G-Tambara Functors are G-commutative Monoids

The interplay between trace, norm and restriction maps connecting the Burnside rings of subgroups of a finite group G has been studied extensively by Tambara. It became apparent that certain relations between those maps are satisfied by the trace, norm and restriction maps between cohomology groups as well as between representation rings of subgroups of G. A collection of abelian groups with trace, norm and restriction maps satisfying such relations is called a TNR-functor in \cite{Tambara} and is now referred to as a Tambara functor. A major motivation for studying Tambara functors is their connection with equivariant stable homotopy theory, i.e. for any $G$-spectrum $R$, with an equivariant $E_{\infty}$ structure, the Mackey functor $\pi_{0}^{G}R$ turns out to be a Tambara functor \cite{Brun}. In \cite{Hill}, Hill and Hopkins introduce the notions of a $G$-symmetric monoidal structure on a symmetric monoidal coefficient system $\underline{\mathcal{C}}$ and of a $G$-commutative monoid in $\underline{\mathcal{C}}(G/G)$. If we think of the unbiased definition of a commutative monoid as an object with a collection of operations indexed by finite sets satisfying certain coherence relations, then a $G$-commutative monoid is an object with a collection of operations indexed by finite $G$-sets subject to appropriate coherence relations. In this thesis we construct a $G$-symmetric monoidal structure on the symmetric monoidal coefficient system formed by the collection of categories $Mackey_H$ of Mackey functors over $H$, with $H \subset G$, together with restriction functors. We then show that Tambara functors are precisely the $G$-commutative monoids in $Mackey_G$.