The additivity of traces in monoidal derivators
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
Cite
@article{arxiv.1212.3277,
title = {The additivity of traces in monoidal derivators},
author = {Moritz Groth and Kate Ponto and Michael Shulman},
journal= {arXiv preprint arXiv:1212.3277},
year = {2014}
}
Comments
56 pages. v2: Major revision. The discussion of monodial and closed structures on derivators is reorganized to develop the structure and verify compatibility with stability simultaneously. The discussion of Mayer-Vietoris sequences in derivators is now arXiv:1306.2072. v3: Final version, to appear in Journal of K-Theory