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

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.

Figures (3)

• Figure 1: Two poset structures on the set of subsets of the cyclic group $C_4$. The left graph is a transfer system. The right graph is not a transfer system as it fails to satisfy closure under intersection.
• Figure 2: Two transfer systems $\mathcal{T}_1\leq \mathcal{T}_2$ on the group $C_2\times C_2$. The group $\Delta$ is the diagonal subgroup given by the image of the diagonal map $C_2\to C_2\times C_2$. This pair of transfer systems is not compatible.
• Figure 3: An example of compatible transfer systems for the alternating group $A_5$ is drawn on the right. The poset of conjugacy classes of subgroups of the alternating group $A_5$ is drawn on the left for reference.

Theorems & Definitions (107)

• Theorem A: Appears below as Theorem \ref{['thm:secondMainTheorem']}
• Theorem B: Appears below as Theorem \ref{['thm:catEqualSystem']}
• Definition 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Lemma 2.5: Proposition 3.1 of BlumbergHillIncomplete
• proof
• Lemma 2.6: Proposition 3.3 of rubin_detecting
• proof