Tambara Functors and Commutative Ring Spectra

Abstract

It is well known that the zeroth stable homotopy group of a genuine equivariant commutative ring spectrum has multiplicative transfers (norms), making it into a Tambara functor. We prove here that all Tambara functors can be obtained in this way. In fact, we prove that the homotopy category of Eilenberg MacLane commutative ring spectra is equivalent to the category of Tambara functors. Several algebraic results on Tambara functors are derived as corollaries. Finally, we rule out the existence of a lax symmetric monoidal construction for Eilenberg MacLane spectra when the group is nontrivial.

30 pages

Cited by in corpus (8)

• Global homotopy theory
• Equivariant symmetric monoidal structures
• Fixed point adjunctions for equivariant module spectra
• Graded Tambara Functors
• $K$-theory of Hermitian Mackey functors and a reformulation of the Novikov Conjecture
• Real topological Hochschild homology of schemes
• Symmetric Powers and Norms of Mackey Functors
• Genuine-commutative ring structure on rational equivariant $K$-theory for finite abelian groups