Portal de Periódicos da CAPES

The additivity of traces in monoidal derivators

2014; Cambridge University Press; Volume: 14; Issue: 3 Linguagem: Inglês

10.1017/is014005011jkt262

1865-5394

Moritz Groth, Kate Ponto, Michael Shulman,

Advanced Topics in Algebra

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.