A new model structure on dendroidal spaces for the theory of $\infty$-operads

TL;DR

This paper develops a new model structure on dendroidal spaces that models the homotopy theory of -operads, extending -category theory to operads with a Quillen equivalence to existing models.

Contribution

It introduces a dendroidal space model structure for -operads, analogous to recent simplicial space models, and proves its Quillen equivalence to established dendroidal models.

Findings

Fibrant objects are dendroidal Segal spaces.

Weak equivalences are Dwyer--Kan equivalences.

Model structure is Quillen equivalent to existing dendroidal models.

Abstract

We introduce a new model structure on the category of dendroidal spaces, designed to provide a further model for the homotopy theory of $\infty$-operads. This model is directly analogous to a recent construction on the category of simplicial spaces by Moser and Nuiten, and can be seen as its dendroidal counterpart. In our new model structure, the fibrant objects are the dendroidal Segal spaces, while the cofibrations form a subclass of those in the Segal space model structure. The weak equivalences between fibrant objects are precisely the Dwyer--Kan equivalences, and the fibrations between them are the isofibrations. We prove that this model structure is Quillen equivalent to the Cisinski--Moerdijk model structures on dendroidal sets and dendroidal spaces, thereby establishing a compatible extension of the theory of $\infty$-categories to the operadic setting.