2010/10/05 by Hiroyuki Nakaoka, Nakaoka, Hiroyuki
Mathematics · #19A22 #Advanced Topics in Algebra #Advanced Topology and Set Theory #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1010.0812
openalex publication_date 2010/10/05 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
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.