Dendroidal sets as models for homotopy operads

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.

💡 Research Summary

The paper “Dendroidal sets as models for homotopy operads” extends the homotopy theory of ∞‑categories, originally developed by Joyal for simplicial sets, to the setting of operads by using dendroidal sets. Dendroidal sets are presheaves on the category Ω of finite rooted trees, and they generalize simplicial sets (which correspond to linear trees). The author first establishes that the category dSet of dendroidal sets is complete and cocomplete, and introduces the representable objects Ω