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

TL;DR

This paper develops a derivator-like system of adjoint triplets on a 2-category of finite sets with group actions, leading to a partial Tambara structure on the Burnside biset functor, enriching biset functor theory.

Contribution

It introduces a derivator-like system of adjoint triplets on the 2-category of finite group actions, inducing a partial Tambara structure on the Burnside biset functor.

Findings

Burnside rings satisfy properties similar to Tambara functors

The 2-category $oldsymbol{S}$ encodes six operations for finite groups

Biset functors are interpreted as Mackey functors on $oldsymbol{S}$

Abstract

In the previous article 'A Mackey-functor theoretic interpretation of biset functors', we have constructed the 2-category $\mathbb{S}$ of 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 $\mathbb{S}$. In this article, we equip $\mathbb{S}$ 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 satify analogous properties to a Tambara functor, in the context of biset functor theory.