a symmetric monoidal category to be "additive". When the category is "stable" in some sense, additivity along cofiber sequences is a question about the interaction of stability and the monoidal structure. May proved such an additivity theorem when the stable structure is a triangulation, based on new axioms for monoidal triangulated categories. in this paper we use stable derivators instead, which are a different model for "stable homotopy theories". We define and study monoidal structures on derivators, providing a context to describe the interplay between stability and monoidal structure using only ordinary category theory and universal properties. We can then perform May's proof of the additivity of traces in a closed monoidal stable derivator without needing extra axioms, as all the needed compatibility is automatic.

Original languageEnglish
Pages (from-to)422-494
Number of pages73
JournalJournal of K-Theory
Volume14
Issue number3
DOIs
StatePublished - Jul 8 2014

Bibliographical note

Publisher Copyright:
Copyright © ISOPP 2014.

Keywords

  • Euler characteristic
  • derivator
  • duality
  • monoidal category
  • trace
  • triangulated category

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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