Enriched $\infty$-operads

TL;DR

This paper develops the theory of enriched $ rightarrow$-operads, introducing models, proving a completion theorem, and establishing a rectification result linking homotopy theories of enriched and strict operads.

Contribution

It introduces multiple models for enriched $ rightarrow$-operads, proves a Rezk-style completion theorem, and establishes a rectification theorem connecting homotopy theories.

Findings

Models for enriched $ rightarrow$-operads are equivalent.

Localization at fully faithful and essentially surjective morphisms yields complete objects.

Homotopy theory of enriched $ rightarrow$-operads is equivalent to that of strictly enriched operads.

Abstract

In this paper we initiate the study of enriched $\infty$-operads. We introduce several models for these objects, including enriched versions of Barwick's Segal operads and the dendroidal Segal spaces of Cisinski and Moerdijk, and show these are equivalent. Our main results are a version of Rezk's completion theorem for enriched $\infty$-operads: localization at the fully faithful and essentially surjective morphisms is given by the full subcategory of complete objects, and a rectification theorem: the homotopy theory of $\infty$-operads enriched in the $\infty$-category arising from a nice symmetric monoidal model category is equivalent to the homotopy theory of strictly enriched operads.