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
-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.
Keywords
Tambara functors, equivariant algebra, transfer systems
