[Paper Review] The genuine operadic nerve

Peter Bonventre|arXiv (Cornell University)|Apr 2, 2019

Homotopy and Cohomology in Algebraic Topology25 references4 citations

TL;DR

This paper introduces the genuine operadic nerve, a functor that translates between genuine equivariant simplicial operads and $G$-symmetric monoidal $Ω$-categories, generalizing Lurie's operadic nerve to the equivariant setting. It establishes that fibrant operads in the genuine equivariant framework correspond precisely to $G$-symmetric monoidal $Ω$-categories and constructs a comparison functor between algebras over operads in both contexts, preserving homotopical structure.

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.

Motivation & Objective

• To extend the classical operadic nerve to the equivariant setting, incorporating group actions and norm maps.
• To define and characterize genuine operadic op-fibrations, generalizing Grothendieck op-fibrations to the equivariant context.
• To show that the genuine operadic nerve sends fibrant genuine equivariant operads to $G$-symmetric monoidal $Ω$-categories.
• To construct a comparison functor between algebras over operads in the simplicial and $Ω$-categorical frameworks.
• To establish a homotopy-theoretic equivalence between genuine equivariant simplicial operads and $G$-symmetric monoidal $Ω$-categories.

Proposed method

• Constructs the genuine operadic nerve as a generalization of Lurie’s operadic nerve, using the category of operators for genuine equivariant operads.
• Introduces the notion of an op-fibration in the context of genuine equivariant operads, extending classical Grothendieck op-fibrations.
• Uses the category of operators of a genuine equivariant operad to define a simplicial category over the category of finite pointed $G$-sets.
• Applies the homotopy coherent nerve to the category of operators to produce an $Ω$-operad, which is shown to be a $G$-symmetric monoidal $Ω$-category.
• Establishes a comparison map between algebras over a genuine equivariant simplicial operad and algebras over its image under the genuine operadic nerve.
• Leverages the adjunction between the homotopy coherent nerve and its left adjoint to construct a functor between $Ω$-categories of algebras.

Experimental results

Research questions

• RQ1 How can the classical operadic nerve be generalized to incorporate equivariant structures and norm maps?
• RQ2 What is the correct notion of fibrancy in the context of genuine equivariant operads, and how does it relate to $G$-symmetric monoidal $Ω$-categories?
• RQ3 How do algebras over genuine equivariant simplicial operads relate to algebras over their images under the genuine operadic nerve?
• RQ4 What is the role of the category of operators in the equivariant setting, and how does it support the construction of the genuine operadic nerve?
• RQ5 Can a comparison functor be constructed between the $Ω$-categories of algebras in the simplicial and $Ω$-categorical models of equivariant homotopy theory?

Key findings

• The genuine operadic nerve sends fibrant genuine equivariant operads to $G$-symmetric monoidal $Ω$-categories, establishing a bridge between two homotopical frameworks.
• The image of a genuine equivariant symmetric monoidal category under the genuine operadic nerve is precisely a $G$-symmetric monoidal $Ω$-category.
• There exists a natural comparison functor between the $Ω$-categories of algebras over a genuine equivariant simplicial operad and its image under the genuine operadic nerve.
• The construction preserves the homotopical structure of algebras, with the comparison map being induced by the adjunction between the homotopy coherent nerve and its left adjoint.
• The genuine operadic nerve is compatible with the operadic nerve in the non-equivariant case, recovering Lurie’s construction when $G$ is trivial.
• The method establishes a Quillen equivalence between the model categories of genuine equivariant simplicial operads and $G$-symmetric monoidal $Ω$-categories, up to homotopy.