Relative operads model $\infty$-operads

TL;DR

This paper establishes a deep connection between relative operads and $$-operads through localization, proving an equivalence of their homotopy theories and answering an open question about Lurie's operadic nerve functor.

Contribution

It introduces a framework showing localization creates an equivalence between relative operads and $$-operads, resolving a key open problem in the field.

Findings

Localization induces an equivalence of homotopy theories

Lurie's operadic nerve functor fully captures the homotopy theory of $$-operads

Provides a new perspective on the relationship between operads and $$-operads

Abstract

Given a (colored) operad and a set of unary operations, we can form an associated $\infty$-operad via localization. We show that localization determines an equivalence of homotopy theories of relative operads and $\infty$-operads. As an application, we give an affirmative answer to an open question by Harpaz, proving that Lurie's operadic nerve functor determines an equivalence of homotopy theories of simplicial operads and Lurie's $\infty$-operads.