Bi-incomplete Tambara functors as $\mathcal{O}$-commutative monoids

TL;DR

This paper generalizes the Hoyer--Mazur theorem to bi-incomplete Tambara functors, which are algebraic structures arising from equivariant ring spectra indexed on incomplete universes, and develops tools for their study.

Contribution

It proves a conjecture by Blumberg and Hill, characterizing bi-incomplete Tambara functors via compatible indexing categories and providing new combinatorial tools.

Findings

Generalization of the Hoyer--Mazur theorem to bi-incomplete setting

Characterization of indexing categories for bi-incomplete Tambara functors

Development of combinatorial criteria for compatibility of indexing categories

Abstract

Tambara functors are an equivariant generalization of rings that appear as the homotopy groups of genuine equivariant commutative ring spectra. In recent work, Blumberg and Hill have studied the corresponding algebraic structures, called bi-incomplete Tambara functors, that arise from ring spectra indexed on incomplete $G$-universes. In this paper, we answer a conjecture of Blumberg and Hill by proving a generalization of the Hoyer--Mazur theorem in the bi-incomplete setting. Bi-incomplete Tambara functors are characterized by indexing categories which parameterize incomplete systems of norms and transfers. In the course of our work, we develop several new tools for studying these indexing categories. In particular, we provide an easily checked, combinatorial characterization of when two indexing categories are compatible in the sense of Blumberg and Hill.