Algebraic & Geometric Topology

Dendroidal sets

Ieke Moerdijk and Ittay Weiss

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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.

Article information

Algebr. Geom. Topol., Volume 7, Number 3 (2007), 1441-1470.

Received: 16 May 2007
Accepted: 15 June 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P48: Loop space machines, operads [See also 18D50] 55U10: Simplicial sets and complexes 55U40: Topological categories, foundations of homotopy theory
Secondary: 18D50: Operads [See also 55P48] 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10]

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


Moerdijk, Ieke; Weiss, Ittay. Dendroidal sets. Algebr. Geom. Topol. 7 (2007), no. 3, 1441--1470. doi:10.2140/agt.2007.7.1441.

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