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On Extension Of Functors
A.Chigogidze defined for each normal functor on the category Comp an
extension which is a normal functor on the category Tych. We consider this
extension for any functor on the category Comp and investigate which properties
it preserves from the definition it preserves from the definition of normal
functor. We investigate as well some topological properties of such extension
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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