Tambara functors

TL;DR

This paper surveys and extends the theory of Tambara functors, algebraic structures that incorporate multiplicative norms and additive transfers, with applications in algebra and stable homotopy theory.

Contribution

It provides a comprehensive overview and new extensions of Tambara functors, connecting them to various mathematical contexts like Burnside rings and equivariant spectra.

Findings

Includes examples such as Burnside rings and representation rings.

Establishes a categorical framework for Tambara functors.

Links Tambara functors to Witt rings and other algebraic structures.

Abstract

We survey and extend the theory of Tambara functors. These are algebraic structures similar to Mackey functors, but with multiplicative norm maps as well as additive transfer maps, and a rule governing their interaction that is most easily formulated in an abstract categorical framework. Examples include Burnside rings, representation rings, and homotopy groups of equivariant E-infinity ring spectra in stable homotopy theory. Some other examples are related to Witt rings in the sense of Dress and Siebeneicher.