Dendroidal sets as models for homotopy operads

TL;DR

This paper develops a homotopy theory framework for infinity-operads using dendroidal sets, establishing a model category structure where fibrant objects are infinity-operads, thus extending the theory of infinity-categories.

Contribution

It introduces a model category structure on dendroidal sets with infinity-operads as fibrant objects, generalizing the Joyal model for infinity-categories.

Findings

Dendroidal sets form a model category for homotopy operads.

Fibrant objects in this model are dendroidal inner Kan complexes.

The structure extends the Joyal model for infinity-categories.

Abstract

The homotopy theory of infinity-operads is defined by extending Joyal's homotopy theory of infinity-categories to the category of dendroidal sets. We prove that the category of dendroidal sets is endowed with a model category structure whose fibrant objects are the infinity-operads (i.e. dendroidal inner Kan complexes). This extends the theory of infinity-categories in the sense that the Joyal model category structure on simplicial sets whose fibrant objects are the infinity-categories is recovered from the model category structure on dendroidal sets by simply slicing over the point.