Dendroidal sets as models for homotopy operads

Abstract

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