Dendroidal sets

Abstract

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.

Enriched ∞-operads

“…The Dendroidal Category. The dendroidal category è was first introduced by Moerdijk and Weiss in [37] as a category of trees whose morphisms are given by maps of free operads. Here we recall a more combinatorial reformulation of this definition due to Kock [31].…”

Dold–Kan correspondence for dendroidal abelian groups

“…The dendroidal category Ω was introduced in [11,16] as an extension of the simplicial category ∆. The category Ω is a category of trees.…”

“…The definition we gave for the composition product of a coloured operad is not the common one in the literature, but it is equivalent to it (see the same definition in [10] for the one-colour case, or an alternative definition for the general case in [1,8,11]). …”

“…The category of planar dendroidal sets is a presheaf category on a certain category of trees Ω p . Dendroidal sets generalize simplicial sets in a way suitable for studying the homotopy theory of coloured operads and their algebras [11]. In the context of non-symmetric coloured operads the role of dendroidal sets is replaced by planar dendroidal sets.…”

An Additivity Theorem for the interchange of En structures

“…Moerdijk and Weiss extended the tensor product of operads to dendroidal sets [22][23][24]29], and suggested an investigation of an Additivity Theorem for n copies of the dendroidal set N d (Ass) associated with the operad Ass of monoid structures. But while Ass ⊗ Ass ∼ = Com (see Corollary 3.9 for more details), the structure of the dendroidal tensor product N d (Ass) ⊗ N d (Ass) is not clear.…”