Genuine equivariant operads

TL;DR

This paper introduces genuine equivariant operads, a new algebraic structure bridging equivariant operads and coefficient systems, and proves their equivalence to existing models, with applications to $N_{ abla}$-operads.

Contribution

It constructs genuine equivariant operads and establishes an Elmendorf-Piacenza type theorem linking them to equivariant operads, providing explicit models for $N_{ abla}$-operads.

Findings

Genuine equivariant operads are equivalent to equivariant operads under certain model structures.

An Elmendorf-Piacenza type theorem is proved for these operads.

Explicit models for $N_{ abla}$-operads are constructed.

Abstract

We build new algebraic structures, which we call genuine equivariant operads, which can be thought of as a hybrid between equivariant operads and coefficient systems. We then prove an Elmendorf-Piacenza type theorem stating that equivariant operads, with their graph model structure, are equivalent to genuine equivariant operads, with their projective model structure. As an application, we build explicit models for the $N_{\infty}$-operads of Blumberg and Hill.