An algebraic model for rational naïve-commutative \(G\)-equivariant ring spectra for finite \(G\)
An \(N_\infty\)-operad is the equivariant analogue of an \(E_\infty\)-operad in the sense that an \(N_\infty\)-algebra is a ring \(G\)-spectrum which is commutative in the equivariant setting. For a finite \(G\), there can be many different \(N_\infty\)-operads and different types of commutativity. This article considers the ``naive'' case for rational \(G\)-spectra, i.e. the \(N_\infty\)-operad with no nontrivial norm maps. While there are explicit algebraic models for rational \(G\)-spectra for finite \(G\), it is not immediate that these can be used to describe algebras for the naive \(N_\infty\)-operad, despite the involved Quillen equivalences being monoidal -- some subtleties first have to be overcome. The authors give a zig-zag of Quillen equivalences between rational \(G\)-spectra and their algebraic model, and carefully verify that these lift to Quillen equivalences between algebras over the relevant operads: the commutative algebra operad for the algebraic model and the naive \(N_\infty\)-operad for rational \(G\)-spectra. The authors also provide checkable sufficient conditions for when the \(E\)-local model category on \(G\)-spectra lifts to the level of algebras over a given operad.
scientific article; zbMATH DE number 6986485