Loday constructions of Tambara functors

TL;DR

This paper introduces an equivariant Loday construction for G-Tambara functors, extending previous work to arbitrary finite groups, and explores its properties, examples, and connections to topological Hochschild homology.

Contribution

It develops a new equivariant Loday construction for G-Tambara functors applicable to any finite group, linking algebraic and topological contexts.

Findings

Construction agrees with twisted cyclic nerve for cyclic groups

Relates to genuine commutative G-ring spectra via $ ext{pi}_0$-functor

Describes Real topological Hochschild homology as a Loday construction

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.