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The additivity of traces in monoidal derivators

Published online by Cambridge University Press:  14 July 2014

Moritz Groth ,
Kate Ponto  and
Michael Shulman
Moritz Groth
Affiliation:
Department of Mathematics, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, Netherlands, M.Groth@math.ru.nl
Kate Ponto
Affiliation:
Department of Mathematics, University of Kentucky, 719 Patterson Office Tower, Lexington, KY, 40506, USA, kate.ponto@uky.edu
Michael Shulman
Affiliation:
Department of Mathematics and Computer Science, University of San Diego, 5998 Alcalá Park San Diego, CA, 92110, USA, shuiman@sandiego.edu
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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.

Keywords

dualitytraceEuler characteristicderivatortriangulated categorymonoidal category18D1018E3018G5555U35

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Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 3 , December 2014 , pp. 422 - 494
Copyright
Copyright © ISOPP 2014 

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