Incomplete Tambara functors

For a "genuine" equivariant commutative ring spectrum $R$, $\pi_0(R)$ admits a rich algebraic structure known as a Tambara functor. This algebraic structure mirrors the structure on $R$ 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 $N_\infty$ operads. In a precise sense, these interpolate between "naive" and "genuine" equivariant ring structures. In this paper, we describe the algebraic analogue of $N_\infty$ ring structures. We introduce and study categories of incomplete Tambara functors, described in terms of certain categories of bispans. Incomplete Tambara functors arise as $\pi_0$ of $N_\infty$ 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 $N_\infty$ operads and also allows us to easily describe the properties of the category of incomplete Tambara functors.