The genuine operadic nerve

Abstract

We construct a generalization of the operadic nerve, providing a translation between the equivariant simplicially enriched operadic world to the parametrized $\infty$-categorical perspective. This naturally factors through genuine equivariant operads, a model for "equivariant operads with norms up to homotopy". We introduce the notion of an op-fibration of genuine equivariant operads, extending Grothendieck op-fibrations, and characterize fibrant operads as the image of genuine equivariant symmetric monoidal categories. Moreover, we show that under the operadic nerve, this image is sent to $G$-symmetric monoidal $G$-$\infty$-categories. Finally, we produce a functor comparing the notion of algebra over an operad in each of these two contexts.

Comments welcome! v2: 39 pages. Strengthened Thm II to include simplicial enrichments, added examples and comparisons of algebras, general revisions. v1: 38 pages

References in corpus (7)

• Higher Topos Theory
• Encoding Equivariant Commutativity via Operads
• The Enriched Grothendieck Construction
• G-symmetric monoidal categories of modules over equivariant commutative ring spectra
• Parametrized higher category theory and higher algebra: A general introduction
• Equivariant dendroidal sets
• Genuine equivariant factorization homology

Cited by in corpus (3)

• Genuine equivariant operads
• Genuine equivariant factorization homology
• Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra