We introduce the concept of a dendroidal set. This is a generalization of the notion of
a simplicial set, specially suited to the study of (coloured) operads in the context of
homotopy theory. We define a category of trees, which extends the category
used
in simplicial sets, whose presheaf category is the category of dendroidal sets. We
show that there is a closed monoidal structure on dendroidal sets which is
closely related to the Boardman–Vogt tensor product of (coloured) operads.
Furthermore, we show that each (coloured) operad in a suitable model category has
a coherent homotopy nerve which is a dendroidal set, extending another
construction of Boardman and Vogt. We also define a notion of an inner Kan
dendroidal set, which is closely related to simplicial Kan complexes. Finally, we
briefly indicate the theory of dendroidal objects in more general monoidal
categories, and outline several of the applications and further theory of dendroidal
sets.
Keywords
operad, homotopy coherent nerve, Kan complex, tensor
product of operads, weak $n$–categories, algebras up to
homotopy
