Two models for the homotopy theory of $\infty$-operads

TL;DR

This paper compares two models for $ abla$-operads, showing their equivalence and unifying different approaches to the homotopy theory of $ abla$-operads.

Contribution

It establishes the equivalence of two prominent models for $ abla$-operads, unifying the homotopy theories of Lurie's $ abla$-operads, dendroidal sets, and simplicial operads.

Findings

Complete Segal operads and complete dendroidal Segal spaces are equivalent models.

All known models for $ abla$-operads are equivalent.

Homotopy theories of different models are shown to be equivalent.

Abstract

We compare two models for $\infty$-operads: the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk. Combining this with comparison results already in the literature, this implies that all known models for $\infty$-operads are equivalent - for instance, it follows that the homotopy theory of Lurie's $\infty$-operads is equivalent to that of dendroidal sets and that of simplicial operads.