## 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 language | English |
---|---|

Pages (from-to) | 422-494 |

Number of pages | 73 |

Journal | Journal of K-Theory |

Volume | 14 |

Issue number | 3 |

DOIs | |

State | Published - Jul 8 2014 |

### Bibliographical note

Funding Information:10.1017/is014005011jkt262 S1865243314000269 00026 The additivity of traces in monoidal derivators M. GROTH, K. PONTO & M. SHULMAN The additivity of traces in monoidal derivators Groth Moritz Ponto Kate Shulman Michael * Department of Mathematics, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, Netherlands, M.Groth@math.ru.nl Department of Mathematics, University of Kentucky, 719 Patterson Office Tower, Lexington, KY, 40506, USA, kate.ponto@uky.edu Department of Mathematics and Computer Science, University of San Diego, 5998 Alcalá Park San Diego, CA, 92110, USA, shuiman@sandiego.edu * The first author was partially supported by the Deutsche Forschungsgemeinschaft within the graduate program ‘Homotopy and Cohomology’ (GRK 1150) and by the Dutch Science Foundation (NWO). The second author was partially supported by NSF grant DMS-1207670. The third author was partially supported by an NSF postdoctoral fellowship and NSF grant DMS-1128155, and appreciates the hospitality of the University of Kentucky. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. 14072014 12 2014 14 3 422 494 18 07 2013 Copyright © ISOPP 2014 2014 ISOPP 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. Key Words: duality trace Euler characteristic derivator triangulated category monoidal category Mathematics Subject Classification 2010: 18D10 18E30 18G55 55U35 pdf S1865243314000269a.pdf

Funding Information:

The first author was partially supported by the Deutsche Forschungsgemeinschaft within the graduate program ‘Homotopy and Cohomology’ (GRK 1150) and by the Dutch Science Foundation (NWO). The second author was partially supported by NSF grant DMS-1207670. The third author was partially supported by an NSF postdoctoral fellowship and NSF grant DMS-1128155, and appreciates the hospitality of the University of Kentucky. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Funding Information:

* The first author was partially supported by the Deutsche Forschungsgemeinschaft within the graduate program ‘Homotopy and Cohomology’ (GRK 1150) and by the Dutch Science Foundation (NWO). The second author was partially supported by NSF grant DMS-1207670. The third author was partially supported by an NSF postdoctoral fellowship and NSF grant DMS-1128155, and appreciates the hospitality of the University of Kentucky. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Funding Information:

"KAG Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology J. K-Theory 1865-2433 1865-5394 Cambridge University Press Cambridge, UK 10.1017/is014005011jkt262 S1865243314000269 00026 The additivity of traces in monoidal derivators M. GROTH, K. PONTO & M. SHULMAN The additivity of traces in monoidal derivators Groth Moritz Ponto Kate Shulman Michael * Department of Mathematics, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, Netherlands, M.Groth@math.ru.nl Department of Mathematics, University of Kentucky, 719 Patterson Office Tower, Lexington, KY, 40506, USA, kate.ponto@uky.edu Department of Mathematics and Computer Science, University of San Diego, 5998 Alcalá Park San Diego, CA, 92110, USA, shuiman@sandiego.edu * The first author was partially supported by the Deutsche Forschungsgemeinschaft within the graduate program ‘Homotopy and Cohomology’ (GRK 1150) and by the Dutch Science Foundation (NWO). The second author was partially supported by NSF grant DMS-1207670. The third author was partially supported by an NSF postdoctoral fellowship and NSF grant DMS-1128155, and appreciates the hospitality of the University of Kentucky. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. 14072014 12 2014 14 3 422 494 18 07 2013 Copyright © ISOPP 2014 2014 ISOPP 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. Key Words: duality trace Euler characteristic derivator triangulated category monoidal category Mathematics Subject Classification 2010: 18D10 18E30 18G55 55U35 pdf S1865243314000269a.pdf

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