On the equivalence between Lurie's model and the dendroidal model for infinity-operads

TL;DR

This paper establishes an equivalence between two models of infinity-operads, dendroidal sets and Lurie's approach, showing they are fundamentally compatible for operads without constants.

Contribution

It proves the equivalence of the dendroidal model and Lurie's model for infinity-operads without constants via a zig-zag of Quillen equivalences.

Findings

The two models are equivalent for operads without constants.

A zig-zag of Quillen equivalences connects the models.

The result clarifies the relationship between different infinity-operad theories.

Abstract

We compare two approaches to the homotopy theory of infinity-operads. One of them, the theory of dendroidal sets, is based on an extension of the theory of simplicial sets and infinity-categories which replaces simplices by trees. The other is based on a certain homotopy theory of marked simplicial sets over the nerve of Segal's category Gamma. In this paper we prove that for operads without constants these two theories are equivalent, in the precise sense of the existence of a zig-zag of Quillen equivalences between the respective model categories.