Dendroidal Sets

By Ieke Moerdijk, Ittay Weiss. 2007

161 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/f56b6c78783b0299132bcf70f87335b76be7edd1

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Citations in the corpus

On an extension of the notion of Reedy category (2008, background): “[30]) is a morphism of generalized Reedy categories, and Theorem 1.”
Dendroidal sets and simplicial operads (2011, background): “set X such that the map from X to the terminal dendroidal set is an inner Kan fibration (such objects are also called inner Kan complexes in [17, 18]).”
Polynomial Functors and Trees (2008, methodology): “This is the natural level of generality from the viewpoint of operad theory, and for the sake of giving a formal construction of the category of trees of Moerdijk and Weiss, which was the original motivation for this work.”
Spaces of smooth embeddings and configuration categories (2015, background): “Recall the concept of a dendroidal space from Moerdijk-Weiss(1) and Cisinski-Moerdijk [28, 9, 10, 11].”
Dendroidal Segal spaces and ∞‐operads (2013, unknown): “We now recall some of the main combinatorial properties in the category of dendroidal sets; see [7, 14, 15] for more details.”
From operator categories to higher operads (2013, background): “an–Vogttensorproducts and weak algebras It is well-known that the classical Boardman–Vogt tensor product [8] exhibits better homotopical properties when it is extended to suitably weak operads, as in [15] and [13]. It can also be externalized over the wreath product construction given the previous s”
Dendroidal sets as models for homotopy operads (2009, methodology): “We recall here from [12] the different kinds of faces of trees in Ω.”
Familial 2-functors and parametric right adjoints (2007, unknown): “In particular one should compare the following example to the discussion in section(3) of [27].”