nLab Tambara functor

Redirected from "Tambara functors".
Note: Tambara functor and Tambara functor both redirect for "Tambara functors".
Contents

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 GG-equivariant context for a finite group GG, 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 GG-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 HKH \subset K of GG. (Hill 17)

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

Examples

Burnside rings, representation rings, zeroth stable homotopy group of a genuine equivariant E 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.

References

Last revised on October 5, 2021 at 09:41:13. See the history of this page for a list of all contributions to it.