Contents

Idea

Tambara functors are algebraic structures similar to Mackey functors, but with multiplicative norm maps as well as additive transfer maps, and a rule governing their interaction. They were introduced by Tambara, as TNR-functors, to encode the relationship between Trace (additive transfer), Norm (multiplicative transfer) and Restriction maps in the representation theory and cohomology theory of finite groups (Tam93).

In the $G$-equivariant context for a finite group $G$, the role of abelian groups in non-equivariant algebra is played by Mackey functors. The category of Mackey functors is a closed symmetric monoidal category with symmetric monoidal product, the box product. In addition to the expected generalization of commutative rings to simply commutative monoids for the box product, there is a poset of generalizations of the notion of commutative rings to the $G$-equivariant context: the incomplete Tambara functors. These interpolate between Green functors, the ordinary commutative monoids for the box product, and Tambara functors. The distinguishing feature for [incomplete] Tambara functors is the presence of certain multiplicative transfer maps, called norm maps. For a Green functor, we have no norm maps; for a Tambara functor, we have norm maps for any pair of subgroups $H \subset K$ of $G$. (Hill 17)

A Mackey functor is represented by a “$G$-equivariant Eilenberg–MacLane spectrum”, … a Tambara functor is represented by a commutative “$G$-equivariant Eilenberg-MacLane ring spectrum” (AB15)

Examples

Burnside rings, representation rings, zeroth stable homotopy group of a genuine equivariant $E_{\infty}$-ring.

the homotopy category of Eilenberg MacLane commutative ring spectra is equivalent to the category of Tambara functors. (Ull13)

Some other examples are related to Witt-Burnside functors, Witt rings in the sense of Dress and Siebeneicher.

Related concepts

• Mackey functor
• polynomial functor
• Witt vectors

References

• Neil Strickland, Tambara Functors, arXiv:1205.2516

• Morten Brun, Witt vectors and Tambara functors, arXiv:math/0304495

• Daisuke Tambara, On multiplicative transfer, Comm. Algebra 21 (1993), no. 4, 1393–1420 (pdf).

• Vigleik Angeltveit and Anna Marie Bohmann, Graded Tambara Functors, (arXiv:1504.00668)

• John Ullman, Tambara Functors and Commutative Ring Spectra, (arXiv:1304.4912)

• Michael Hill, Derived Equivariant Algebraic Geometry, (lecture and notes)

• Kristen Mazur, On the Structure of Mackey Functors and Tambara Functors, (thesis)

• Michael Hill, On the Andre-Quillen homology of Tambara functors, (arXiv:1701.06219)

• Andrew Blumberg, Michael Hill, Incomplete Tambara functors, (arXiv:1603.03292)