Symmetric Powers and Norms of Mackey Functors

Abstract: In this paper we give detailed algebraic descriptions of the derived symmetric power and norm constructions on categories of Mackey functors, as well as the derived G-symmetric monoidal structure. We build on the results of [Ull2], in which it is shown that every Tambara functor over a finite group G arises as the zeroth stable homotopy group of a commutative ring G-spectrum. The norm / restriction adjunctions on categories of Tambara functors promised in [Ull2] are demonstrated algebraically. Finally, we give a new characterization of Tambara functors in terms of multiplicative push forwards of Mackey functors, and use this to obtain an appealing new description of the free Tambara functor on a Mackey functor which closely matches the structure of equivariant extended powers.