Partial Tambara structure on the Burnside biset functor, induced from a derivator-like system of adjoint triplets

Abstract

In the previous article “A Mackey-functor theoretic interpretation of biset functors”, we have constructed the 2-category Sof finite sets with variable finite group actions, in which bicoproducts and bipullbacks exist. As shown in it, biset functors can be regarded as a special class of Mackey functors on S.In this article, we equip S with a system of adjoint triplets, which satisfies properties analogous to a derivator. This system encodes the six operations for finite groups. As a corollary, we show that the associated Burnside rings satisfy analogous properties to a Tambara functor, in the context of biset functor theory.