Bi-incomplete Tambara Functors

DOI10.1017/9781108942874.009zbMATH Open1484.55008arXiv2104.10521OpenAlexW4207081266MaRDI QIDQ5027304

Andrew J. Blumberg, Michael A. Hill

Publication date: 4 February 2022

Abstract: For an equivariant commutative ring spectrum $R$, $p{i}_{0}R$ 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 $N_{i}nfty$ ring spectrum in the category of genuine $G$-spectra, then all possible additive transfers are present and $p{i}_{0}R$ has the structure of an incomplete Tambara functor. However, if $R$ is an $N_{i}nfty$ 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.

Full work available at URL: https://arxiv.org/abs/2104.10521

zbMATH Keywords

equivariant homotopy theoryMackey functorTambara functor

Mathematics Subject Classification ID

Stable homotopy theory, spectra (55P42) Equivariant homotopy theory in algebraic topology (55P91)

Related Items (7)

Encoding equivariant commutativity via operads ⋮ Biset transformations of Tambara functors ⋮ Composition closed premodel structures and the Kreweras lattice ⋮ Incomplete Tambara functors ⋮ Goerss-Hopkins obstruction theory for \(\infty \)-categories ⋮ Counting compatible indexing systems for \(C_{p^n}\) ⋮ Equivariant stable categories for incomplete systems of transfers

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