The additivity of traces in monoidal derivators

Moritz Groth; Kate Ponto; Michael Shulman

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arXiv · 1212.3277

The additivity of traces in monoidal derivators

Moritz Groth·Kate Ponto·Michael Shulman

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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.

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BibTeX RIS

Moritz Groth, Kate Ponto, Michael Shulman. 2014-03-06. The additivity of traces in monoidal derivators. https://arxiv.org/abs/1212.3277

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