nLab Tambara functor




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)


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.


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