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Dendroidal sets

Ieke Moerdijk and Ittay Weiss

Algebraic & Geometric Topology 7 (2007) 1441–1470

DOI: 10.2140/agt.2007.7.1441

arXiv: math.AT/0701293


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 Δ 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.


operad, homotopy coherent nerve, Kan complex, tensor product of operads, weak n–categories, algebras up to homotopy

Mathematical Subject Classification

Primary: 55P48, 55U10, 55U40

Secondary: 18D10, 18D50, 18G30


Received: 16 May 2007
Accepted: 15 June 2007
Published: 14 November 2007

Ieke Moerdijk
Department of Mathematics
Utrecht University
P O Box 80010
3508 TA Utrecht
The Netherlands
Ittay Weiss
Department of Mathematics
Utrecht University
P O Box 80010
3508 TA Utrecht
The Netherlands