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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.
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Bi-incomplete Tambara functors
For an equivariant commutative ring spectrum R, \pi_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_\infty ring spectrum in the category of genuine G-spectra, then all possible additive transfers are present and \pi_0 R has the structure of an incomplete Tambara functor. However, if R is an N_\infty ring spectrum in a category of incomplete G-spectra, the situation is more subtle. In this chapter, 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.
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- Award ID(s):
- 1812064
- PAR ID:
- 10328638
- Editor(s):
- Balchin, S.; Barnes, D.; Kędziorek, M.; Szymik, M.
- Date Published:
- Journal Name:
- Equivariant Topology and Derived Algebra
- Page Range / eLocation ID:
- 276 - 313
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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