The additivity of traces in monoidal derivators

TL;DR

This paper proves that traces in monoidal stable derivators are additive along cofiber sequences, providing a universal categorical framework that simplifies and generalizes previous results in stable homotopy theory.

Contribution

It introduces monoidal structures on derivators and demonstrates additivity of traces without extra axioms, extending May's theorem to a broader context.

Findings

Additivity of traces proved in monoidal stable derivators.

Monoidal structures on derivators are characterized using universal properties.

Compatibility of stability and monoidal structure is automatic in this framework.

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.