Bi-incomplete Tambara functors

TL;DR

This paper develops the algebraic theory of bi-incomplete Tambara functors, describing how additive transfers and multiplicative norms interact in incomplete G-spectrum contexts, extending the understanding of equivariant ring spectra.

Contribution

It introduces and characterizes bi-incomplete Tambara functors, detailing the algebraic constraints governing their transfer and norm interactions in incomplete G-spectrum settings.

Findings

Complete description of interactions between transfers and norms.

Algebraic constraints for bi-incomplete Tambara functors.

Extension of incomplete Tambara functor theory.

Abstract

For an equivariant commutative ring spectrum $R$, $\pi_0 R$ has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If $R$ is an $N_\infty$ ring spectrum in the category of genuine $G$-spectra, then all possible additive transfers are present and $\pi_0 R$ has the structure of an incomplete Tambara functor. However, if $R$ is an $N_\infty$ ring spectrum in a category of incomplete $G$-spectra, the situation is more subtle. In this paper, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures.