For a “genuine” equivariant commutative ring spectrum
,
admits a
rich algebraic structure known as a Tambara functor. This algebraic structure mirrors the
structure on
arising from the existence of multiplicative norm maps. Motivated by the surprising
fact that Bousfield localization can destroy some of the norm maps, in previous
work we studied equivariant commutative ring structures parametrized by
operads. In a precise sense, these interpolate between “naive” and “genuine”
equivariant ring structures.
In this paper, we describe the algebraic analogue of
ring
structures. We introduce and study categories of incomplete Tambara functors,
described in terms of certain categories of bispans. Incomplete Tambara functors arise as
of
algebras,
and interpolate between Green functors and Tambara functors. We classify all
incomplete Tambara functors in terms of a basic structural result about polynomial
functors. This classification gives a conceptual justification for our prior description
of
operads and also allows us to easily describe the properties of the category of
incomplete Tambara functors.
