Dendroidal sets and simplicial operads

TL;DR

This paper establishes a deep connection between different models of higher operad theories, showing equivalences between Segal operads, simplicial operads, and dendroidal sets, and relates these to classical categories and operads.

Contribution

It proves a Quillen equivalence between the homotopy theories of Segal operads and simplicial operads, unifying various models of infinity-operads and classical operads.

Findings

Homotopy coherent nerve functor is a right Quillen equivalence.

Quillen equivalences between Segal categories, simplicial categories, and quasi-categories.

Application of theory to classical operads in spaces.

Abstract

We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model category of simplicial operads to the model category structure for infinity-operads on the category of dendroidal sets. By slicing over the monoidal unit, this also gives the Quillen equivalence between Segal categories and simplicial categories proved by J. Bergner, as well as the Quillen equivalence between quasi-categories and simplicial categories proved by A. Joyal and J. Lurie. We also explain how this theory applies to the usual notion of operad (i.e. with a single colour) in the category of spaces.