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Tambarization of a Mackey functor and its application to the Witt-Burnside construction
For an arbitrary group , a (semi-)Mackey functor is a pair of covariant
and contravariant functors from the category of -sets, and is regarded as a
-bivariant analog of a commutative (semi-)group. In this view, a
-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 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 . 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 -bivariant analog of the monoid-ring functor.
In the latter part, when is finite, we invsetigate relations with other
Mackey-functorial constructions ---crossed Burnside ring, Elliott's ring of
-strings, Jacobson's -Burnside ring--- all these lead to the study of the
Witt-Burnside construction.Comment: 31 page
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