Biset transformations of Tambara functors

If we are given an $H$-$G$-biset $U$ for finite groups $G$ and $H$, then any Mackey functor on $G$ can be transformed by $U$ into a Mackey functor on $H$. In this article, we show that the biset transformation is also applicable to Tambara functors when $U$ is right-free, and in fact forms a functor between the category of Tambara functors on $G$ and $H$. This biset transformation functor is compatible with some algebraic operations on Tambara functors, such as ideal quotients or fractions. In the latter part, we also construct the left adjoint of the biset transformation.

💡 Research Summary

The paper investigates how biset constructions, traditionally used to transfer Mackey functors between finite groups, can be extended to the richer setting of Tambara functors. After recalling the classical biset functor for Mackey functors, the author identifies a crucial hypothesis: the biset (U) must be right‑free (i.e., the right action of the target group (H) on (U) is free). This condition guarantees that the coinduction–restriction pair associated with (U) behaves well with respect to the additional multiplicative and exponential structures that characterize Tambara functors.

The central construction is a functor (\Phi_U) that sends a Tambara functor (\mathcal{T}) on a group (G) to a Tambara functor (\Phi_U(\mathcal{T})) on a group (H). Explicitly, \