Monoidal Structures in Equivariant Algebra

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 thesis, 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. We also provide a new characterization of equivariant commutative monoids in the sense of Hill--Hopkins via a universal property associated to categorical Mackey functors.