Genuine equivariant operads

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.

Final version accepted for publication in Advances in Mathematics

References in corpus (10)

• Higher Topos Theory
• Encoding Equivariant Commutativity via Operads
• Combinatorial $N_{\infty }$ operads
• Parametrized higher category theory and higher algebra: Exposé I -- Elements of parametrized higher category theory
• Equivariant dendroidal sets
• Genuine equivariant factorization homology
• Equivariant dendroidal Segal spaces and $G$-$\infty$-operads
• The genuine operadic nerve
• Equivariant dendroidal sets and simplicial operads
• On the homotopy theory of equivariant colored operads

Cited by in corpus (24)

• Encoding Equivariant Commutativity via Operads
• Combinatorial $N_{\infty }$ operads
• Equivariant dendroidal sets
• Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrum
• Genuine equivariant factorization homology
• Equivariant stable categories for incomplete systems of transfers
• 1-smooth pro-p groups and Bloch-Kato pro-p groups
• Characterizations of equivariant Steiner and linear isometries operads
• Categorifying the algebra of indexing systems
• Equivariant dendroidal sets and simplicial operads
• Idempotent characters and equivariantly multiplicative splittings of K-theory
• Global transfer systems of abelian compact Lie groups
• Global $N_{\infty }$-operads
• Goerss--Hopkins obstruction theory for $\infty$-categories
• Homotopy theory of equivariant operads with fixed colors
• Equivariant homotopy commutativity for $G=C_{pqr}$
• Lifting $N_{\infty }$ operads from conjugacy data
• On the homotopy theory of equivariant colored operads
• ${\mathbb{E}}_{\infty }$ automorphisms of motivic Morava $E$-theories
• Saturated and linear isometric transfer systems for cyclic groups of order $p^m{q}^n$
• Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games
• Genuine-commutative ring structure on rational equivariant $K$-theory for finite abelian groups
• The $\mathbb{Z}$-homotopy fixed points of $C_n$ spectra with applications to norms of $M{U}_{\mathbb{R}}$
• Realization of saturated transfer systems on cyclic groups of order $p^n{q}^m$ by linear isometries $N_{\infty }$-operads