Loday constructions of Tambara functors

Abstract

Building on work of Hill, Hoyer and Mazur we propose an equivariant version of a Loday construction for $G$-Tambara functors where $G$ is an arbitrary finite group. For any finite simplicial $G$-set and any $G$-Tambara functor, our Loday construction is a simplicial $G$-Tambara functor. We study its properties and examples. For a circle with rotation action by a finite cyclic group our construction agrees with the twisted cyclic nerve of Blumberg, Gerhardt, Hill, and Lawson. We also show how the Loday construction for genuine commutative $G$-ring spectra relates to our algebraic one via the $\underline{\pi}_0$-functor. We describe Real topological Hochschild homology as such a Loday construction.