Incomplete Tambara functors

TL;DR

This paper introduces incomplete Tambara functors as algebraic structures that generalize Green and Tambara functors, providing a framework to understand equivariant ring spectra with partial norm maps, and classifies them using polynomial functors.

Contribution

It defines and studies categories of incomplete Tambara functors, connecting them to $N_ abla$-algebras and classifying them via polynomial functors, advancing the understanding of equivariant algebraic structures.

Findings

Incomplete Tambara functors interpolate between Green and Tambara functors.

Classification of incomplete Tambara functors via polynomial functors.

Provides a conceptual framework linking $N_ abla$ operads and algebraic structures.

Abstract

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.