Tambarization of a Mackey functor and its application to the Witt-Burnside construction
For an arbitrary group $G$, a (semi-)Mackey functor is a pair of covariant and contravariant functors from the category of $G$-sets, and is regarded as a $G$-bivariant analog of a commutative (semi-)group. In this view, a $G$-bivariant analog of a (semi-)ring should be a (semi-)Tambara functor. A Tambara functor is firstly defined by Tambara, which he called a TNR-functor, when $G$ is finite. As shown by Brun, a Tambara functor plays a natural role in the Witt-Burnside construction. It will be a natural question if there exist sufficiently many examples of Tambara functors, compared to the wide range of Mackey functors. In the first part of this article, we give a general construction of a Tambara functor from any Mackey functor, on an arbitrary group $G$. In fact, we construct a functor from the category of semi-Mackey functors to the category of Tambara functors. This functor gives a left adjoint to the forgetful functor, and can be regarded as a $G$-bivariant analog of the monoid-ring functor. In the latter part, when $G$ is finite, we invsetigate relations with other Mackey-functorial constructions ---crossed Burnside ring, Elliott's ring of $G$-strings, Jacobson's $F$-Burnside ring--- all these lead to the study of the Witt-Burnside construction.