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

David Chan

Vol. 6 (2024), No. 1, 1–47

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. 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 parametrize 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.

Tambara functors, equivariant algebra, transfer systems
Mathematical Subject Classification
Primary: 55P91
Secondary: 18M05, 55N91
Received: 19 October 2022
Revised: 30 May 2023
Accepted: 9 July 2023
Published: 20 January 2024
David Chan
Vanderbilt University
Nashville, TN
United States